Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Nuclear moments of indium isotopes reveal abrupt change at magic number 82

## Abstract

In spite of the high-density and strongly correlated nature of the atomic nucleus, experimental and theoretical evidence suggests that around particular ‘magic’ numbers of nucleons, nuclear properties are governed by a single unpaired nucleon1,2. A microscopic understanding of the extent of this behaviour and its evolution in neutron-rich nuclei remains an open question in nuclear physics3,4,5. The indium isotopes are considered a textbook example of this phenomenon6, in which the constancy of their electromagnetic properties indicated that a single unpaired proton hole can provide the identity of a complex many-nucleon system6,7. Here we present precision laser spectroscopy measurements performed to investigate the validity of this simple single-particle picture. Observation of an abrupt change in the dipole moment at N = 82 indicates that, whereas the single-particle picture indeed dominates at neutron magic number N = 82 (refs. 2,8), it does not for previously studied isotopes. To investigate the microscopic origin of these observations, our work provides a combined effort with developments in two complementary nuclear many-body methods: ab initio valence-space in-medium similarity renormalization group and density functional theory (DFT). We find that the inclusion of time-symmetry-breaking mean fields is essential for a correct description of nuclear magnetic properties, which were previously poorly constrained. These experimental and theoretical findings are key to understanding how seemingly simple single-particle phenomena naturally emerge from complex interactions among protons and neutrons.

This is a preview of subscription content, access via your institution

## Access options

\$32.00

All prices are NET prices.

## Data availability

Examples of spectra source data for the previously unmeasured 129,131In isotopes, most relevant to this work, are included in this article. The full datasets generated and/or analysed during the current study are available in the Zenodo repository, https://doi.org/10.5281/zenodo.6406949. The code used to analyse the data is also included in the repository.

The data files related to the DFT calculations are available at https://webfiles.york.ac.uk/HFODD/Projects/Magnetic_and_electric_moments_in_Indium/.

## Code availability

The code used to analyse the data is included in the Zenodo repository, https://doi.org/10.5281/zenodo.6406949.

The code used to perform the DFT calculations is available at https://webfiles.york.ac.uk/HFODD/Projects/hf301m/.

## References

1. Schmidt, T. The electric quadrupole moment of the nucleus. Nature 138, 404 (1936).

2. Jones, K. L. et al. The magic nature of 132Sn explored through the single-particle states of 133Sn. Nature 465, 454–457 (2010).

3. Taniuchi, R. et al. 78Ni revealed as a doubly magic stronghold against nuclear deformation. Nature 569, 53–58 (2019).

4. Togashi, T., Tsunoda, Y., Otsuka, T., Shimizu, N. & Honma, M. Novel shape evolution in Sn isotopes from magic numbers 50 to 82. Phys. Rev. Lett. 121, 062501 (2018).

5. Jenkins, D. G. Recent advances in nuclear physics through on-line isotope separation. Nat. Phys. 10, 909–913 (2014).

6. Heyde, K. L. G. The Nuclear Shell Model. Springer Series in Nuclear and Particle Physics (Springer, 1990).

7. Eberz, J. et al. Spins, moments and mean square charge radii of 104–127In determined by laser spectroscopy. Nucl. Phys. A 464, 9–28 (1987).

8. Rosiak, D. et al. Enhanced quadrupole and octupole strength in doubly magic 132Sn. Phys. Rev. Lett. 121, 252501 (2018).

9. Gysbers, P. et al. Discrepancy between experimental and theoretical β-decay rates resolved from first principles. Nat. Phys. 15, 428–431 (2019).

10. Morris, T. D. et al. Structure of the lightest tin isotopes. Phys. Rev. Lett. 120, 152503 (2018).

11. Schunck, N. (ed.) Energy Density Functional Methods for Atomic Nuclei 2053–2563 (IOP Publishing, 2019).

12. Rodríguez, L. V. et al. Doubly-magic character of 132Sn studied via electromagnetic moments of 133Sn. Phys. Rev. C 102, 051301 (2020).

