Skip to main content
SpringerLink
Log in

Predictions of nuclear charge radii based on the convolutional neural network

  • Published:
Nuclear Science and Techniques Aims and scope Submit manuscript

Abstract

In this study, we developed a neural network that incorporates a fully connected layer with a convolutional layer to predict the nuclear charge radii based on the relationships between four local nuclear charge radii. The convolutional neural network (CNN) combines the isospin and pairing effects to describe the charge radii of nuclei with \(A \ge\) 39 and \(Z \ge\) 20. The developed neural network achieved a root mean square (RMS) deviation of 0.0195 fm for a dataset with 928 nuclei. Specifically, the CNN reproduced the trend of the inverted parabolic behavior and odd–even staggering observed in the calcium isotopic chain, demonstrating reliable predictive capability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data availability

The data that support the findings of this study are openly available in Science Data Bank at https://www.doi.org/10.57760/sciencedb.j00186.00238 and https://cstr.cn/31253.11.sciencedb.j00186.00238.

References

  1. A. Bohr, B.R. Mottelson, Nuclear Structure, vol. One Benjamin, New York (1969). https://doi.org/10.1063/1.3022342

  2. H. Limura, F. Buchinger, Charge radii in macroscopic microscopic mass models of reflection asymmetry. Phys. Rev. C 78, 067301 (2008). https://doi.org/10.1103/PhysRevC.78.067301

    Article  ADS  Google Scholar 

  3. S. Geldhof, M. Kortelainen, O. Beliuskina et al., Impact of nuclear deformation and pairing on the charge radii of palladium isotopes Phys. Rev. Lett. 128, 152501 (2022). https://doi.org/10.1103/PhysRevLett.128.152501

    Article  ADS  Google Scholar 

  4. L.B. Wang, P. Mueller, K. Bailey et al., Laser spectroscopic determination of the \({}^{6}\)He nuclear charge radius. Phys. Rev. Lett. 93, 142501 (2004). https://doi.org/10.1103/PhysRevLett.93.142501

    Article  ADS  Google Scholar 

  5. N. Wang, T. Li, Shell and isospin effects in nuclear charge radii. Phys. Rev. C 88, 011301 (2013). https://doi.org/10.1103/PhysRevC.88.011301

    Article  ADS  Google Scholar 

  6. T.Q. Liang, J. Liu, Z.Z. Ren et al., Elastic electron scattering form factors of deformed exotic Xe isotopes. Phys. Rev. C 98, 044310 (2018). https://doi.org/10.1103/PhysRevC.98.044310

    Article  ADS  Google Scholar 

  7. H. De Vries, C.W. De Jager, C. De Vries, Nuclear charge-density-distribution parameters from elastic electron scattering. At. Data Nucl. Data Tables 36, 495 (1987). https://doi.org/10.1016/0092-640X(87)90013-1

    Article  ADS  Google Scholar 

  8. K. Heilig, A. Steudel, Changes in mean-square nuclear charge radii from optical isotope shifts. At. Data Nucl. Data Tables 14, 613 (1974). https://doi.org/10.1016/S0092-640X(74)80006-9

    Article  ADS  Google Scholar 

  9. P. Aufmuth, K. Heilig, A. Steudel, Changes in mean-square nuclear charge radii from optical isotope shifts. At. Data Nucl. Data Tables 37, 455 (1987). https://doi.org/10.1016/0092-640X(87)90028-3

    Article  ADS  Google Scholar 

  10. R. Engfer, H. Schneuwly, J.L. Vuileumier et al., Charge-distribution parameters, isotope shifts, isomer shifts, and magnetic hyperfine constants from muonic atoms. At. Data Nucl. Data Tables 14, 509 (1974). https://doi.org/10.1016/S0092-640X(74)80003-3

    Article  ADS  Google Scholar 

  11. G. Fricke, C. Bernhardt, K. Heilig et al., Nuclear ground state charge radii from electromagnetic interactions. At. Data Nucl. Data Tables 60, 177 (1995). https://doi.org/10.1006/adnd.1995.1007

