Determination of neutron-skin thickness using configurational information entropy

Determination of neutron-skin thickness using configurational information entropy 1 2 corresp first-author machunwang@126.com machunwang@126.com machunwang@126.com https://orcid.org/0000-0001-9372-518X 1 1 1 https://orcid.org/0000-0001-7169-4044 1 1 School of Physics, Henan Normal University, Xinxiang 453007, China School of Physics, Henan Normal University, Xinxiang 453007, China Key Laboratory Optoelectronic Sensing Integrated Application of Henan Province, Henan Normal University, Xinxiang 453007, China Key Laboratory Optoelectronic Sensing Integrated Application of Henan Province, Henan Normal University, Xinxiang 453007, China Configurational information entropy (CIE) theory was employed to determine the neutron skin thickness of neutron-rich calcium isotopes. The nuclear density distributions and fragment cross-sections in 350 MeV/u 40-60 Ca + 9 Be projectile fragmentation reactions were calculated using a modified statistical abrasion-ablation model. CIE quantities were determined from the nuclear density, isotopic, mass, and charge distributions. The linear correlations between the CIE determined using the isotopic, mass, and charge distributions and the neutron skin thickness of the projectile nucleus show that CIE provides new methods to extract the neutron skin thickness of neutron-rich nuclei. Neutron-skin thickness Configurational information entropy Cross section distribution Projectile fragmentation introduction 1 Next generation radioactive nuclear beam facilities will provide new opportunities to explore extreme nuclei near and beyond drip lines. Nuclei with a large neutron excess can form exotic neutron skin or halo structures, which have attracted significant interest experimentally and theoretically for the past 30 years. The neutron skin thickness is defined as δ np = δ n - δ p , which denotes the difference between the point neutron and point proton root-mean-square (RMS) radii of a nucleus. Many methods have been developed to experimentally determine the neutron skin thickness. However, most of the models are indirect measurements and model dependent. Typical methods used to determine neutron skin thickness include the reaction cross section ( <math id="M1"><mrow><msub><mi>σ</mi><mtext>R</mtext></msub></mrow></math> ), charge-changing cross section ( <math id="M2"><mrow><msub><mi>σ</mi><mrow><mtext>cc</mtext></mrow></msub></mrow></math> ) [ 1 1 , 2 2 ], electric dipole polarizability [ 3 3 ], photon multiplicity [ 4 4 ], the π - / π + ratio or ∑ - /∑ + [ 5 5 , 6 6 ], 3 H/ 3 He ratio [ 7 7 ], and α decay half-life time [ 8 8 ]. The projectile fragmentation reaction, which is the main experimental approach for studying rare isotopes, is suitable for determining the neutron skin thickness owing to the obvious experimental phenomena induced by the neutron skin structure [ 9 9 , 10 10 ]. For example, isospin effects in the isotopic cross section[ 11 11 ], neutron-abrasion cross section ( <math id="M3"><mrow><msub><mi>σ</mi><mrow><mtext>nabr</mtext></mrow></msub></mrow></math> ) [ 12 12 ], neutron removal cross section [ 13 13 ], mirror nuclei ratio or isobaric ratio [ 14 14 ], and isoscaling parameter ( α ) [ 15 15 ]. Parity-violating electron scattering (PVS) is the only method used to determine neutron skin thickness that is model-independent. Ref. [ 16 16 ] reports a theoretical investigate of PVS for 48 Ca and 208 Pb, and a theoretical investigation of Bayesian approach is given in Ref. [ 17 17 ]. Determining the neutron skin thickness of 48 Ca and 208 Pb is presently of significant interest and is listed in the U.S. 2015 Long Range Plan for Nuclear Science [ 18 18 ]. The lead radius experiment (PREX) has been previously used to determine the neutron skin thickness of 208 Pb [ 19 19 ]. A recent PREX result indicates a much thicker neutron skin than previously predicted [ 20 20 ]. Determining the neutron skin thickness of nuclei near the neutron drip line remains an important research outcome and is one of the most