# Progress of Machine Learning Studies on the Nuclear Charge Radii

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Machine Learning for Nuclear Charge Radii

## 3. CNN Method

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Shaginyan, V.R. Coulomb Energy of Nuclei. Phys. At. Nucl.
**2001**, 64, 471–476. [Google Scholar] [CrossRef] - Mayer, M.G. On Closed Shells in Nuclei. Phys. Rev.
**1948**, 74, 235–239. [Google Scholar] [CrossRef] - Haxel, O.; Jensen, J.H.D.; Suess, H.E. On the magic numbers in nuclear structure. Phys. Rev.
**1949**, 75, 1766. [Google Scholar] [CrossRef] - Brown, B.A. Mirror charge radii and the neutron equation of state. Phys. Rev. Lett.
**2017**, 119, 122502. [Google Scholar] [CrossRef] [PubMed] - Yang, J.J.; Piekarewicz, J. Difference in proton radii of mirror nuclei as a possible surrogate for the neutron skin. Phys. Rev. C
**2018**, 97, 014314. [Google Scholar] [CrossRef] - Sammarruca, F. Proton skins, Neutron skins and proton radii of mirror nuclei. Front. Phys.
**2018**, 6, 90. [Google Scholar] [CrossRef] - Vries, H.D.; Jager, C.; Vries, C.D. Nuclear charge-density-distribution parameters from elastic electron scattering. At. Data Nucl. Data Tables
**1987**, 36, 495–536. [Google Scholar] [CrossRef] - Fricke, G.; Bernhardt, C.; Heilig, K.; Schaller, L.A.; Schellenberg, L.; Shera, E.B.; Dejager, C.W. Nuclear ground state charge radii from electromagnetic interactions. At. Data Nucl. Data Tables
**1995**, 60, 177–285. [Google Scholar] [CrossRef] - Lee, F. Changes of mean-square nuclear charge radii from isotope shifts of electronic K
_{α}X-rays. At. Data Nucl. Data Tables**1974**, 14, 605–611. [Google Scholar] - Tran, D.T.; Ong, H.J.; Nguyen, T.T.; Tanihata, I.; Aoi, N.; Ayyad, Y.; Chan, P.Y.; Fukuda, M.; Hashimoto, T.; Hoang, T.H.; et al. Charge-changing-cross-section measurements of
^{12–16}C at around 45A MeV and development of a Glauber model for incident energies 10–2100A MeV. Phys. Rev. C**2016**, 94, 064604. [Google Scholar] [CrossRef] - Kanungo, R.; Horiuchi, W.; Hagen, G.; Jansen, G.R.; Navratil, P.; Ameil, F.; Atkinson, J.; Ayyad, Y.; Cortina-Gil, D.; Dillmann, I.; et al. Proton distribution radii of
^{12–19}C illuminate features of neutron halos. Phys. Rev. Lett.**2016**, 117, 102501. [Google Scholar] [CrossRef] [PubMed] - Li, T.; Luo, Y.N.; Wang, N. Compilation of recent nuclear ground state charge radius measurements and tests for models. At. Data Nucl. Data Tables
**2021**, 140, 101440. [Google Scholar] [CrossRef] - Sheng, Z.Q.; Fan, G.W.; Qian, J.F.; Hu, J.G. An effective formula for nuclear charge radii. Eur. Phys. J. A
**2015**, 51, 40. [Google Scholar] [CrossRef] - Brown, B.A.; Bronk, C.; Hodgson, P.E. Systematics of Nuclear RMS Charge Radii. J. Phys. G Nucl. Phys.
**1984**, 10, 1683–1701. [Google Scholar] [CrossRef] - Nerlo-Pomorska, B.; Pomorski, K. A simple formula for nuclear charge radius. Z. Phys. A
**1994**, 384, 169–172. [Google Scholar] [CrossRef] - Angeli, I.; Marinova, K.P. Table of experimental nuclear ground state charge radii: An update. At. Data Nucl. Data Tables
**2013**, 99, 69–95. [Google Scholar] [CrossRef] - Casten, R.F.; Brenner, D.S.; Haustein, P.E. Valence p-n interactions and the development of collectivity in heavy nuclei. Phys. Rev. Lett.
