Configurational information entropy analysis of fragment mass cross distributions to determine the neutron skin thickness of projectile nuclei

Configurational information entropy analysis of fragment mass cross distributions to determine the neutron skin thickness of projectile nuclei 1 corresp first-author htuwhling@126.com htuwhling@126.com htuwhling@126.com https://orcid.org/0000-0002-9067-2462 1 1 College of Physics, Henan Normal University, Xinxiang 453007, China College of Physics, Henan Normal University, Xinxiang 453007, China Configurational information entropy (CIE) analysis has been shown to be applicable for determining the neutron skin thickness ( δ np ) of neutron-rich nuclei from fragment production in projectile fragmentation reactions. The BNN + FRACS machine learning model was adopted to predict the fragment mass cross-sections ( σ A ) of the projectile fragmentation reactions induced by calcium isotopes from 36 Ca to 56 Ca on a 9 Be target at 140 MeV/u. The fast Fourier transform was adopted to decompose the possible information compositions in σ A distributions and determine the quantity of CIE ( S A [ f ]). It was found that the range of fragments significantly influences the quantity of S A [ f ], which results in different trends of S A [ f ] ~ δ np correlation. The linear S A [ f ] ~ δ np correlation in a previous study [Nucl. Sci. Tech. 33, 6 (2022)] could be reproduced using fragments with relatively large mass fragments, which verifies that S A [ f ] determined from fragment σ A is sensitive to the neutron skin thickness of neutron-rich isotopes. Neutron skin thickness Mass cross-section distribution Configurational information entropy Projectile fragmentation reaction introduction 1 In neutron-rich nuclei, valence neutrons may have a large spatial extension, causing them to form a thick skin structure or even an exotic giant neutron halo for near-drip-line nuclei [ 1 1 - 3 3 1-3 ]. Defined as the difference between the point neutron and proton root-mean-square radii ( δ np = δ n - δ p ) of a nucleus, the neutron skin thickness reflects the difference between the nuclear density distributions of neutrons and protons. The neutron skin thickness is an important parameter for constraining the nuclear symmetry energy and nuclear equation of state. In the newly opened Facility for Rare Ion Beams (FRIB), USA, and other building factories, such as the High Intensity Heavy Ion Accelerator Facility (HIAF), China, a variety of new isotopes are expected to be created with advanced technologies in beam intensity and particle identification, which makes it possible to employ unstable nuclear beams to study neutron halos or even neutron clusters [ 4 4 ]. Because of difficulties in the direct measurement of neutrons, the neutron skin thickness is always d etermined using many probes in nuclear reactions, such as the reaction cross-section or interaction cross-section, ratios of charged particles, nucleon removal cross-sections, or others sensitive to the change in neutron density distribution (see recent reviews in [ 4 4 , 5 5 ]). The projectile fragmentation (PF) reaction is a violent reaction induced by heavy ions at incident energies above a few tens of MeV/u. It is generally believed to have three processes in transport models (or two processes in some simpler models), that is, the collision, expansion, and subsequent secondary decay process, which sees a corresponding change in information entropy along different collision processes and can be reflected by fragment distributions [ 5 5 ]. The isospin effect in fragment production, that is, the neutron-rich fragments, will be enhanced in a more neutron-rich reaction system, making it useful for determining the neutron density distribution in the projectile nucleus [ 6 6 - 8 8 6-8 ] or nuclear symmetry energy [ 9 9 , 10 10 ]. In a recent study [Nucl.Sci.Tech. 33, 6 (2022)] employing a modified statistical abrasion-ablation (SAA) model to predict fragment production, configurational information entropy (CIE) analysis was adopted to study the neutron skin thickness of neutron-rich nuclei through fragment distributions in PF reactions [ 11 11 ]. It has been illustrated that the quantities of CIE determined from mass distributions or charge distributions linearly decrease with increasing neutron skin thickness of neutron-rich nuclei. Considering that the SAA model cannot adequately reproduce the mass distribution for light fragments, the CIE analysis of the fragment distribution has been limited to fragments with mass number A f ≥ 10 or charge number Z f ≥ 10. On the other hand, light fragments also have different production mechanisms than large mass fragments, which mainly account for peripheral collisions. In this study, both