# Magic number (physics)

In nuclear physics, a **magic number** is a number of protons or neutrons at which a nucleus is exceptionally stable because a shell of nucleons is completely filled. The canonical magic numbers are 2, 8, 20, 28, 50, 82, and 126, and they are the same for neutrons and protons.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup><sup> • </sup><sup>[2](https://www.nobelprize.org/uploads/2018/06/mayer-lecture.pdf)</sup> Nuclei with a magic number of protons or neutrons are locally lighter than their neighbors or unusually abundant in nature, and nuclei that are magic in both, such as calcium-48, tin-100, and nickel-78, are called doubly magic.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup><sup> • </sup><sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup>

| Key fact | Detail |
|---|---|
| Magic numbers | 2, 8, 20, 28, 50, 82, 126, identical for protons and neutrons<sup>[2](https://www.nobelprize.org/uploads/2018/06/mayer-lecture.pdf)</sup> |
| Origin | A self-induced attractive mean field with strong spin-orbit coupling; the numbers 28, 50, 82, and 126 arise from the spin-orbit term<sup>[4](https://www.mdpi.com/2624-8174/4/1/18)</sup> |
| Discovery | Explained in 1949, independently by Maria Goeppert Mayer and by Haxel, Jensen, and Suess; the term is attributed to Eugene Wigner<sup>[5](https://journals.aps.org/pr/abstract/10.1103/PhysRev.75.1766.2)</sup><sup> • </sup><sup>[6](https://www.aps.org/apsnews/2008/08/goeppert-mayer-nuclear-shell-model)</sup> |
| Recognition | Mayer and J. Hans D. Jensen shared the 1963 Nobel Prize in Physics for the shell model<sup>[2](https://www.nobelprize.org/uploads/2018/06/mayer-lecture.pdf)</sup> |
| Doubly magic examples | 48Ca (Z=20, N=28), 100Sn, 78Ni; only 4He, 16O, and 40Ca are doubly magic in the pure harmonic-oscillator (double-LS) sense<sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup><sup> • </sup><sup>[7](https://par.nsf.gov/servlets/purl/10333174)</sup> |
| Shell evolution | Far from stability, the N=8, 20, and 28 closures can disappear in some neutron-rich nuclei, while new closures such as N=16, 32, and 34 emerge<sup>[8](https://arxiv.org/html/2411.15562)</sup> |
| Nucleosynthesis | Closed shells at N=82 and N=126 produce the r-process abundance peaks at A=130 and A=195<sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup> |

## Physical origin in the shell model

The shell model treats protons and neutrons as moving independently in a self-induced attractive mean field, like electrons in an atom but with a different potential shape. Filling shells in a simple harmonic-oscillator potential reproduces the first three magic numbers, 2, 8, and 20, but predicts nothing special beyond that. The observed numbers 28, 50, 82, and 126 appear only when a strong spin-orbit interaction is added, and the effect of that term grows as the orbital angular momentum j becomes larger.<sup>[4](https://www.mdpi.com/2624-8174/4/1/18)</sup> In 1949 Mayer, Haxel, Suess, and Jensen showed that a spin-orbit interaction with a sign opposite to that known from the fine structure in atoms produces the observed gaps between nuclear shells.<sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup> This sign difference is the sharpest break with the atomic analogy: in nuclei the strong nuclear force reverses the ordering known from atomic fine structure, and the resulting gaps appear at 28, 50, 82, and 126.<sup>[4](https://www.mdpi.com/2624-8174/4/1/18)</sup><sup> • </sup><sup>[9](https://academic.oup.com/ptep/article-2026/4/04A101/8258960)</sup>