13. Hinke, C. B. et al. Superallowed Gamow–Teller decay of the doubly magic nucleus 100Sn. Nature 486, 341–345 (2012).

14. Manea, V. et al. First glimpse of the N = 82 shell closure below Z = 50 from masses of neutron-rich cadmium isotopes and isomers. Phys. Rev. Lett. 124, 092502 (2020).

15. Neyens, G. Nuclear magnetic and quadrupole moments for nuclear structure research on exotic nuclei. Rep. Prog. Phys. 66, 633–689 (2003).

16. Stroberg, S. R., Bogner, S. K., Hergert, H. & Holt, J. D. Nonempirical interactions for the nuclear shell model: an update. Annu. Rev. Nucl. Part. Sci. 69, 307–362 (2019).

17. Hebeler, K., Bogner, S. K., Furnstahl, R. J., Nogga, A. & Schwenk, A. Improved nuclear matter calculations from chiral low-momentum interactions. Phys. Rev. C 83, 031301 (2011).

18. de Groote, R. et al. Precision measurement of the magnetic octupole moment in 45Sc as a test for state-of-the-art atomic-and nuclear-structure theory. Phys. Lett. B 827, 136930 (2022).

19. Bennaceur, K., Dobaczewski, J., Haverinen, T. & Kortelainen, M. Properties of spherical and deformed nuclei using regularized pseudopotentials in nuclear DFT. J. Phys. G Nucl. Part. Phys. 47, 105101 (2020).

20. Dobaczewski, J. et al. Mean-field description of ground-state properties of drip-line nuclei: pairing and continuum effects. Phys. Rev. C 53, 2809–2840 (1996).

21. Dobaczewski, J., Flocard, H. & Treiner, J. Hartree-Fock-Bogolyubov description of nuclei near the neutron-drip line. Nucl. Phys. A 422, 103–139 (1984).

22. Sheikh, J. A., Dobaczewski, J., Ring, P., Robledo, L. M. & Yannouleas, C. Symmetry restoration in mean-field approaches. J. Phys. G Nucl. Part. Phys. 48, 123001 (2021).

23. Cocolios, T. E. et al. High-resolution laser spectroscopy with the Collinear Resonance Ionisation Spectroscopy (CRIS) experiment at CERN-ISOLDE. Nucl. Instrum. Methods Phys. Res. B Beam Interact. Mater. Atoms 376, 284–287 (2016).

24. Sahoo, B. K. et al. Analytic response relativistic coupled-cluster theory: the first application to indium isotope shifts. New J. Phys. 22, 012001 (2020).

25. Vernon, A. et al. Simulation of the relative atomic populations of elements 1<=Z<=89 following charge exchange tested with collinear resonance ionization spectroscopy of indium. Spectrochim. Acta Part B At. Spectrosc. 153, 61–83 (2019).

26. Garcia Ruiz, R. F. et al. High-precision multiphoton ionization of accelerated laser-ablated species. Phys. Rev. X 8, 041005 (2018).

27. Epelbaum, E., Hammer, H.-W. & Meißner, U.-G. Modern theory of nuclear forces. Rev. Mod. Phys. 81, 1773–1825 (2009).

28. Machleidt, R. & Entem, D. R. Chiral effective field theory and nuclear forces. Phys. Rep. 503, 1–75 (2011).

29. Jiang, W. G. et al. Accurate bulk properties of nuclei from A = 2 to ∞ from potentials with ∆ isobars. Phys. Rev. C 102, 054301 (2020).

30. Engel, Y., Brink, D., Goeke, K., Krieger, S. & Vautherin, D. Time-dependent Hartree-Fock theory with Skyrme’s interaction. Nucl. Phys. A 249, 215–238 (1975).

31. Perlińska, E., Rohoziński, S. G., Dobaczewski, J. & Nazarewicz, W. Local density approximation for proton-neutron pairing correlations: formalism. Phys. Rev. C 69, 014316 (2004).

32. Dobaczewski, J., Engel, J., Kortelainen, M. & Becker, P. Correlating Schiff moments in the light actinides with octupole moments. Phys. Rev. Lett. 121, 232501 (2018).