light and large fragments were analyzed using the CIE method, based on predictions of a newly developed machine learning model, and the fragment size dependence of CIE in PF reactions was studied. To obtain the fragment mass cross-section ( σ A ) distribution in PF reactions, a precise model prediction of fragment production is required. Among the many parameterizations used to predict fragment cross-sections, FRACS has been proven to be of good quality [ 12 12 ], which incorporates the incident energy dependence of the reaction and odd-even staggering (OES) effects in the fragments. Moreover, machine learning models, such as the Bayesian neural network (BNN), have also been suggested for constructing new models to predict fragment production in PF reactions [ 13 13 ] as well as spallation reactions [ 14 14 - 16 16 14-16 ]. It was found that, under the guidance of physical models, BNN technology can improve the quality of physical models and the simple BNN model [ 13 13 , 15 15 , 16 16 ]. The recently proposed BNN + FRACS model shows a high prediction quality for fragment cross-sections in PF reactions [ 13 13 ] based on the advantages of the BNN and FRACS models, which makes it possible to simulate both small-mass and large-mass fragments well. In this study, the BNN + FRACS model was adopted to predict the fragment σ A distributions in calcium isotope-induced PF reactions, and CIE analysis was used to determine the quantity of CIE for fragment distributions. A further study on the correlation between CIE and the neutron skin thickness of calcium isotopes was performed. These models are described in Sect. 2. In Sect. 3, the results and discussion are shown. The conclusions of this study are presented in Sect. 4. model description 1 In this section, the adopted models are described briefly. CIE analysis can be divided into two types, considering that the distribution is discrete or continuous. Because fragments produced in PF reactions are discrete particles, only the CIE analysis of the discrete distribution will be introduced in Sect. 2.1. In Sect. 2.2 and 2.3, the main characteristics of the FRACS and BNN + FRACS models will be briefly introduced, respectively. Method of determining CIE from a physical distribution 2 The quantity of CIE determines chaotic information from a distribution, which usually partially reflects that of the system [ 17 17 ]. Fragment production is considered localized as clusters in heavy-ion collisions, and this method focuses on determining the CIE of a system with spatially localized clusters. The first step starts with the Fourier transform (FT) of a set of functions describing the system <math display="block" id="M1"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>L</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>R</mi></mstyle><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><msup><mrow><mo>|</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow><mn>2</mn></msup><mtext>d</mtext><mi>x</mi><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><msup><mrow><mo>|</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow><mn>2</mn></msup><mtext>d</mtext><mi>k</mi><mn>,</mn></mtd></mtr></mtable></math> which obeys Plancherel’s theorem [ 18 18 ]. f ( x ) must be a bounded squared-integrable function. Thus, { F ( k )} denotes the frequency-domain signal sequence of the original signal sequence { X n }. From F ( k ), the fraction f ( k ) in the k mode can be defined [ 18 18 ] as <math display="block" id="M2"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mrow><mrow><mo>|</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup><msup><mtext>d</mtext><mi>d</mi></msup><mi>k</mi></mrow></mfrac><mn>,</mn></mrow></math> and the integration is over all k in which F ( k ) is defined. { f ( k )} is the normalized frequency-domain signal sequence, for which f ( k ) denotes the normalized fraction in the k mode. d denotes the number of spatial dimensions. From this definition, it is known that f ( k ) measures the relative weight of a given mode k and is always smaller than 1. The CIE is defined based on f ( k ) according to the form of Shannon information entropy [ 19 19 ], <math display="block" id="M3"><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mi>f</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi>ln</mi><mi>f</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In this manner, the quantity of CIE is composed of the information entropy for configurations compatible with certain constraints on a physical system. When each mode k is even in the CIE, f ( m )=1/ N , and the discrete configuration entropy has a maximum at S [ f ]= ln N . Under extreme conditions in which there is only one mode of k , S [ f ]=0. For a continuous