## Experimental signatures

Several independent observables mark a shell closure. Nuclei at magic numbers show locally enhanced binding energies and rapid changes of one- and two-nucleon separation energies; the AME2020 mass evaluation tabulates separation energies and decay energies precisely so that such discontinuities can be read off.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup><sup> • </sup><sup>[10](https://beta.iopscience.iop.org/article/10.1088/1674-1137/abddaf)</sup> The liquid drop model, which treats the nucleus as a uniform fluid, underestimates the actual binding energies of magic nuclei, which is direct evidence for shell structure beyond the smooth fluid picture.<sup>[11](https://personal.soton.ac.uk/ab1u06/teaching/phys3002/course/05_shell.pdf)</sup>

Spectroscopy adds a second test: the energy of the first 2+ excited state of an even-even nucleus rises sharply at a closed shell, so observed 2+ energies plotted against neutron number are used to test which magic gaps are realized.<sup>[4](https://www.mdpi.com/2624-8174/4/1/18)</sup> Kinks in charge radii and spectroscopic factors provide further signatures.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup> Neutron reactions give a third: nuclei with magic neutron numbers have much lower neutron absorption cross sections, measured in millibarns, making neutron capture less likely.<sup>[12](http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/shell2.html)</sup>

## Shell evolution far from stability

The magic numbers are not fixed constants of the nuclear chart. In short-lived exotic nuclei with large neutron-to-proton asymmetry, the traditional magic numbers, viewed through observables, were shown to change.<sup>[13](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.92.015002)</sup> Radioactive-ion-beam experiments over the last three decades established that the neutron magic numbers 8, 20, and 28 can disappear in some neutron-rich isotopes and the emergence of new ones such as 16 and 34.<sup>[8](https://arxiv.org/html/2411.15562)</sup> The N=20 and N=28 gaps can disappear in some neutron-rich nuclei, with experimental evidence for deformed structures below 44S and 32Mg.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup>

**New closures around calcium.** Around 48Ca, new neutron magic numbers 32 and 34 appear, with N=34 significant only around Z=20.<sup>[8](https://arxiv.org/html/2411.15562)</sup> A 2013 spectroscopic study of 54Ca (20 protons, 34 neutrons) using proton [knockout reactions](https://www.edgepedia.org/knockout-reactions) at RIKEN provided direct experimental evidence for a sizable subshell closure at N=34 and highlighted the doubly magic nature of 54Ca.<sup>[14](https://www.nature.com/articles/nature12522)</sup> The first direct mass measurements of 55Ca, 56Ca, and 57Ca at RIKEN gave atomic mass excesses of −18650(160), −13510(250), and −7370(990) keV respectively, evidence for a sizable energy gap between the neutron 2p1/2 and 1f5/2 orbitals in 54Ca, while 56Ca shows open-shell neutron behavior.<sup>[15](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.022506)</sup> Local subshell closures identified from binding energies also include Z=14 in silicon isotopes, Z=64 in 146Gd, N=16 in 36Ca and 24O, N=32 in 52Ca, N=56 in 96Zr, and N=64 in tin isotopes.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup>

**Where theorists disagree.** The mechanism behind this shell migration is contested. The tensor part of the nuclear force was long invoked to explain monopole shifts of single-particle orbitals, but that effect falls short of a systematic understanding of shell evolution; a recent proposal, the Dirac Confluence Mechanism, attributes magicity evolution to the interplay between spin-orbit and pseudo-spin-orbit partner orbitals.<sup>[8](https://arxiv.org/html/2411.15562)</sup> The persistence of N=34 away from calcium is also unsettled: precision mass measurements of 13 neutron-rich Ca, Ti, and V isotopes at TRIUMF and LEBIT show no peaklike behavior at N=34 and do not support a shell closure there toward higher proton number, while the same data show the two-neutron separation-energy delta at N=32 decreasing from calcium toward vanadium, signaling the disappearance of the N=32 closure with increasing proton number.<sup>[16](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevC.106.024312/fulltext)</sup> By contrast, a 2026 review of a decade of RIKEN experiments reports that direct reactions on liquid hydrogen targets prove the double magicity of 52,54Ca and support persistence of the N=34 subshell closure below calcium, in argon, potassium, calcium, and scandium isotopes.<sup>[9](https://academic.oup.com/ptep/article-2026/4/04A101/8258960)</sup> The two lines of evidence agree that N=32 and N=34 are strongest at calcium and weaken with proton number, but they differ on how far each closure extends.