**1987**, 58, 658. [Google Scholar] [CrossRef] - Virender, T.; Shashi, K.D. A study of charge radii and neutron skin thickness near nuclear drip line. Nucl. Phys. A
**2019**, 992, 121623. [Google Scholar] - Warda, M.; Nerlo-Pomorska, B.; Pomorski, K. Isospin Dependence of Proton and Neutron Radii within Relativistic Mean Field Theory. Nucl. Phys. A
**1998**, 635, 484–494. [Google Scholar] [CrossRef] - Wang, J.S.; Shen, W.Q.; Zhu, Z.Y.; Feng, J.; Guo, Z.Y.; Zhan, W.L.; Xiao, G.Q.; Cai, X.Z.; Fang, D.Q.; Zhang, H.Y.; et al. RMF calculation and phenomenological formulas for the rms radii of light nuclei. Nucl. Phys. A
**2001**, 691, 618–630. [Google Scholar] [CrossRef] - Boehnlein, A.; Diefenthaler, M.; Sato, N.; Schram, M.; Ziegler, V.; Fanelli, C.; Hjorth-Jensen, M.; Horn, T.; Kuchera, M.P.; Lee, D.; et al. Colloquium: Machine learning in nuclear physics. Rev. Mod. Phys.
**2022**, 94, 031003. [Google Scholar] [CrossRef] - Bedaque, P.; Boehnlein, A.; Cromaz, M.; Diefenthaler, M.; Elouadrhiri, L.; Horn, T.; Kuchera, M.; Lawrence, D.; Lee, D.; Lidia, S.; et al. AI for nuclear physics. Eur. Phys. J. A
**2021**, 57, 100. [Google Scholar] [CrossRef] - Schwartz, M.D. Modern Machine Learning and Particle Physics. Harv. Data Sci. Rev.
**2021**, 3, 2. [Google Scholar] [CrossRef] - Akkoyun, S.; Bayram, T.; Kara, S.O.; Sinan, A. An artificial neural network application on nuclear charge radii. J. Phys. G Nucl. Part. Phys.
**2013**, 40, 055106. [Google Scholar] [CrossRef] - Utama, R.; Chen, W.C.; Piekarewicz, J. Nuclear charge radii: Density functional theory meets Bayesian neural networks. J. Phys. G Nucl. Part. Phys.
**2016**, 43, 114002. [Google Scholar] [CrossRef] - Wu, D.; Bai, C.L.; Sagawa, H.; Zhang, H.Q. Calculation of nuclear charge radii with a trained feed-forward neural network. Phys. Rev. C
**2020**, 102, 054323. [Google Scholar] [CrossRef] - Ma, Y.F.; Su, C.; Liu, J.; Ren, Z.Z.; Xu, C.; Gao, Y.H. Predictions of nuclear charge radii and physical interpretations based on the naive Bayesian probability classifier. Phys. Rev. C
**2020**, 101, 014304. [Google Scholar] [CrossRef] - Dong, X.X.; An, R.; Lu, J.X.; Geng, L.S. Novel Bayesian neural network based approach for nuclear charge radii. Phys. Rev. C
**2022**, 105, 014308. [Google Scholar] [CrossRef] - Ma, J.Q.; Zhang, Z.H. Improved phenomenological nuclear charge radius formulae with kernel ridge regression. Chin. Phys. C
**2022**, 46, 074105. [Google Scholar] [CrossRef] - Shang, T.S.; Li, J.; Niu, Z.M. Prediction of nuclear charge density distribution with feedback neural network. Nucl. Sci. Tech.
**2022**, 33, 153. [Google Scholar] [CrossRef] - Dong, X.X.; An, R.; Lu, J.X.; Geng, L.S. Nuclear charge radii in Bayesian neural networks revisited. Phys. Lett. B
**2023**, 838, 137726. [Google Scholar] [CrossRef] - Wang, Y.; Zhang, Q.H.; Yao, Q.X.; Huo, Y.G.; Zhou, M.; Lu, Y.F. Multiple radionuclide identification using deep learning with channel attention module and visual explanation. Front. Phys.