distribution, the continuous CIE can be defined [ 11 11 ]. The flowchart in Fig. 1 Fig. 1 shows the method of obtaining CIE from an original signal sequence X n according to the procedures described in Sec. 2.1. The fast Fourier transform (FFT) was chosen to analyze the fragment cross-section distribution, <math display="block" id="M4"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></munderover></mstyle><msub><mi>X</mi><mi>n</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mn>2</mn><mi>π</mi><mi>k</mi><mi>n</mi></mrow></msup><mn>,</mn></mrow></math> where k = m/N , 0 ≤ m ≤ N-1, and N is the amount of data in the X n distribution. FRACS model 2 Considering the numerous physical parameters that influence fragment production in PF reactions, the FRACS model can yield good predictions for fragments in PF reactions above 100 MeV/u [ 12 12 ]. The FRACS model improves upon the well-known EPAX3 [ 20 20 ] model by including the energy dependence of fragment production as well as the OES phenomenon in fragments. For a fragment with specific mass and charge numbers ( A f , Z f ), its cross-section is predicted by <math display="block" id="M5"><mrow><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>f</mtext></msub><mn>,</mn><msub><mi>Z</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><mi>Y</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><mtext>prob</mtext></mrow></msub><mo>−</mo><msub><mi>Z</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><msub><mi>δ</mi><mrow><mtext>OES</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>f</mtext></msub><mn>,</mn><msub><mi>Z</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where Y ( A f ) refers to the fragment mass cross-section, Y ( Z prob - Z f ) describes the isobaric distribution, and δ OES ( A f , Z f denotes OES in fragment production. In this study, Y ( A f ) was focused on to extract CIE. According to FRACS, Y ( A f ) is given by the following formula: <math display="block" id="M6"><mrow><mi>Y</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>σ</mi><mtext>R</mtext></msub><mi>P</mi><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mtext>p</mtext></msub><mo>−</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where σ R is the geometrical scaling factor, and P is the slope of this exponential. The physical quantities of σ R and P are related to the masses of the projectile and target nuclei, as well as the incident energy of the reaction. The parameters for σ R and P make FRACS agree well with the experimental data. A detailed description of the FRACS model is provided in [ 12 12 ]. BNN + FRACS machine learning model 2 In a recent study, the BNN + FRACS machine learning model was proposed to predict the fragment cross-section in PF reactions. A detailed description is provided in [ 13 13 ]. In short, the BNN + FRACS model combines the advantages of the FRACS model and the strong learning ability of the BNN to big data, which improves the prediction abilities of both the BNN and FRACS models. The main characteristics of the BNN + FRACS model are as follows: Under the supervision of the FRACS model, a predictive model was constructed using BNN technology based on massive learning of the difference between the measured fragments and the FRACS prediction in various reactions, <math display="block" id="M7"><mrow><mi>Δ</mi><mo>=</mo><mi>lg</mi><msub><mi>σ</mi><mrow><mtext>exp</mtext></mrow></msub><mo>−</mo><mi>lg</mi><msub><mi>σ</mi><mrow><mtext>FRACS</mtext></mrow></msub><mn>,</mn></mrow></math> where σ exp and σ FRACS denote the measured data and FRACS prediction, respectively. After a careful comparison of different neural nodes in the hidden layer, the optimized structure of the BNN + FRACS model was found to be formed by one input layer with seven input parameters, one hidden layer with 46 neural units, and one output layer with one output parameter. The input parameters were ( E , A p , Z p , A t , Z t , A f , Z f ), where E is the incident energy in MeV/u, A and Z are the mass and charge numbers, and the sub-indexes p, t, f refer to the projectile, target, and fragment, respectively. A total of 6393 fragments from 53 measured PF reactions were included in the learning dataset, which made it possible for the BNN + FRACS model to reproduce a wide range of fragments with incident energies from 40 to 1 GeV/u and reactions induced by projectile nuclei from 40 Ar to 208 Pb. The BNN + FRACS model was selected to generate fragment mass cross-sections in PF reactions for its good reproduction of a wide range of fragments. Refer to [ 13 13 ] for a detailed description of the BNN + FRACS model. Results and discussion 1 The fragment mass cross-sections for 140 MeV/u 36-56 Ca + 9 Be reactions were predicted using the