interesting topics in the new era of radioactive beam facilities. Information entropy theory was established by C.E. Shannon [ 21 21 ]. The theory makes it possible to transform variables in a system into an exact information quantity [ 22 22 ] and has been used in various applications [ 23 23 , 24 24 ]. The first application of information entropy theory in heavy-ion reactions can be traced to the study of nuclear liquid-gas transition in nuclear multifragmentation [ 25 25 ]. Recent studies have extended it to study the information entropy carried by a single fragment produced in projectile fragmentation reactions, and revealed the scaling phenomenon of fragments covering a wide range of neutron excess [ 26 26 - 28 28 26-28 ]. Configurational information entropy (CIE) was developed to quantify the information entropy of a physical distribution [ 29 29 ], which connects the dynamical and informational contents of a physical system with localized configurations. Many applications of CIE methods can be found in Korteweg-de Vries (KdV) solitions, compact astrophysical systems, and scalar glueballs (see a brief introduction in Ref. [ 23 23 ]), theoretical research of new Higgs boson decay channels [ 30 30 ], deploying heavier eta meson states in AdS/QCD [ 31 31 ], confinement/deconfinement transition in QCD [ 32 32 ], quarkonium in a finite density plasma [ 33 33 ], time evolution in physical systems [ 34 34 , 35 35 ], etc. In projectile fragmentation reactions, fragment distributions show a sensitive dependence on the change in neutron density [ 36 36 - 38 38 36-38 ], which makes it possible to determine the neutron skin thickness of neutron-rich nuclei. In this study, the CIE method was adopted to quantify the CIE of nuclear density and fragment distributions in projectile fragmentation reactions. The analyzed data were generated using a modified statistical abrasion ablation (SAA) model, which is known to be a good model for describing the fragment cross sections of projectile fragmentation reactions [ 39 39 , 40 40 ]. Theories 1 Modified Statistical Abrasion-Ablation Model 2 The modified statistical abrasion-ablation (SAA) model [ 39 39 , 40 40 ] can be used for projectile fragmentation reactions at both intermediate and high energies, which improves the original SAA model by Brohm and Schmidt [ 41 41 ]. In quasi-free nucleon-nucleon collisions, the reaction is described as a two-step process: In the initial stage, the nucleons are described by a Glauber-type model as "participants" and "spectators", where the participants interact strongly in an overlapping region between the projectile and target, while the spectators move virtually undisturbed [ 42 42 ]. In the second stage, the excitation energy is compared to the separation energies of protons, neutrons, and α to determine the type of particle the prefragment can emit according to min ( <math id="M4"><mrow><msub><mi>s</mi><mtext>p</mtext></msub><mn>,</mn><msub><mi>s</mi><mtext>n</mtext></msub><mn>,</mn><msub><mi>s</mi><mi>α</mi></msub></mrow></math> ). After the de-excitation calculation, the cross sections of final fragments that are comparable to the measured fragments were obtained. The description of the modified SAA model is presented in Refs. [ 23 23 , 39 39 , 40 40 ]. The colliding nuclei are composed of many parallel tubes oriented along the beam direction. Their transverse motion is neglected, and the interactions between the tube pairs are independent. For a specific pair of interacting tubes, the absorption of the projectile neutrons and protons is assumed to be in a binomial distribution. At a given impact parameter b, the transmission probabilities of neutrons (protons) of an infinitesimal tube in the projectile are calculated using <math display="block" id="M5"><mrow><msub><mi>t</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>{</mo><mo>−</mo><mo stretchy="false">[</mo><msubsup><mi>D</mi><mtext>n</mtext><mtext>T</mtext></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo><msub><mi>σ</mi><mrow><mtext>n</mtext><mi>i</mi></mrow></msub><mo>+</mo><msubsup><mi>D</mi><mtext>p</mtext><mtext>T</mtext></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo><msub><mi>σ</mi><mrow><mtext>p</mtext><mi>i</mi></mrow></msub><mo