**2022**, 10, 1036557. [Google Scholar] [CrossRef] - Niu, Z.M.; Liang, H.Z. Nuclear mass predictions based on Bayesian neutral network approach. Nucl. Phys. Lett. B
**2018**, 778, 48–53. [Google Scholar] [CrossRef] - Wu, X.H.; Guo, L.H.; Zhao, P.W. Nuclear masses in extended kernel ridge regression with odd-even effects. Phys. Lett. B
**2021**, 819, 136387. [Google Scholar] [CrossRef] - Saxena, G.; Sharma, P.K.; Saxena, P. Modified empirical formulas and machine learning for α-decay systematics. J. Phys. G Nucl. Part. Phys.
**2021**, 48, 055103. [Google Scholar] [CrossRef] - Wang, Z.A.; Pei, J.C.; Liu, Y.; Qiang, Y. Bayesian Evaluation of incomplete fission yields. Phys. Rev. Lett.
**2019**, 123, 122501. [Google Scholar] [CrossRef] [PubMed] - Michael, N. Neural Networks and Deep Learning; Determination Press: San Francisco, CA, USA, 2015. [Google Scholar]
- Kevin, P.M. Probabilistic Machine Learning: An Introduction; The MIT Press: Cambridge, MA, USA; London, UK, 2022; pp. 463–497. [Google Scholar]
- Angeli, I. A consistent set of nuclear rms charge radii: Properties of the radius surface R(N,Z). At. Data Nucl. Data Tables
**2004**, 87, 185–206. [Google Scholar] [CrossRef] - Day Goodacre, T.; Afanasjev, A.V.; Barzakh, A.E.; Marsh, B.A.; Sels, S.; Ring, P.; Nakada, H.; Andreyev, A.N.; Van Duppen, P.; Althubiti, N.A.; et al. Laser Spectroscopy of Neutron-Rich
^{207,208}Hg Isotopes: Illuminating the Kink and Odd-Even Staggering in Charge Radii across the N = 126 Shell Closure. Phys. Rev. Lett.**2021**, 126, 032502. [Google Scholar] [CrossRef] - Dong, C.; Loy, C.C.; He, K.; Tang, X.O. Image Super-Resolution Using Deep Convolutional Networks. IEEE T-PAMI
**2016**, 38, 295–307. [Google Scholar] [CrossRef] - Wang, M.; Huang, W.J.; Kondev, F.G.; Audi, G.; Naimi, S. The AME2020 atomic mass evaluation (II). Tables, graphs and references. Chin. Phys. C
**2021**, 45, 030003. [Google Scholar] [CrossRef] - Wang, N.; Li, T. Shell and isospin effects in nuclear charge radii. Phys. Rev. C
**2013**, 88, 011301. [Google Scholar] [CrossRef] - An, R.; Jiang, X.; Cao, L.G.; Zhang, F.S. Odd-even staggering and shell effects of charge radii for nuclei with even Z from 36 to 38 and from 52 to 62. Phys. Rev. C
**2022**, 105, 014325. [Google Scholar] [CrossRef] - Cejnar, P.; Jolie, J.; Casten, R.F. Quantum phase transitions in the shapes of atomic nuclei. Rev. Mod. Phys.
**2010**, 82, 2155. [Google Scholar] [CrossRef] - Heyde, K.; Wood, J.L. Shape coexistence in atomic nuclei. Rev. Mod. Phys.
**2011**, 83, 1467. [Google Scholar] [CrossRef] - Silverans, R.E.; Lievens, P.; Vermeeren, L.; Arnold, E.; Neu, W.; Neugart, R.; Wendt, K.; Buchinger, F.; Ramsay, E.B.; Ulm, G. Nuclear Charge Radii of
^{70–100}Sr by Nonoptical Detection in Fast-Beam Laser Spectroscopy. Phys. Rev. Lett.**1988**, 60, 2607–2610. [Google Scholar] [CrossRef] - Rodriguez-Guzman, R.; Sarriguren, P.; Robledo, L.M.; Perez-Martin, S. Charge radii and structural evolution in Sr, Zr and Mo isotopes. Phys. Lett. B
**2010**, 691, 202–207. [Google Scholar] [CrossRef]