BNN + FRACS model. To establish the correlation between CIE and the neutron skin thickness, the double-parameter Fermi-type nuclear density distribution was adopted for neutrons and protons. <math display="block" id="M8"><mrow><msub><mi>ρ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>ρ</mi><mi>i</mi><mn>0</mn></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>r</mi><mo>−</mo><msub><mi>C</mi><mi>i</mi></msub></mrow><mrow><msub><mi>t</mi><mi>i</mi></msub><mo>/</mo><mn>4.4</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac><mn>,</mn><mi> </mi><mi> </mi><mi> </mi><mi>i</mi><mo>=</mo><mtext>n</mtext><mn>,</mn><mtext>p</mtext></mrow></math> where i denotes neutrons or protons, <math id="M9"><mrow><msubsup><mi>ρ</mi><mi>i</mi><mn>0</mn></msubsup></mrow></math> is the normalization constant, t i is the diffuseness parameter, and C i is half the density radius of the nuclear density distribution. The neutron skin thickness of the projectile nucleus can be calculated from the ρ n and ρ p distributions according to the following definition: <math display="block" id="M10"><mrow><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>r</mi><mtext>n</mtext><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow></mrow></msqrt><mo>−</mo><msqrt><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>r</mi><mtext>p</mtext><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow></mrow></msqrt></mrow></math> The fragment σ A distributions for a series of 140 MeV/u 36-56 Ca + 9 Be reactions are shown in Fig. 2 Fig. 2 . More results have been predicted, but only some of them were plotted for concise discussion. Because the mass number of fragments reflects the colliding area, the ratio of the fragment mass to that of the projectile nucleus A f / A p was adopted as a cut for the upper limitation when selecting the fragment range for the FFT analysis. In addition, to better discuss the mass distribution, a cut was made to A f ≤ 90Compared to the experimental results for the 140 MeV/u 40 Ca + 9 Be (open squares) and 48 Ca + 9 Be (open circles) reactions [ 21 21 ], the BNN + FRACS model effectively reproduced the measured σ A distributions, which made it possible to carry out a CIE analysis of the mass distribution for reactions induced by calcium isotopes with different isospins. Taking the 140 MeV/u 44 Ca + 9 Be reaction as an example, the FFT analysis of the σ A distribution is shown. In the analysis, the upper limitations on the fragments were set as 60% A p to 90% A p (in steps of 5% A p ) to observe the influence of fragment size, which indicates the percentage of projectiles involved in the collision and the neutron skin effect. The results are presented in Fig. 3 Fig. 3 . In general, except for the first peak, the amplitude F ( k ) decreased as the mode k increased. At approximately k =0.23, a second weak peak was found with upper limitations on the fragments of A f ≤ 70 Using Eq. (3), the CIE for fragment σ A distributions was calculated from the FFT analysis, which is labeled as S A [ f ]. The results of S A [ f ] and their correlation with δ np for the 140 MeV/u <math id="M12"><mrow><msup><mrow/><mrow><msub><mi>A</mi><mtext>p</mtext></msub></mrow></msup></mrow></math> Ca + 9 Be reactions are plotted in Fig. 4 Fig. 4 , in which A p ranges from 36 to 56, and the upper limitations on fragments masses are changed from 60% A p to 90% A p . δ np of the projectile nucleus was calculated using the nuclear density distribution according to Eq. (8). Around 43,44 Ca, δ np ~0, indicating proton skins in 36-42 Ca, and neutron skins in 45-60 Ca. The quantity of S A [ f ] determined from the fragment σ A distribution was not well correlated with δ np . In general, with different upper limitations on fragment mass, S A [ f ] was mainly influenced by the relatively neutron-deficient 36-43 Ca reactions, showing an increase with the upper limitation on the fragment mass. It was also observed that S A [ f ] gradually increased with δ np for 36-42 Ca, whereas it rapidly increased with δ np for 45-56 Ca. Note that with a fragment upper limitation of 80% A p , S A [ f ] decreased with increasing δ np when δ np < 0.03 fm, whereas the opposite trend occurred when δ np > 0.03 fm. More clearly, with the upper limitations of A f ≤ 85% A p and A f ≤ 90% A p , S A [ f ] decreased with δ np when δ np < 0.02 and then increased with δ np in the more neutron-rich nuclei-induced reactions. Remembering the S A [ f ]∼ δ np correlation predicted by the modified SAA model in [ 11 11 ], S A [ f ] was found to linearly decrease with increasing δ np of neutron-rich calcium isotopes from 40 Ca to 60 Ca. The results do not seem to agree with previous findings. While exploring how this disagreement could have emerged, it was found