    Article  ADS  Google Scholar 

  12. E. Boehm, P.L. Lee, Changes of mean-square nuclear charge radii from isotope shifts of electronic \(K_{\alpha }\) X-rays. At. Data Nucl. Data Tables 14, 605 (1974). https://doi.org/10.1016/S0092-640X(74)80005-7

    Article  ADS  Google Scholar 

  13. I. Angeli, K.P. Marinova, Table of experimental nuclear ground state charge radii: an update. At. Data Nucl. Data Tables 99, 69 (2013). https://doi.org/10.1016/j.adt.2011.12.006

    Article  ADS  Google Scholar 

  14. T. Li, Y. Luo, N. Wang, Compilation of recent nuclear ground state charge radius measurements and tests for models. At. Data Nucl. Data Tables 140, 101440 (2021). https://doi.org/10.1016/j.adt.2021.101440

    Article  Google Scholar 

  15. S. Goriely, S. Hilaire, M. Girod et al., First Gogny–Hartree–Fock–Bogoliubov nuclear mass model. Phys. Rev. Lett. 102, 242501 (2009). https://doi.org/10.1103/PhysRevLett.102.242501

    Article  ADS  Google Scholar 

  16. S. Goriely, N. Chamel, J.M. Pearson, Further explorations of Skyrme–Hartree–Fock–Bogoliubov mass formulas. XII. Stiffness and stability of neutron-star matter. Phys. Rev. C 82, 035804 (2009). https://doi.org/10.1103/PhysRevC.82.035804

    Article  ADS  Google Scholar 

  17. L.S. Geng, H. Toki, S. Sugimoto et al., Relativistic mean field theory for deformed nuclei with pairing correlations. Prog. Theor. Phys. 110, 921 (2003). https://doi.org/10.1143/PTP.110.921

    Article  MATH  ADS  Google Scholar 

  18. P.W. Zhao, Z.P. Li, J.M. Yao et al., New parametrization for the nuclear covariant energy density functional with a point-coupling interaction. Phys. Rev. C 82, 054319 (2010). https://doi.org/10.1103/PhysRevC.82.054319

    Article  ADS  Google Scholar 

  19. Y. Funaki, T. Yamada, H. Horiuchi et al., \(\alpha\)-particle condensation in \({}^{16}\)O studied with a full four-body orthogonality condition model calculation. Phys. Rev. Lett. 101, 082502 (2008). https://doi.org/10.1103/PhysRevLett.101.082502

    Article  ADS  Google Scholar 

  20. S.Q. Zhang, J. Meng, S.G. Zhou et al., Isospin and \(Z^{1/3}\)-dependence of the nuclear charge radii. Eur. Phys. J. A 13, 285 (2002). https://doi.org/10.1007/s10050-002-8757-6

    Article  ADS  Google Scholar 

  21. Y.A. Lei, Z.H. Zhang, J.Y. Zeng, Improved \(Z^{1/3}\) law of nuclear charge radius. Commun. Theor. Phys. 51, 123 (2009). https://doi.org/10.1088/0253-6102/51/1/23

    Article  ADS  Google Scholar 

  22. B. Nerlo-Pomorska, K. Pomorski, Isospin dependence of nuclear radius. Z. Phys. A 344, 359 (1993). https://doi.org/10.1007/BF01283190

    Article  ADS  Google Scholar 

  23. B. Nerlo-Pomorska, K. Pomorski, Simple formula for nuclear charge radius. Z. Phys. A 348, 169 (1994). https://doi.org/10.1007/BF01291913

    Article  ADS  Google Scholar 

  24. G. Royer, R. Rousseau, On the liquid drop model mass formulae and charge radii. Eur. Phys. J. A 42, 541 (2009). https://doi.org/10.1140/epja/i2008-10745-8

    Article  ADS  Google Scholar 

  25. Z.Q. Sheng, G.W. Fan, J.F. Qian et al., An effective formula for nuclear charge radii. Eur. Phys. J. A 51, 4 (2015). https://doi.org/10.1140/epja/i2015-15040-1

