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

Hui-Ling Wei• Xun Zhu • Chen Yuan

Abstract Configurational information entropy (CIE)analysis has been shown to be applicable for determining the neutron skin thickness (δnp) 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(σA) of the projectile fragmentation reactions induced by calcium isotopes from 36Ca to 56Ca on a 9Be target at 140 MeV/u. The fast Fourier transform was adopted to decompose the possible information compositions in σA distributions and determine the quantity of CIE (SA[f]). It was found that the range of fragments significantly influences the quantity of SA[f],which results in different trends of SA[f]～δnp correlation. The linear SA[f]～δnp correlation in a previous study [Nucl. Sci. Tech. 33, 6 (2022)]could be reproduced using fragments with relatively large mass fragments, which verifies that SA[f] determined from fragment σA is sensitive to the neutron skin thickness of neutron-rich isotopes.

Keywords Neutron skin thickness ∙Mass cross-section distribution ∙Configurational information entropy ∙Projectile fragmentation reaction

1 Introduction

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].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[6–8]or nuclear symmetry energy[9,10].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]. 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 Af ≥10 or charge number Zf ≥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 (σA) 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],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] as well as spallation reactions [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, 15, 16]. The recently proposed BNN + FRACS model shows a high prediction quality for fragment cross-sections in PF reactions[13]based on the advantages of the BNN and FRACS models, which makes it possible to simulate both smallmass and large-mass fragments well.

In this study,the BNN+FRACS model was adopted to predict the fragment σAdistributions in calcium isotopeinduced 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.

2 Model description

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.

2.1 Method of determining CIE from a physical distribution

The quantity of CIE determines chaotic information from a distribution, which usually partially reflects that of the system [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 f(x)∊L2(R),

which obeys Plancherel’s theorem [18]. f(x) must be a bounded squared-integrable function. Thus, {F(k)}denotes the frequency-domain signal sequence of the original signal sequence {Xn}. From F(k), the fraction f(k)in the k mode can be defined [18] as

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],

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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]. The flowchart in Fig. 1 shows the method of obtaining CIE from an original signal sequence Xnaccording to the procedures described in Sec. 2.1.

The fast Fourier transform(FFT)was chosen to analyze the fragment cross-section distribution,

where k=m/N, 0 ≤m ≤N-1, and N is the amount of data in the Xndistribution.

2.2 FRACS model

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]. The FRACS model improves upon the well-known EPAX3 [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 (Af,Zf), its crosssection is predicted by

Fig. 1 (Color online) Flow chart of CIE extraction from the original signal sequence {Xn} of a physical distribution. The main procedure includes the FFT of the original signal sequence, obtaining the frequency-domain signal sequence {F(k)}, the normalization to the frequency-domain signal sequence for {f(k)}, and the calculation of configurational information entropy (S[f]) using the Shannon information entropy formula

where σR is the geometrical scaling factor, and P is the slope of this exponential.The physical quantities of σR 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 σR and P make FRACS agree well with the experimental data. A detailed description of the FRACS model is provided in [12].

2.3 BNN + FRACS machine learning model

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].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,

where σexp and σFRACS 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,Ap,Zp,At,Zt,Af,Zf),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 from40Ar to208Pb.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]for a detailed description of the BNN + FRACS model.

3 Results and discussion

The fragment mass cross-sections for 140 MeV/u36-56Ca +9Be 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.

where i denotes neutrons or protons,ρ0iis the normalization constant, tiis the diffuseness parameter, and Ciis half the density radius of the nuclear density distribution. The neutron skin thickness of the projectile nucleus can be calculated from the ρn and ρp distributions according to the following definition:

The fragment σAdistributions for a series of 140 MeV/u36-56Ca +9Be reactions are shown in 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 Af/Ap 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 Af ≤90%Ap to avoid imprecise fragment predictions in peripheral collisions by the BNN + FRACS model. After the fragment cuts, the{σA}sequence served as data group{Xn}in Eq.(4),which was performed using the FFT analysis. From the reaction of36Ca to48Ca, the σAdistributions generally exhibited similar trends,which can be divided into two parts:the first for the light mass fragment, where σAdecreased rapidly within the range Af ≤～14, and the second, where σAincreased with Af within the range ～14 ≤A ≤90%Ap.There was an additional part in the σAdistribution of the52Ca-induced reaction and in the reactions of projectile nuclei larger than52Ca. Compared to the experimental results for the 140 MeV/u40Ca +9Be (open squares) and48Ca +9Be (open circles) reactions [21], the BNN +FRACS model effectively reproduced the measured σAdistributions, 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/u44Ca +9Be reaction as an example, the FFT analysis of the σAdistribution is shown.In the analysis,the upper limitations on the fragments were set as 60%Ap to 90%Ap (in steps of 5%Ap)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. 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 Af ≤70%Ap. With an increase in the upper mass limitation, the distribution of the FFT spectrum gradually decreased.

