II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Nuclear charge radii with a trained feed-forward neural network†
D. Wu,∗1 C. L. Bai,∗1 H. Sagawa,∗2,∗3 and H. Q. Zhang∗4
During the last few decades, theoretical and experimental studies of the isotopic changes of ground- andlow-lying states have been performed intensively to elucidate the evolution of the shell structure, shape coexistence phenomena, and shape transitions. In particular, charge radii and electromagnetic moments are verysensitive quantities from which precise information of
the nuclear structure can be extracted.
Machine learning (ML) is one of the most popular algorithms for dealing with complex systems owing to its powerful and convenient inference abilities.
The neural network, which is an algorithm of machinelearning, has been widely used in diﬀerent ﬁelds: artiﬁcial intelligence (AI), medical treatment, and physics
of complex systems. In this work, we attempt to train
a ML model for a description of the nuclear charge
radii based directly on some experimental or quasiexperimental data, such as proton and neutron numbers, shell eﬀect, and deformation. We search for any
other physical quantities that are correlated to thecharge radii.
To this end, we employ a standard fully connected
feed-forward neural network (FNN), which can build acomplex mapping between the input space and output
space through multiple compounding of simple nonlinear functions. The FNN is a multilayer neural network with an input layer, hidden layers, and an output
layer. The structure of the neural network is labelled
as [N1 , N2 , . . ., Nn ], where Ni denotes neuron numbers of the ith layer and i = 1 and n represent theinput and output layer, respectively. In this study, weadopt the input layer N1 = 3 and the output layer
Nn = 1. The model is trained with the input data setof proton number Z, neutron number N , the excitation energy of the ﬁrst 2+ state E2+ , and the symmetry
1energy. The deformation and shell eﬀects on charge radius are included by the E2+ values. We adopt as the1data set all the nuclei for which the experimental values of both E2+ , and charge radii are available. The
1data set includes 347 nuclei in total. As the data set
is not large enough, we must choose a small network
structure, which includes 44 neurons and involves 201
parameters.
In the present ML study, all the charge radii of Ca,
Sm, and Pb isotopes are included in the testing set
to check the prediction power of models. The model
reproduces well not only the slope of isotopic depen†
∗1
∗2
∗3
∗4

Fig. 1. Charge radii of Ca isotopes calculated by the modeltrained with and without taking the symmetry energy
input into account. The ML results obtained with andwithout the symmetry energy input are labelled by thered stars and green diamonds, respectively. The experimental data taken from Refs. 1), 2) are labelled by thefilled black circles.

dence, but also the kink of charge radii at the magic
numbers N = 82 of Sm isotopes and N = 126 of Pb
isotopes. The obtained charge radii of Ca isotopes
with and without symmetry energy input are shownin Fig. 1. Experimental results taken from Refs. 1),2) show a sharp kink structure at N = 28 and a peak
between two closed shells at N = 20 and 28, followed
by a rapid increase after N = 28. This ﬁgure indicatesthat the symmetry energy input is critical for the qualitative and quantitative description of the Ca isotopes.
We also perform Hartree-Fock-Bogolyubov (HFB)
calculations for the radii of Ca isotopes. The HFB
calculations show a clear correlation between the symmetry energy and charge radii in Ca isotopes as far as
the absolute magnitude is concerned. Whereas the MLshows that the symmetry energy input has a remarkable eﬀect on the precise descriptions of charge radii ofCa isotopes including the kink structure, the HFB calculation shows that the symmetry energy changes theabsolute magnitude of charge radii, but the kink structure at N = 28 is not well reproduced. The physicalimplication of the present successful ML study remains
an open question and needs to be studied in the future.
References

Condensed from the article in Phys. Rev. C. 102, 054323(2020)
College of Physics, Sichuan University
RIKEN Nishina Center
Center for Mathematics and Physics, University of AizuChina Institute of Atomic Energy

1) R. F. Garcia Ruiz et al., Nat. Phys. 12, 594 (2016).
2) I. Angeli, K. P. Marinova, At. Data Nucl. Data Tables 99, 69 (2013).

- 42 -1