The proton side evolves too. The Z=82 proton shell gap is large for N≥126 but fades in neutron-deficient lead isotopes, consistent with shape coexistence, in which a nucleus alternates between spherical and deformed configurations at similar energy.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup> Mass models such as UNEDF1 predict that the magic gaps at Z=50 and Z=82 are significantly weakened around N=106 and N=150 respectively, in regions beyond experimentally known masses.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup>

## Magic numbers and nucleosynthesis

Closed neutron shells act as waiting points in the r-process, the rapid neutron-capture sequence that builds the heavy elements. At N=82 and N=126, low neutron-capture cross sections stall the flow until beta decay moves the material to the next element, and beta-decay half-lives drop just after shell closure; the result is the broad abundance peaks at mass numbers A=130 and A=195.<sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup><sup> • </sup><sup>[12](http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/shell2.html)</sup> The shape of these peaks therefore encodes the size of the shell gaps far from stability: a reduction of the N=82 gap is expected to shift the r-process path and change the shape of the A=130 peak accordingly.<sup>[17](https://ar5iv.labs.arxiv.org/html/1401.1378)</sup> Half-lives matter as well: shell-model half-lives for the N=126 waiting-point nuclei are shorter than older finite-range-droplet-model values, shifting the third abundance peak toward higher mass and increasing predicted thorium and uranium production in core-collapse supernova models, while new experimental half-lives for N=82 isotones obtained in 2015 are rather consistently reproduced across models.<sup>[18](https://beta.iopscience.iop.org/article/10.3847/1538-4357/aabfde)</sup>

## Comparison with atomic shells

The analogy with electron shells is real but limited. In both systems, filling a shell produces exceptional stability, and the nucleus just above a closed neutron shell has weakly bound valence neutrons, reminiscent of the alkali metals in the chemical shell structure.<sup>[12](http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/shell2.html)</sup> The breakdowns are equally instructive. The central potential differs (a mean field from the strong force rather than Coulomb attraction to a nucleus), the spin-orbit coupling has the opposite sign, and the magic numbers themselves migrate with neutron-proton asymmetry.<sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup><sup> • </sup><sup>[13](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.92.015002)</sup> A further distinction concerns which nuclei count as doubly magic. Under the standard spin-orbit shell model the doubly magic list includes 48Ca, 100Sn, and 78Ni, with 48Ni showing evidence of being unbound to two-[proton emission](https://www.edgepedia.org/proton-emission); under a stricter classification requiring pure harmonic-oscillator (LS) closure, only three doubly magic nuclei are known, 4He, 16O, and 40Ca, because the next candidate, 80Zr, fails since its Z=N=40 gap is too small due to the lowering of the 0g9/2 orbital.<sup>[3](https://ar5iv.labs.arxiv.org/html/1006.2520)</sup><sup> • </sup><sup>[7](https://par.nsf.gov/servlets/purl/10333174)</sup>

**Semi-magic and deformed nuclei.** Nuclei with one magic number, such as 120Sn (Z=50, N=70), are called semi-magic, but quadrupole correlations can erode apparent magicity; an updated analysis of empirical shell gaps using ISOLTRAP precision masses at ISOLDE/CERN emphasizes the role of quadrupole correlations in explaining the full range of experimental information on mutually enhanced magicity.<sup>[1](https://arxiv.org/html/2309.16871v2)</sup><sup> • </sup><sup>[19](https://link.springer.com/article/10.1140/epja/s10050-023-00929-5)</sup>