    Article  Google Scholar 

  26. J. Piekarewicz, M. Centelles, X. Roca-Maza et al., Garvey–Kelson relations for nuclear charge radii. Eur. Phys. J. A 46, 379 (2010). https://doi.org/10.1140/epja/i2010-11051-8

    Article  ADS  Google Scholar 

  27. B.H. Sun, Y. Lu, J.P. Peng et al., New charge radius relations for atomic nuclei. Phys. Rev. C 90, 054318 (2014). https://doi.org/10.1103/PhysRevC.90.054318

    Article  ADS  Google Scholar 

  28. M. Bao, Y.Y. Zong, Y.M. Zhao et al., Local relations of nuclear charge radii. Phys. Rev. C 102, 014306 (2020). https://doi.org/10.1103/PhysRevC.102.014306

    Article  ADS  Google Scholar 

  29. Y.G. Ma, Hypernuclei as a laboratory to test hyperon-nucleon interactions. Nucl. Sci. Tech. 34, 97 (2023). https://doi.org/10.1007/s41365-023-01248-6

    Article  Google Scholar 

  30. R. Wang, Y.G. Ma, R. Wada et al., Nuclear liquid–gas phase transition with machine learning. Phys. Rev. Res. 2, 043202 (2020). https://doi.org/10.1103/PhysRevResearch.2.043202

    Article  Google Scholar 

  31. J. Steinheimer, L. Pang, K. Zhou et al., A machine learning study to identify spinodal clumping in high energy nuclear collisions. JHEP 12, 122 (2019). https://doi.org/10.1007/JHEP12%282019%29122

  32. Y.G. Ma, Effects of \(\alpha\)-clustering structure on nuclear reaction and relativistic heavy-ion collisions. Nuclear Techniques 46(8), 080001 (2023). https://doi.org/10.11889/j.0253-3219.2023.hjs.46.080001 (in Chinese)

  33. R. Utama, J. Piekarewicz, H.B. Prosper, Nuclear mass predictions for the crustal composition of neutron stars: a Bayesian neural network approach. Phys. Rev. C 93, 014311 (2016). https://doi.org/10.1103/PhysRevC.93.014311

    Article  ADS  Google Scholar 

  34. R. Utama, J. Piekarewicz, Refining mass formulas for astrophysical applications: a Bayesian neural network approach. Phys. Rev. C 96, 044308 (2017). https://doi.org/10.1103/PhysRevC.96.044308

    Article  ADS  Google Scholar 

  35. Z.M. Niu, H.Z. Liang, B.H. Sun et al., High precision nuclear mass predictions towards a hundred kilo-electron-volt accuracy. Sci. Bull. 63, 759 (2018). https://doi.org/10.1016/j.scib.2018.05.009

    Article  Google Scholar 

  36. W.B. He, Y.G. Ma, L.G. Pang et al., High-energy nuclear physics meets machine learning. Nucl. Sci. Tech. 34, 88 (2023). https://doi.org/10.1007/s41365-023-01233-z

    Article  Google Scholar 

  37. W.B. He, Q.F. Li, Y.G. Ma et al., Machine learning in nuclear physics at low and intermediate energies. Sci. China Phys. Mech. Astron. 66, 282001 (2023). https://doi.org/10.1007/s11433-023-2116-0

    Article  ADS  Google Scholar 

  38. J.J. He, W.B. He, Y.G. Ma et al., Machine-learning-based identification for initial clustering structure in relativistic heavy-ion collisions. Phys. Rev. C 104, 044902 (2021). https://doi.org/10.1103/PhysRevC.104.044902

    Article  ADS  Google Scholar 

  39. Y.L. Cheng, S.Z. Shi, Y.G. Ma et al., Examination of nucleon distribution with Bayesian imaging for isobar collisions. Phys. Rev. C 107, 064909 (2023). https://doi.org/10.1103/PhysRevC.107.064909

    Article  ADS  Google Scholar 

  40. S. Akkoyun, T. Bayram, S.O. Kara et al., An artificial neural network application on nuclear charge radii. J. Phys. G 40, 055106 (2013). https://doi.org/10.1088/0954-3899/40/5/055106