Using Eq.(3),the CIE for fragment σAdistributions was calculated from the FFT analysis,which is labeled as SA[f].The results of SA[f] and their correlation with δnp for the 140 MeV/uApCa +9Be reactions are plotted in Fig. 4, in which Ap ranges from 36 to 56, and the upper limitations on fragments masses are changed from 60%Ap to 90%Ap.δnp of the projectile nucleus was calculated using the nuclear density distribution according to Eq. (8). Around43,44Ca, δnp ～0, indicating proton skins in36-42Ca, andneutron skins in45-60Ca. The quantity of SA[f] determined from the fragment σAdistribution was not well correlated with δnp. In general, with different upper limitations on fragment mass, SA[f] was mainly influenced by the relatively neutron-deficient36-43Ca reactions, showing an increase with the upper limitation on the fragment mass. It was also observed that SA[f] gradually increased with δnp for36-42Ca, whereas it rapidly increased with δnp for45-56Ca. Note that with a fragment upper limitation of 80%Ap, SA[f] decreased with increasing δnp when δnp＜0.03 fm, whereas the opposite trend occurred when δnp ＞0.03 fm.More clearly,with the upper limitations of Af ≤85%Ap and Af ≤90%Ap, SA[f] decreased with δnp when δnp＜0.02 and then increased with δnp in the more neutron-rich nuclei-induced reactions.

Fig. 4 (Color online) Correlation between SA[f] obtained from the fragment σA distribution and δnp of the projectile nucleus for the 140 MeV/u ApCa+ 9Be reactions.Ap ranges from 36 to 56,which covers both the neutron-deficient and neutron-rich calcium isotopes. The bottom x and upper axes denote the neutron skin thickness δnp and corresponding projectile nuclei, respectively. The fragments mass,with upper limitations from 60%Ap to 90%Ap(in steps of 5%Ap), of projectile nuclei are plotted with solid symbols. Except for the 80%Ap, 85%Ap, and 90%Ap upper limitations, the lines denote the exponential fittings to the SA[f]～δnp correlations, except those of Af ≤85%Ap and Af ≤90%Ap

Remembering the SA[f]～δnp correlation predicted by the modified SAA model in [11], SA[f] was found to linearly decrease with increasing δnp of neutron-rich calcium isotopes from40Ca to60Ca. 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]only included fragments with Af ≥10, whereas in this study, Af was extended to fragments as light as Af =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 Af ≤90%Ap, that is, Af ≤14 and 15 ≤Af ≤90%Ap. The determined SA[f] and its correlation with δnp of the projectile nuclei are plotted in Fig. 5. Within the range Af ≤14, SA[f] decreased very slowly when δnp＜0 and increased with δnp for the more neutron-rich projectiles with thicker neutron skins. An opposite trend is found for the fragments within 15 ≤Af ≤90%Ap,that is,for δnp＜0,SA[f]increased with δnp and decreased with increasing δnp for projectiles with thicker neutron skins. For fragments within the range Af ≤90%Ap, the trend of the SA[f] distribution was similar to that of Af ≤14, indicating that SA[f] was significantly influenced by the selected range of fragments. Compared with the SA[f]～δnp correlation reported in [11], the results for 15 ≤Af ≤90%Ap are in good agreement, in which SA[f] decreased linearly with increasing δnp of the neutron-rich projectile nucleus. It is suggested that the adopted range of fragments significantly influences SA[f]as well as the SA[f]～δnp 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.

Fig. 5 (Color online) CIE (SA[f]) correlation to neutron skin thickness (δnp) of projectile nuclei 36-56Ca. The quantities of SA[f]are determined from different ranges of fragments, for which the squares, circles, and triangles denote SA[f] determined from the fragment mass within Af ≤90%Ap, Af ≤14, and 15 ≤Af ≤90%Ap,respectively. Linear fittings (lines) are performed on the SA[f]～δnp correlations according to δnp＜0 and δnp ＞0

4 Conclusion

CIE analysis was adopted to quantify the CIE incorporated in the fragment σAdistributions of PF reactions. The newly proposed BNN+FRACS model was used to predict the fragment σAdistributions in the 140 MeV/u36-56Ca +9Be 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 σAdistributions, 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 Af/Ap. 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 SA[f] was not sensitive to the neutron skin thickness if the upper limitations on the fragments were relatively small,for example,Af ≤75%Ap 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 SA[f], for neutron-deficient projectile nuclei,SA[f] exhibited a decreasing trend with increasing δnp,whereas it increased with δnp for neutron-rich projectile nuclei.

• By dividing the fragments into two groups according to their mass numbers, that is, the relatively light fragments of Af ≤14 and the relatively large fragments of 15 ≤Af ≤90%Ap, SA[f]was found to have different types of correlations with δnp of the projectile nucleus.

• The CIE determined from fragments of 15 ≤Af ≤90%Ap reproduced the trend of a previous correlation obtained in [11], showing a good linear SA[f]～δnp correlation in the neutron-rich projectile nuclei of calcium isotopes.

These results indicate that although the SA[f]～δnp relationship may be significantly influenced by light mass fragments, the correlation between them is very weak.Thus,the use of large-mass fragments is proposed to study the SA[f]～δnp correlation for neutron-rich projectile nuclei.

AcknowledgementsWe thank Prof. Chun-Wang Ma for his suggestion to study this topic.

Author ContributionsAll authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Xun 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.