33. Chupp, T. E., Fierlinger, P., Ramsey-Musolf, M. J. & Singh, J. T. Electric dipole moments of atoms, molecules, nuclei, and particles. Rev. Mod. Phys. 91, 015001 (2019).

34. Dolinski, M. J., Poon, A. W. & Rodejohann, W. Neutrinoless double-beta decay: status and prospects. Annu. Rev. Nucl. Part. Sci. 69, 219–251 (2019).

35. Engel, J., Pittel, S. & Vogel, P. Nuclear physics of dark matter detection. Int. J. Mod. Phys. E 1, 1–37 (1992).

36. Co’, G., Donno, V. D., Anguiano, M., Bernard, R. N. & Lallena, A. M. Electric quadrupole and magnetic dipole moments of odd nuclei near the magic ones in a self-consistent approach. Phys. Rev. C 92, 024314 (2015).

37. Krane, K. S. & Halliday, D. Introductory Nuclear Physics (Wiley, 1988).

38. Henderson, J. et al. Testing microscopically derived descriptions of nuclear collectivity: Coulomb excitation of 22Mg. Phys. Lett. B 782, 468–473 (2018).

39. Schmidt, T. Über die magnetischen Momente der Atomkerne. Z. Phys. 106, 358–361 (1937).

40. Taprogge, J. et al. 1p3/2 proton-hole state in 132Sn and the shell structure along N = 82. Phys. Rev. Lett. 112, 132501 (2014).

41. Fogelberg, B. et al. Decays of 131In, 131Sn, and the position of the h11/2 neutron hole state. Phys. Rev. C 70, 034312 (2004).

42. Fogelberg, B. & Blomqvist, J. Single-hole and three-quasiparticle levels in 131Sn observed in the decay of 131g,m1,m2In. Nucl. Phys. A 429, 205–217 (1984).

43. Vaquero, V. et al. Fragmentation of single-particle strength around the doubly magic nucleus 132Sn and the position of the 0f5/2 proton-hole state in 131In. Phys. Rev. Lett. 124, 022501 (2020).

44. Lechner, S. et al. Probing the single-particle behavior above 132Sn via electromagnetic moments of 133,134Sb and n = 82 isotones. Phys. Rev. C 104, 014302 (2021).

45. Weiffenbach, C. V. & Tickle, R. Structure of odd-a indium isotopes determined by the (d, 3He) reaction. Phys. Rev. C 3, 1668–1678 (1971).

46. Kay, B. The SOLARIS spectrometer. In APS April Meeting Abstracts Q13.002 (APS, 2019).

47. Tang, T. L. et al. First exploration of neutron shell structure below lead and beyond n = 126. Phys. Rev. Lett. 124, 062502 (2020).

48. Bender, M., Dobaczewski, J., Engel, J. & Nazarewicz, W. Gamow-Teller strength and the spin-isospin coupling constants of the Skyrme energy functional. Phys. Rev. C 65, 054322 (2002).

49. Sassarini, P. L., Dobaczewski, J., Bonnard, J. & Garcia Ruiz, R. F. Global analysis of electromagnetic moments in odd near doubly magic nuclei. Preprint at https://arxiv.org/abs/2111.04675 (2021).

50. Borzov, I. N., Tolokonnikov, S. V. & Fayans, S. Spin-dependent effective nucleon-nucleon interaction in nuclei. Sov. J. Nucl. Phys. 40, 732–739 (1984).

51. Wakasa, T., Ichimura, M. & Sakai, H. Unified analysis of spin isospin responses of nuclei. Phys. Rev. C 72, 067303 (2005).

52. Roca-Maza, X., Colò, G. & Sagawa, H. New Skyrme interaction with improved spin-isospin properties. Phys. Rev. C 86, 031306 (2012).

53. Davesne, D., Pastore, A. & Navarro, J. Linear response theory with finite-range interactions. Prog. Part. Nucl. Phys. 120, 103870 (2021).

54. Pastore, S., Pieper, S. C., Schiavilla, R. & Wiringa, R. B. Quantum Monte Carlo calculations of electromagnetic moments and transitions in A ≤ 9 nuclei with meson-exchange currents derived from chiral effective field theory. Phys. Rev. C 87, 035503 (2013).

