introduction

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 [1-3]. 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]. Because of difficulties in the direct measurement of neutrons, the neutron skin thickness is always determined 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, 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]. 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 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 (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 small-mass and large-mass fragments well.

In this study, the BNN + FRACS model was adopted to predict the fragment σA 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

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 $\begin{array}{l}f(x)\in L^2(R),\\ {\displaystyle {\int }_{-\infty }^{\infty }}{\left|f(x)\right|}^2\text{d}x={\displaystyle {\int }_{-\infty }^{\infty }}{\left|F(k)\right|}^2\text{d}k\mathrm{,}\end{array}$ (1) 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 $f(k)=\frac{{\left|F(k)\right|}^2}{{\displaystyle \int }{\left|F(k)\right|}^2{\text{d}}^{d}k}\mathrm{,}$ (2) 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], $S[f]=-{\displaystyle \sum _{m=1}^k}f(m)\ln f(m)\mathrm{.}$ (3) 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 Xn according to the procedures described in Sec. 2.1.

Fig. 1Full-screen View

(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.

The fast Fourier transform (FFT) was chosen to analyze the fragment cross-section distribution, $F(k)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}}{X}_n{e}^{-i2\pi kn}\mathrm{,}$ (4) where k=m/N, 0 ≤ m ≤ N-1, and N is the amount of data in the Xn distribution.

Results and discussion

The fragment mass cross-sections for 140 MeV/u ^36-56Ca + 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. ${\rho }_i(r)=\frac{{\rho }_i^0}{1+\exp \left(\frac{r-C_i}{t_i/4.4}\right)}\mathrm{,}i=\text{n}\mathrm{,}\text{p}$ (8) where i denotes neutrons or protons, ${\rho }_i^0$ is the normalization constant, ti is the diffuseness parameter, and Ci 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: ${\delta }_{\text{np}}=\sqrt{\langle r_{\text{n}}^2\rangle }-\sqrt{\langle r_{\text{p}}^2\rangle }$ (9) The fragment σA distributions for a series of 140 MeV/u ^36-56Ca + 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 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 ⁴⁰Ca + ⁹Be (open squares) and ⁴⁸Ca + ⁹Be (open circles) reactions [21], the BNN + FRACS model effectively reproduced the measured σA distributions, which made it possible to carry out a CIE analysis of the mass distribution for reactions induced by calcium isotopes with different isospins.

Fig. 2Full-screen View

(Color online) Predicted mass cross-section (σA) distributions using the BNN + FRACS model (in solid symbols) for 140 MeV/u ${}^{A_{\text{p}}}$Ca + ⁹Be reactions. The y-axis denotes σA in mb, and the x-axis shows the mass numbers of fragments. The projectile nucleus ranges from the proton-rich ³⁶Ca to the neutron-rich ⁵⁶Ca, and A_p is changed from 36 to 56 in steps of 4. The measured data in [21] are plotted as open symbols.

Taking the 140 MeV/u ⁴⁴Ca + ⁹Be reaction as an example, the FFT analysis of the σA 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. 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

Fig. 3Full-screen View

(Color online) Fast Fourier transform (FFT) of σA distributions for the 140 MeV/u ⁴⁴Ca + ⁹Be reactions, as plotted in Fig. 2. The x-axis denotes the mode of k, and F(k) is the amplitude of mode k. FFT is performed on fragment cuts from 60%A_p to 90%A_p to study the influence of fragment range and fragment size on CIE.

Using Eq. (3), the CIE for fragment σA distributions was calculated from the FFT analysis, which is labeled as SA[f]. The results of SA[f] and their correlation with δ_np for the 140 MeV/u ${}^{A_{\text{p}}}$Ca + ⁹Be reactions are plotted in Fig. 4, in which A_p ranges from 36 to 56, and the upper limitations on fragments masses are changed from 60%Ap to 90%A_p. δ_np of the projectile nucleus was calculated using the nuclear density distribution according to Eq. (8). Around ^43,44Ca, δ_np~0, indicating proton skins in ^36-42Ca, and neutron skins in ^45-60Ca. The quantity of SA[f] determined from the fragment σA distribution was not well correlated with δ_np. In general, with different upper limitations on fragment mass, SA[f] was mainly influenced by the relatively neutron-deficient ^36-43Ca reactions, showing an increase with the upper limitation on the fragment mass. It was also observed that SA[f] gradually increased with δ_np for ^36-42Ca, whereas it rapidly increased with δ_np for ^45-56Ca. Note that with a fragment upper limitation of 80%A_p, SA[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.

Fig. 4Full-screen View

(Color online) Correlation between SA[f] obtained from the fragment σA distribution and δ_np of the projectile nucleus for the 140 MeV/u ${}^{A_{\text{p}}}$Ca + ⁹Be reactions. A_p 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 δ_np and corresponding projectile nuclei, respectively. The fragments mass, with upper limitations from 60%A_p to 90%A_p(in steps of 5%A_p), of projectile nuclei are plotted with solid symbols. Except for the 80%A_p, 85%A_p, and 90%A_p upper limitations, the lines denote the exponential fittings to the SA[f] ~ δ_np correlations, except those of A_f ≤ 85%A_p, and 90%A_p

Remembering the SA[f]∼δ_np correlation predicted by the modified SAA model in [11], SA[f] was found to linearly decrease with increasing δ_np of neutron-rich calcium isotopes from ⁴⁰Ca to ⁶⁰Ca. 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 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, Af ≤ 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. Within the range A_f ≤ 14, S_A[f] decreased very slowly when δnp &lt; 0 and increased with δnp for the more neutronrich projectiles with thicker neutron skins. An opposite trend is found for the fragments within 15 ≤ A_f ≤ 90% Ap, that is, for δnp &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], 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.

Fig. 5Full-screen View

(Color online) CIE (SA[f]) correlation to neutron skin thickness (δ_np) 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 A_f ≤ 90%A_p, Af ≤ 14, and 15 ≤ A_f ≤ 90%A_p, respectively. Linear fittings (lines) are performed on the S_A[f] ~ δ_np correlations according to δ_np < 0 and δ_np > 0.

Conclusion

CIE analysis was adopted to quantify the CIE incorporated in the fragment σA distributions of PF reactions. The newly proposed BNN + FRACS model was used to predict the fragment σA distributions in the 140 MeV/u ^36-56Ca + 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 σA 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 SA[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 SA[f], for neutron-deficient projectile nuclei, SA[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 SA[f] was found to have different types 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], showing a good linear SA[i]~δ_np dnp correlation in the neutron-rich projectile nuclei of calcium isotopes.

These results indicate that although the SA[f] ~ δ_np 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] ~δ_np correlation for neutron-rich projectile nuclei.