## Recent developments

The calcium region has been the most active frontier. High-precision mass measurements of 13 neutron-rich Ca, Ti, and V isotopes at TRIUMF's TITAN MR-ToF-MS and the LEBIT Penning trap refined the mass surface around N=32 and N=34, with 8 of the masses improving on previous precision.<sup>[16](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevC.106.024312/fulltext)</sup> At the [Facility for Rare Isotope Beams](https://www.edgepedia.org/facility-for-rare-isotope-beams), the FRIB Decay Station initiator measured 15 previously unknown beta-decay half-lives for neutron-rich isotopes near calcium-54, revealing an unexpected change in half-life trends at 34 neutrons, with nuclei beyond this point living longer than anticipated; the results feed models of processes in neutron-star crusts.<sup>[20](https://frib.msu.edu/news-center/news/new-measurements-reveal-longer-expected-lifetimes-neutron-rich-nuclei)</sup> The 2026 PTEP review consolidates the RIKEN program's conclusion that 52,54Ca are doubly magic while leaving the reach of N=34 an open measurement question across isotopic chains.<sup>[9](https://academic.oup.com/ptep/article-2026/4/04A101/8258960)</sup>

## History and practical significance

[Maria Goeppert Mayer](https://www.edgepedia.org/maria-goeppert-mayer), working with [Edward Teller](https://www.edgepedia.org/edward-teller) at the University of Chicago and [Argonne National Laboratory](https://www.edgepedia.org/argonne-national-laboratory), compiled isotope abundances and observed that nuclei with 2, 8, 20, 28, 50, 82, or 126 protons or neutrons were especially stable; she published the evidence for the shell model in August 1948.<sup>[6](https://www.aps.org/apsnews/2008/08/goeppert-mayer-nuclear-shell-model)</sup> The explanatory breakthrough came in 1949, when the Haxel–Jensen–Suess paper appeared in Physical Review volume 75 on 1 June 1949, independently and simultaneously with Mayer's work; Mayer, who had never met the three men, was convinced by the coincidence, met Jensen in 1950, and the two later collaborated.<sup>[5](https://journals.aps.org/pr/abstract/10.1103/PhysRev.75.1766.2)</sup><sup> • </sup><sup>[21](http://nobelprize.org/nobel_prizes/physics/laureates/1963/mayer-bio.html)</sup> The term "magic numbers" is thought to have been coined by Eugene Wigner, who was somewhat skeptical about the shell model.<sup>[6](https://www.aps.org/apsnews/2008/08/goeppert-mayer-nuclear-shell-model)</sup> Mayer and Jensen shared the 1963 Nobel Prize in Physics.<sup>[2](https://www.nobelprize.org/uploads/2018/06/mayer-lecture.pdf)</sup>

The numbers remain operational anchors for nuclear data. NUBASE2020, the authoritative evaluated compilation of masses, half-lives, and decay modes underlying the AME2020 mass evaluation, is maintained for the radioactive-beam facilities that probe shell evolution, including FAIR, ISOLDE, SPIRAL2, ATLAS, FRIB, HIAF, and RIKEN RIBF.<sup>[22](https://www-nds.iaea.org/amdc/ame2020/NUBASE2020.pdf)</sup> On the applied side, the IAEA's Evaluated Nuclear Data File aggregates neutron cross sections from the major national libraries (ENDF/B-7.1, JEFF-3.2, CENDL-3.1, JENDL-4.0u2, BROND-2) used for neutron transport, shielding calculations, and nuclear reactors.<sup>[23](https://www.iaea.org/resources/databases/evaluated-nuclear-data-file)</sup> The landscape these data describe is large: a [Bayesian model averaging](https://www.edgepedia.org/bayesian-model-averaging) analysis predicts 7759 nuclei with Z≤119 have a probability of existence of at least 0.5.<sup>[24](https://link.aps.org/doi/10.1103/PhysRevC.101.044307)</sup>