55. Rice, M. & Pound, R. V. Ratio of the magnetic moments of In115 and In113. Phys. Rev. 106, 953–953 (1957).

56. Childs, W. J. & Goodman, L. S. Nuclear spin and hyper-fine Interaction of In113m. Phys. Rev. 118, 1578–1581 (1960).

57. Flynn, C. P. & Seymour, E. F. W. Knight shift of the nuclear magnetic resonance in liquid indium. Proc. Phys. Soc. 76, 301–303 (1960).

58. Cameron, J. A., King, H. J., Eastwood, H. K. & Summers-Gill, R. G. The magnetic moment of indium-115m. Can. J. Phys. 40, 931–942 (1962).

59. Köster, U. Intense radioactive-ion beams produced with the ISOL method. Eur. Phys. J. A 15, 255–263 (2002).

60. Dillmann, I. et al. Selective laser ionization of N>=82 indium isotopes: the new r-process nuclide 135In. Eur. Phys. J. A 13, 281–284 (2002).

61. Rothe, S., Marsh, B. A., Mattolat, C., Fedosseev, V. N. & Wendt, K. A complementary laser system for ISOLDE RILIS. J. Phys. Conf. Ser. 312, 052020 (2011).

62. Mané, E. et al. An ion cooler-buncher for high-sensitivity collinear laser spectroscopy at ISOLDE. Eur. Phys. J. A 42, 503–507 (2009).

63. Frånberg, H. et al. Off-line commissioning of the ISOLDE cooler. Nucl. Instrum. Methods Phys. Res. B Beam Interact. Mater. Atoms 266, 4502–4504 (2008).

64. Vernon, A. R. et al. Optimising the collinear resonance ionisation spectroscopy (CRIS) experiment at CERN-ISOLDE. Nucl. Instrum. Methods Phys. Res. B Beam Interact. Mater. Atoms 463, 384–389 (2020).

65. Vernon, A. et al. Simulation of the relative atomic populations of elements 1 ≤ Z ≤ 89 following charge exchange tested with collinear resonance ionization spectroscopy of indium. Spectrochim. Acta Part B At. Spectrosc. 153, 61–83 (2019).

66. Sonnenschein, V. et al. Characterization of a pulsed injection-locked Ti:sapphire laser and its application to high resolution resonance ionization spectroscopy of copper. Laser Phys. 27, 085701 (2017).

67. Bass, M., Franken, P. A., Hill, A. E., Peters, C. W. & Weinreich, G. Optical mixing. Phys. Rev. Lett. 8, 18 (1962).

68. Persson, J. R. Table of hyperfine anomaly in atomic systems. At. Data Nucl. Data Tables 99, 62–68 (2013).

69. Cheal, B. et al. Nuclear spins and moments of Ga isotopes reveal sudden structural changes between N = 40 and N = 50. Phys. Rev. Lett. 104, 252502 (2010).

70. Morris, T. D., Parzuchowski, N. M. & Bogner, S. K. Magnus expansion and in-medium similarity renormalization group. Phys. Rev. C 92, 34331 (2015).

71. Simonis, J., Stroberg, S. R., Hebeler, K., Holt, J. D. & Schwenk, A. Saturation with chiral interactions and consequences for finite nuclei. Phys. Rev. C 96, 014303 (2017).

72. Shimizu, N., Mizusaki, T., Utsuno, Y. & Tsunoda, Y. Thick-restart block Lanczos method for large-scale shell-model calculations. Comput. Phys. Commun. 244, 372–384 (2019).

73. Schunck, N. et al. Solution of the Skyrme-Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis. (VIII) HFODD (v2.73y): a new version of the program. Comput. Phys. Commun. 216, 145–174 (2017).

74. Dobaczewski, J. et al. Solution of universal nonrelativistic nuclear DFT equations in the Cartesian deformed harmonic-oscillator basis. (IX) HFODD (v3.06h): a new version of the program. J. Phys. G Nucl. Part. Phys. 48, 102001 (2021).