**Figure 1.**An example of a convolutional layer with the 2 × 3 × 3 × 3 kernel. The notation * indicates the convolution operation. 3 channels in the input layer are mapped to 2 channels in the hidden layer with the stride length of 1 pixel.

**Figure 2.**The structure of the constructed CNNs. C1 consists of four convolutional layers with one residual block, while C2 is a simple network with three layers.

**Figure 3.**The matrix layout of nuclear isotopes with 102 rows and 158 columns. The 102 × 158 matrix is filled for each of six physical quantities ($Z,N,{R}_{c}^{rms},{B}_{av},I,P$), so the 6 × 102 × 158 numerical matrix can be obtained. For each nucleus, only the 13 × 13 square matrix framed by the green dashed line is exactly as the input of CNNs in practical calculations. The heaviest one in the collected experimental data is ${}^{248}$Cm ($Z=96$). When this region is centered on it, 102 × 158 matrices are required.

**Figure 4.**The example of the inputting of CNNs. Input 1 consists of the ${R}_{c}^{rms}$ channel without central data and the filled ($Z,N,{B}_{av},I,P$) channels, while Input 2 composes of the same ${R}_{c}^{rms}$ channel and ($Z,N,{B}_{av},I,P$) channels where only central data are remained.

**Figure 5.**The deviations between the experimental charge radii (${R}_{\mathrm{exp}}$) and the results of C2 as Input 2 are inputted (${R}_{\mathrm{C}2}$). The left panels are for the training sets, while the results about test sets are shown in the right ones. Additionally, the labels 1 and 2 correspond to the groups of data division 1 and 2 in Table 4.

**Figure 6.**The differences between the experimental data and the results of C2 with Input 2 fed. The data group is from the random division 2 in Table 4.

**Figure 7.**The comparison of C2 predictions with the experimental charge radii for isotones with $N=$ 64, 118 and Sr, Ba isotopes.

**Figure 8.**The differences between predicted results and experimental values of the nuclear charge radii of Ca, Zn, Zr and Pb isotopic chains.

**Table 1.**The comparisons of different ML models for charge radii. The number of nuclei involved in the corresponding model is marked as (Count). The RMS deviation of the interpolation and extrapolation for data sets are signed as ${\sigma}_{in}$ and ${\sigma}_{out}$, respectively, and $\Delta R={R}_{c}^{exp}-{R}_{c}^{th}$, which is the residuals between the experimental data and the calculated values by theoretical models or phenomenological formulae.

Reference | ML Method | Data Range (Count) | Input | Output | ${\mathit{\sigma}}_{\mathit{i}\mathit{n}}$ (fm) | ${\mathit{\sigma}}_{\mathit{o}\mathit{u}\mathit{t}}$ (fm) | ||||
---|---|---|---|---|---|---|---|---|---|---|

Train | Test | Entire | Train | Test | Entire | |||||

Ref. [24] | ANN | $A\ge 6$ (900) | Z, N | ${R}_{c}^{rms}$ | 0.036 | 0.025 | ||||

Ref. [25] * | BNN | $Z\ge 20,A\ge 40$ (820) | Z, A | $\Delta R$ | 0.0171 | 0.0163 | 0.0169 | 0.0210 | 0.0262 | 0.0217 |

Ref. [26] | ANN | (347) | Z, N, $g\left(B\right(E2),\delta )$ | ${R}_{c}^{rms}$ | 0.0266 | 0.0231 | ||||

Ref. [27] * | NBPc | $A\ge 3$ (896) | Z, N | $\Delta R$ | 0.0195 | 0.0200 | 0.0196 | 0.0195 | ||

Ref. [28] | BNN | $Z\ge 20,A\ge 40$ (933) | Z, A, $\delta $, P | $\Delta R$ | 0.0143 | 0.0187 | 0.0149 | |||

Ref. [29] * | KRR | $Z\ge 8,N\ge 8$ (884) | Z, N | $\Delta R$ | 0.0123 | 0.0268 | 0.0168 | |||

Ref. [30] | FNN | (370) | Z, N, ${Z}^{1/3}$, ${A}^{1/3}$ | c, z | 0.0769 | |||||

Ref. [31] | BNN | $Z\ge 20,A\ge 40$ (933) | Z, A, $\delta $, P, ${I}^{2}$, $LI$ | $\Delta R$ | 0.0140 | 0.0139 |

**Table 2.**The extrapolating standard deviations $\sigma $ (fm) obtained from different neural network models using nuclei beyond ${}^{40}$Ca as input data. The 722 nuclei beyond ${}^{40}$Ca in the 2004 compilation [39] are chosen as the train set, and the remnant 98 nuclei with $Z\ge 20,A\ge 40$ in the 2013 compilation [16] are divided into the test set. The results of BNN are from Ref. [25].

Method | Data | $\mathit{\sigma}$ (fm) | |
---|---|---|---|

Train (Count) | Test (Count) | ||

BNN [25] | $Z\ge 20,A\ge 40$ | 0.0210 (722) | 0.0262 (98) |

C1 | Input 1 | 0.0113 (722) | 0.0145 (98) |

Input 2 | 0.0114 (722) | 0.0160 (98) | |

C2 | Input 1 | 0.0148 (722) | 0.0145 (98) |

Input 2 | 0.0118 (722) | 0.0180 (98) |

**Table 3.**The extrapolating standard deviations $\sigma $ (fm) obtained from different machine learning models as nuclei from the more global mass number regions are inputted. The data of the NBP method are from Ref. [27].