    Article  ADS  Google Scholar 

  41. D. Wu, C.L. Bai, H. Sagawa et al., Calculation of nuclear charge radii with a trained feed-forward neural network. Phys. Rev. C 102, 054323 (2020). https://doi.org/10.1103/PhysRevC.102.054323

    Article  ADS  Google Scholar 

  42. R. Utama, W.C. Chen, J. Piekarewicz, Nuclear charge radii: density functional theory meets Bayesian neural networks. J. Phys. G 43, 114002 (2016). https://doi.org/10.1088/0954-3899/43/11/114002

    Article  ADS  Google Scholar 

  43. Z.M. Niu, H.Z. Liang, B.H. Sun et al., Predictions of nuclear \(\beta\)-decay half-lives with machine learning and their impact on r-process nucleosynthesis. Phys. Rev. C 99, 064307 (2019). https://doi.org/10.1103/PhysRevC.99.064307

    Article  ADS  Google Scholar 

  44. L. Neufcourt, Y.C. Cao, W. Nazarewicz et al., Bayesian approach to model-based extrapolation of nuclear observables. Phys. Rev. C 98, 034318 (2018). https://doi.org/10.1103/PhysRevC.98.034318

    Article  ADS  Google Scholar 

  45. X.X. Dong, A. Rong, J.X. Lu et al., Novel Bayesian neural network based approach for nuclear charge radii. Phys. Rev. C 105, 014308 (2022). https://doi.org/10.1103/PhysRevC.105.014308

    Article  ADS  Google Scholar 

  46. Z.M. Niu, Z.L. Zhu, Y.F. Niu et al., Radial basis function approach in nuclear mass predictions. Phys. Rev. C 88, 024325 (2013). https://doi.org/10.1103/PhysRevC.88.024325

    Article  ADS  Google Scholar 

  47. J.S. Zheng, N.Y. Wang, Z.Y. Wang et al., Mass predictions of the relativistic mean-field model with the radial basis function approach. Phys. Rev. C 90, 014303 (2014). https://doi.org/10.1103/PhysRevC.90.014303

    Article  ADS  Google Scholar 

  48. Z.P. Gao, Y.J. Wang, H.L. Lü et al., Machine learning the nuclear mass. Nucl. Sci. Tech. 32, 109 (2021). https://doi.org/10.1007/s41365-021-00956-1

    Article  Google Scholar 

  49. N. Wang, M. Liu, Nuclear mass predictions with a radial basis function approach. Phys. Rev. C 84, 051303(R) (2011). https://doi.org/10.1103/PhysRevC.84.051303

    Article  ADS  Google Scholar 

  50. T.S. Shang, J. Li, Z.M. Liu, Prediction of nuclear charge density distribution with feedback neural network. Nucl. Sci. Tech. 33, 153 (2022). https://doi.org/10.1007/s41365-022-01140-9

    Article  Google Scholar 

  51. X.X. Dong, A. Rong, J.X. Lu et al., Nuclear charge radii in Bayesian neural networks revisited. Phys. Lett. B 838, 137726 (2023). https://doi.org/10.1016/j.physletb.2023.137726

    Article  Google Scholar 

  52. Y.Y. Li, F. Zhang, J. Su, Improvement of the Bayesian neural network to study the photoneutron yield cross sections. Nucl. Sci. Tech. 33, 135 (2022). https://doi.org/10.1007/s41365-022-01131-w

    Article  Google Scholar 

  53. K. Mills, M. Spanner, I. Tamblyn, Deep learning and the Schrödinger equation. Phys. Rev. A 96, 042113 (2017). https://doi.org/10.1103/PhysRevA.96.042113

    Article  ADS  Google Scholar 

  54. K. Ryczko, D.A. Strubbe, I. Tamblyn, Deep learning and density-functional theory. Phys. Rev. A 100, 022512 (2019). https://doi.org/10.1103/PhysRevA.100.022512

    Article  ADS  Google Scholar 

  55. G.T. Garvey, I. Kelson, New nuclidic mass relationship. Phys. Rev. Lett. 16, 197 (1966). https://doi.org/10.1103/PhysRevLett.16.197