75. Kortelainen, M. et al. Nuclear energy density optimization: large deformations. Phys. Rev. C 85, 024304 (2012).

76. Dobaczewski, J. & Dudek, J. Solution of the Skyrme–Hartree–Fock equations in the Cartesian deformed harmonic oscillator basis II. The program HFODD. Comput. Phys. Commun. 102, 183–209 (1997).

77. Schunck, N. et al. One-quasiparticle states in the nuclear energy density functional theory. Phys. Rev. C 81, 024316 (2010).

78. Satuła, W., Bączyk, P., Dobaczewski, J. & Konieczka, M. No-core configuration-interaction model for the isospin- and angular-momentum-projected states. Phys. Rev. C 94, 024306 (2016).

79. Varshalovich, D., Moskalev, A. & Khersonskii, V. Quantum Theory of Angular Momentum (World Scientific, 1988).

80. Ring, P. & Schuck, P. The Nuclear Many-body Problem (Springer, 1980).

## Acknowledgements

This work was supported by ERC Consolidator Grant no. 648381 (FNPMLS); STFC grants ST/L005794/1, ST/L005786/1, ST/P004423/1, ST/M006433/1 and ST/P003885/1, and Ernest Rutherford grant no. ST/L002868/1; the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under grant DE-SC0021176; GOA 15/010 from KU Leuven, BriX Research Program No. P7/12; the FWO-Vlaanderen (Belgium); the European Unions Grant Agreement 654002 (ENSAR2); National Key R&D Program of China (contract no. 2018YFA0404403); the National Natural Science Foundation of China (no. 11875073); the Polish National Science Centre under contract no. 2018/31/B/ST2/02220. TRIUMF receives funding by a contribution through the National Research Council of Canada. The theoretical work was further supported by NSERC and the U.S. Department of Energy under contract DE-FG02-97ER41014. The VS-IMSRG computations were performed with an allocation of computing resources on Cedar at WestGrid and Compute Canada, and on the Oak Cluster at TRIUMF managed by the University of British Columbia department of Advanced Research Computing (ARC). We would also like to thank the ISOLDE technical group for their support and assistance and the University of Jyväskylä for the use of the injection-locked cavity. We acknowledge the CSC – IT Center for Science Ltd., Finland, for the allocation of computational resources. This project was partly undertaken on the Viking Cluster, which is a high-performance computing facility provided by the University of York. We are grateful for computational support from the University of York High Performance Computing service, Viking and the Research Computing team.

## Author information

Authors

### Contributions

A.R.V. prepared the manuscript with input from all authors, especially R.F.G.R., J.Bo., J.D., J.D.H., T.M., G.N., K.T.F., T.E.C., R.P.deG. and S.R.S. R.F.G.R., J.Bi., C.L.B., M.L.B., T.E.C., K.T.F., W.G., R.P.deG., A.K., K.M.L., G.N., S.G.W., A.R.V. and X.F.Y. proposed the experiment(s), A.R.V., C.L.B., M.L.B., T.E.C., K.T.F., G.J.F.-S., G.G., W.G., R.P.deG., R.H., A.K., D.L., K.M.L., R.F.G.R., S.G.W., X.F.Y. and D.Y. conducted the experiment(s), A.R.V., C.L.B., R.F.G.R. and J.H. analysed the results, J.Bo. and J.D. performed theoretical (DFT) nuclear calculations, J.D.H., T.M. and S.R.S. performed theoretical (VS-IMSRG) nuclear calculations. All authors reviewed the manuscript.

### Corresponding authors

Correspondence to A. R. Vernon or R. F. Garcia Ruiz.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Gianluca Colo and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Example hyperfine spectra of the 129In and 131In isotopes.

a, b, Example spectra measured using the 246.8-nm (5p 2P3/2 → 9s 2S1/2) transition (a), and using the 246.0-nm (5p 2P1/2 → 8s 2S1/2) transition (b). The 9/2+ ground and 1/2 isomer states are indicated.

## Rights and permissions

Reprints and Permissions

Vernon, A.R., Garcia Ruiz, R.F., Miyagi, T. et al. Nuclear moments of indium isotopes reveal abrupt change at magic number 82. Nature 607, 260–265 (2022). https://doi.org/10.1038/s41586-022-04818-7

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41586-022-04818-7