## References

1. [Nucleonic shells and nuclear masses (arXiv:2309.16871)](https://arxiv.org/html/2309.16871v2)
2. [Maria Goeppert Mayer, Nobel Lecture (1963)](https://www.nobelprize.org/uploads/2018/06/mayer-lecture.pdf)
3. [New nuclear magic numbers (arXiv:1006.2520)](https://ar5iv.labs.arxiv.org/html/1006.2520)
4. [Emerging Concepts in Nuclear Structure Based on the Shell Model, MDPI Nuclear Physics](https://www.mdpi.com/2624-8174/4/1/18)
5. [Haxel, Jensen, Suess, On the "Magic Numbers" in Nuclear Structure, Phys. Rev. 75, 1766 (1949)](https://journals.aps.org/pr/abstract/10.1103/PhysRev.75.1766.2)
6. [August 1948: Maria Goeppert Mayer and the Nuclear Shell Model, APS News](https://www.aps.org/apsnews/2008/08/goeppert-mayer-nuclear-shell-model)
7. [The Nuclear Shell Model towards the Drip Lines (NSF PAR)](https://par.nsf.gov/servlets/purl/10333174)
8. [A unified mechanism for the origin and evolution of nuclear magicity (arXiv:2411.15562)](https://arxiv.org/html/2411.15562)
9. [Shell Migration at N = 32, 34 around Ca Region, PTEP (2026)](https://academic.oup.com/ptep/article-2026/4/04A101/8258960)
10. [The AME 2020 atomic mass evaluation (II), Chinese Physics C](https://beta.iopscience.iop.org/article/10.1088/1674-1137/abddaf)
11. [Shell Model lecture notes, University of Southampton](https://personal.soton.ac.uk/ab1u06/teaching/phys3002/course/05_shell.pdf)
12. [Shell Model of Nucleus, HyperPhysics, Georgia State University](http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/shell2.html)
13. [Evolution of shell structure in exotic nuclei, Rev. Mod. Phys. 92, 015002 (2020)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.92.015002)
14. [Evidence for a new nuclear 'magic number' from the level structure of 54Ca, Nature (2013)](https://www.nature.com/articles/nature12522)
15. [Magic Nature of Neutrons in 54Ca: First Mass Measurements of 55–57Ca, Phys. Rev. Lett. 121, 022506](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.022506)
16. [Precision mass measurements of neutron-rich Ca, Ti, and V isotopes, Phys. Rev. C 106, 024312](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevC.106.024312/fulltext)
17. [Shell Evolutions and Nuclear Forces (arXiv:1401.1378)](https://ar5iv.labs.arxiv.org/html/1401.1378)
18. [β-decay Rates for Exotic Nuclei and r-process Nucleosynthesis, Astrophysical Journal](https://beta.iopscience.iop.org/article/10.3847/1538-4357/aabfde)
19. [The empirical shell gap revisited, Eur. Phys. J. A (2023)](https://link.springer.com/article/10.1140/epja/s10050-023-00929-5)
20. [New measurements reveal longer-than-expected lifetimes in neutron-rich nuclei, FRIB](https://frib.msu.edu/news-center/news/new-measurements-reveal-longer-expected-lifetimes-neutron-rich-nuclei)
21. [Maria Goeppert Mayer – Biographical, NobelPrize.org](http://nobelprize.org/nobel_prizes/physics/laureates/1963/mayer-bio.html)
22. [The NUBASE2020 evaluation of nuclear physics properties, AMDC/IAEA](https://www-nds.iaea.org/amdc/ame2020/NUBASE2020.pdf)
23. [Evaluated Nuclear Data File, IAEA](https://www.iaea.org/resources/databases/evaluated-nuclear-data-file)
24. [Quantified limits of the nuclear landscape, Phys. Rev. C 101, 044307](https://link.aps.org/doi/10.1103/PhysRevC.101.044307)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Particles and nuclei › Nuclear physics › Nuclear structure and models › Nuclear models › Nuclear shell model*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license