that the mass distributions adopted in [ 11 11 ] only included fragments with A f ≥ 10, whereas in this study, A f was extended to fragments as light as A f = 2. The inclusion of light fragments may be the reason for this disagreement. Considering this difference, the CIE analysis was re-performed by dividing the fragments into two groups for the case of A f ≤ 90% A p , that is, A f ≤ 14 and 15 ≤ Af ≤ 90%Ap. The determined S A [ f ] and its correlation with δ np of the projectile nuclei are plotted in Fig. 5 Fig. 5 . Within the range A f ≤ 14, S A [ f ] decreased very slowly when δ np & lt; 0 and increased with δ np for the more neutronrich projectiles with thicker neutron skins. An opposite trend is found for the fragments within 15 ≤ A f ≤ 90% A p, that is, for δ np & lt; 0, S A [ f ] increased with δ np and decreased with increasing δ np for projectiles with thicker neutron skins. For fragments within the range A f ≤ 90% A p , the trend of the S A [ f ] distribution was similar to that of A f ≤ 14, indicating that S A [ f ] was significantly influenced by the selected range of fragments. Compared with the S A [ f ] ~ δ np correlation re ported in [ 11 11 ], the results for 15 ≤ A f ≤ 90% A p are in good agreement, in which S A [ f ] decreased linearly with increasing δ np of the neutron-rich projectile nucleus. It is suggested that the adopted range of fragments significantly influences S A [ f ] as well as the S A [ f ] ~ δ np correlation. If the fragments are limited to relatively small sizes, the obtained CIE may be insensitive to the neutron skin thickness if it is not very thick. Large fragments, which are produced in peripheral collisions and are influenced by neutron skin structures, better reflect information about neutron skin thickness. Conclusion 1 CIE analysis was adopted to quantify the CIE incorporated in the fragment σ A distributions of PF reactions. The newly proposed BNN + FRACS model was used to predict the fragment σ A distributions in the 140 MeV/u 36-56 Ca + 9 Be reactions. The neutron skin thickness of the projectile nuclei was calculated using Fermi-type nuclear density distributions for protons and neutrons. By performing an FFT analysis of fragment σ A distributions, F ( k ) distributions were obtained and further used to determine the CIE included in the fragment distributions. The influence of fragment size on the CIE was investigated by setting different upper limitations on the fragment mass according to different percentages of A f / A p . It is concluded that the range of fragments influences the quantity of CIE and exhibits different correlations with the neutron skin thickness of the projectile nucleus, which are as follows: • The CIE S A [ f ] was not sensitive to the neutron skin thickness if the upper limitations on the fragments were relatively small, for example, A f ≤ 75% A p and less, for which the roots in the neutron skin were mainly influenced by the diffuseness of the nucleus. • When relatively large mass fragments were included to determine S A [ f ], for neutron-deficient projectile nuclei, S A [ f ] exhibited a decreasing trend with increasing δ np , whereas it increased with δ np for neutron-rich projectile nuclei. • By dividing the fragments into two groups according to their mass numbers, that is, the relatively light fragments of A f ≤ 14 and the relatively large fragments of 15 ≤ A f ≤ 90% A p S A [ f ] was found to have differenttypes of correlations with δ np of the projectile nucleus. • The CIE determined from fragments of 15 ≤ A f ≤ 90% A p reproduced the trend of a previous correlation obtained in [ 11 11 ], showing a good linear S A [ i ]~ δ np dnp correlation in the neutron-rich projectile nuclei of calcium isotopes. 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DOI: 10.1103/PhysRevC.74.054612 http://doi.org/10.1103/PhysRevC.74.054612 Projectile fragmentation of 40Ca, 48Ca, 58Ni, and 64Ni at 140 MeV/nucleon All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Xu Zhu, Chen Yuan and Hui-Ling Wei. The first draft of the manuscript was written by Hui-Ling Wei and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. We thank Prof. Chun-Wang Ma for his suggestion to study this topic. 10.1007/s41365-022-01096-w This work was supported by the National Natural Science Foundation of China (No. 11975091) and the Program for Innovative Research Team (in Science and Technology) in the University of Henan Province, China (No. 21IRTSTHN011). 2022-07-29 2022-08-20 2022-08-27 2022-09-13 2022-10-24 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01096-w http://dx.doi.org/10.1007/s41365-022-01096-w . Springer Nature or its licensor 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. 2022 Nuclear Science and Techniques 09 33