    Article  ADS  Google Scholar 

  56. G.T. Garvey, W.J. Gerace, R.L. Jaffe et al., Set of nuclear-mass relations and a resultant mass table. Rev. Mod. Phys. 41, S1 (1969). https://doi.org/10.1103/RevModPhys.41.S1

    Article  ADS  Google Scholar 

  57. Y. LeCun, B. Boser, J.S. Denker et al., Backpropagation applied to handwritten zip code recognition. Neural Comput. 1, 541 (1989). https://doi.org/10.1162/neco.1989.1.4.541

    Article  Google Scholar 

  58. I. Angeli, Effect of valence nucleons on RMS charge radii and surface thickness. J. Phys. G: Nucl. Part. Phys. 17, 439 (1991). https://doi.org/10.1088/0954-3899/17/4/006

    Article  ADS  Google Scholar 

  59. R. An, X.X. Dong, L.G. Cao et al., Local variations of charge radii for nuclei with even Z from 84 to 120. Commun. Theor. Phys. 75, 035301 (2023). https://doi.org/10.1088/1572-9494/acb58b

    Article  ADS  Google Scholar 

  60. G.A. Lalazissis, M.M. Sharma, P. Ring, Rare-earth nuclei: radii, isotope-shifts and deformation properties in the relativistic mean-field theory. Nucl. Phys. A 597, 35 (1996). https://doi.org/10.1016/0375-9474(95)00436-X

    Article  ADS  Google Scholar 

  61. R.F. Casten, Possible unified interpretation of heavy nuclei. Phys. Rev. Lett. 54, 1991 (1985). https://doi.org/10.1103/PhysRevLett.54.1991

    Article  ADS  Google Scholar 

  62. T. Togashi, Y. Tsunoda, T. Otsuka et al., Quantum phase transition in the shape of Zr isotopes. Phys. Rev. Lett. 117, 172502 (2016). https://doi.org/10.1103/PhysRevLett.117.172502

    Article  ADS  Google Scholar 

  63. B.A. Marsh, T.D. Goodacre, S. Sels et al., Characterization of the shape-staggering effect in mercury nuclei. Nature Phys. 14, 1163 (2018). https://doi.org/10.1038/s41567-018-0292-8

    Article  ADS  Google Scholar 

  64. S. Péru, S. Hilaire, S. Goriely et al., Description of magnetic moments within the Gogny Hartree–Fock-Bogolyubov framework: application to Hg isotopes. Phys. Rev. C 104, 024328 (2021). https://doi.org/10.1103/PhysRevC.104.024328

    Article  ADS  Google Scholar 

  65. R. An, L.S. Geng, S.S. Zhang, Novel ansatz for charge radii in density functional theories. Phys. Rev. C 102, 024307 (2020). https://doi.org/10.1103/PhysRevC.102.024307

    Article  ADS  Google Scholar 

  66. P.G. Reinhard, W. Nazarewicz, Toward a global description of nuclear charge radii: exploring the Fayans energy density functional. Phys. Rev. C 95, 064328 (2017). https://doi.org/10.1103/PhysRevC.95.064328

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai, 200433, China

    Ying-Yu Cao & Bo Zhou

  2. School of Physics and Optoelectronics Engineering, Anhui University, Hefei, 230601, China

    Jian-You Guo

  3. Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai, 200438, China

    Bo Zhou

Authors
  1. Ying-Yu Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian-You Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bo Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Ying-Yu Cao, Jian-You Guo and Bo Zhou. The first draft of the manuscript was written by Ying-Yu Cao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Bo Zhou.

Ethics declarations

Bo Zhou is an editorial board member for Nuclear Science and Techniques and was not involved in the editorial review, or the decision to publish this article.

Conflict of interests

All authors declare that there are no competing interests.

Additional information

This work was supported by Shanghai “Science and Technology Innovation Action Plan” Project (No. 21ZR140950).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, YY., Guo, JY. & Zhou, B. Predictions of nuclear charge radii based on the convolutional neural network. NUCL SCI TECH 34, 152 (2023). https://doi.org/10.1007/s41365-023-01308-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41365-023-01308-x

Keywords

Search

Navigation