Method | Data | $\mathit{\sigma}$ (fm) | |
---|---|---|---|

Train (Count) | Test (Count) | ||

NBP [27] | A > 3 | 0.0200 (787) | 0.0196 (82) |

C1 | A > 8 (input 1) | 0.0133 (786) | 0.0161 (109) |

A > 8 (input 2) | 0.0133 (786) | 0.0178 (109) | |

C2 | A > 8 (input 1) | 0.0114 (786) | 0.0149 (109) |

A > 8 (input 2) | 0.0115 (786) | 0.0157 (109) |

**Table 4.**The interpolating standard deviations $\sigma $ (fm) obtained from the C1 and C2 model both with the two inputting forms. Overall, 1027 nuclei with $A>8$ from Refs. [12,16] are chosen as the entire set. Then, 80% of the data are randomly selected from it five times as 5 training sets, and the corresponding remaining data are divided into test sets.

Model | Data Set | 1 | 2 | 3 | 4 | 5 | Average |
---|---|---|---|---|---|---|---|

C1 Input 1 | ${\sigma}_{\mathrm{train}}$ | 0.0141 | 0.0124 | 0.0111 | 0.0130 | 0.0126 | 0.0126 |

${\sigma}_{\mathrm{test}}$ | 0.0206 | 0.0144 | 0.0167 | 0.0211 | 0.0163 | 0.0178 | |

C1 Input 2 | ${\sigma}_{\mathrm{train}}$ | 0.0132 | 0.0128 | 0.0130 | 0.0135 | 0.0138 | 0.0133 |

${\sigma}_{\mathrm{test}}$ | 0.0181 | 0.0140 | 0.0140 | 0.0165 | 0.0166 | 0.0158 | |

C2 Input 1 | ${\sigma}_{\mathrm{train}}$ | 0.0149 | 0.0115 | 0.0125 | 0.0124 | 0.0128 | 0.0128 |

${\sigma}_{\mathrm{test}}$ | 0.0204 | 0.0132 | 0.0183 | 0.0190 | 0.0175 | 0.0177 | |

C2 Input 2 | ${\sigma}_{\mathrm{train}}$ | 0.0127 | 0.0101 | 0.0110 | 0.0112 | 0.0128 | 0.0116 |

${\sigma}_{\mathrm{test}}$ | 0.0178 | 0.0132 | 0.0138 | 0.0138 | 0.0179 | 0.0153 |

**Table 5.**The summary of the CNN method in this work for charge radii. As in Table 1, the RMS deviation of the interpolation and extrapolation for data sets are signed as ${\sigma}_{in}$ and ${\sigma}_{out}$, respectively. The RMS charge radii of surrounding nuclei are input in this work, which is signed as ${R}_{c}^{o}$.

ML Method | Data Range | Input | Output | ${\mathit{\sigma}}_{\mathit{i}\mathit{n}}$ (fm) | ${\mathit{\sigma}}_{\mathit{o}\mathit{u}\mathit{t}}$ (fm) | |||||
---|---|---|---|---|---|---|---|---|---|---|

Train | Test | Entire | Train | Test | Entire | |||||

this work | CNN | $A\ge 8$ | $Z,N$, ${R}_{c}^{o}$, ${B}_{av},I,P$ | ${R}_{c}^{rms}$ | 0.0116 | 0.0153 | 0.0114 | 0.0149 | 0.0118 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Su, P.; He, W.-B.; Fang, D.-Q.
Progress of Machine Learning Studies on the Nuclear Charge Radii. *Symmetry* **2023**, *15*, 1040.
https://doi.org/10.3390/sym15051040

**AMA Style**

Su P, He W-B, Fang D-Q.
Progress of Machine Learning Studies on the Nuclear Charge Radii. *Symmetry*. 2023; 15(5):1040.
https://doi.org/10.3390/sym15051040

**Chicago/Turabian Style**

Su, Ping, Wan-Bing He, and De-Qing Fang.
2023. "Progress of Machine Learning Studies on the Nuclear Charge Radii" *Symmetry* 15, no. 5: 1040.
https://doi.org/10.3390/sym15051040