stretchy="false">]</mo><mo>}</mo><mn>,</mn></mrow></math> where D T is the normalized integrated nuclear density distribution of the target along the beam direction for protons <math id="M6"><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><mi>s</mi><msubsup><mi>D</mi><mtext>p</mtext><mtext>T</mtext></msubsup><mo>=</mo><msup><mi>Z</mi><mtext>T</mtext></msup></mrow></math> and neutrons <math id="M7"><mrow><mo>∫</mo><msup><mtext>d</mtext><mn>2</mn></msup><mi>s</mi><msubsup><mi>D</mi><mtext>n</mtext><mtext>T</mtext></msubsup><mo>=</mo><msup><mi>N</mi><mtext>T</mtext></msup></mrow></math> ( N T and Z T are the neutron and proton numbers of the target, respectively). s and b are defined in a plane perpendicular to the beam direction, and <math id="M8"><mrow><msub><mi>σ</mi><mrow><msup><mi>i</mi><mo>′</mo></msup><mi>i</mi></mrow></msub></mrow></math> denotes the free-space nucleon-nucleon cross sections ( i ', i =n for neutrons and i ', i =p for protons) [ 44 44 ]. The average absorbed mass in the infinitesimal tube limit at a given b is <math display="block" id="M9"><mtable columnalign="left"><mtr><mtd><mo>〈</mo><mi>Δ</mi><mi>A</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><mi>s</mi><msubsup><mi>D</mi><mtext>n</mtext><mtext>T</mtext></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo stretchy="false">)</mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>t</mi><mtext>n</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><mi>s</mi><msubsup><mi>D</mi><mtext>p</mtext><mtext>P</mtext></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo stretchy="false">)</mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>t</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> For a specific fragment, the production cross section can be calculated using <math display="block" id="M10"><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>Δ</mi><mi>N</mi><mn>,</mn><mi>Δ</mi><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><mi>b</mi><mi>P</mi><mo stretchy="false">(</mo><mi>Δ</mi><mi>N</mi><mn>,</mn><mi>b</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Δ</mi><mi>Z</mi><mn>,</mn><mi>b</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where P (Δ N,b ) and P (Δ Z, b ) are the probability distributions of the abraded neutrons and protons at a given impact parameter b , respectively. (Δ N , Δ Z ) is the residual fragment after the abrasion stage, which is called the prefragment. The excitation energy of the prefragment is calculated as E * =13.3〈Δ A ( b )〉 MeV, where <math id="M11"><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>〉</mo></mrow></math> is the number of abraded nucleons from the projectile, and 13.3 MeV is the mean excitation energy owing to an abraded nucleon [ 43 43 ]. In the second stage, the excitation energy is compared to the separation energies of protons, neutrons, and α to determine the type of particle the prefragment can emit according to min ( <math id="M12"><mrow><msub><mi>s</mi><mtext>p</mtext></msub><mn>,</mn><msub><mi>s</mi><mtext>n</mtext></msub><mn>,</mn><msub><mi>s</mi><mi>α</mi></msub></mrow></math> ). After the de-excitation calculation, the cross sections of final fragments that were comparable to the measured fragments were obtained. Fermi-type density distributions were adopted for protons and neutrons in a nucleus, as shown in the equation below: <math display="block" id="M13"><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><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mtext> </mtext><mo stretchy="false">(</mo><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><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn><mi> </mi><mi> </mi><mi> </mi><mi>i</mi><mo>=</mo><mtext>n,</mtext><mtext> </mtext><mtext>p</mtext></mrow></math> where <math id="M14"><mrow><msubsup><mi>ρ</mi><mi>i</mi><mn>0</mn></msubsup></mrow></math> is the normalization constant of neutrons ( <math id="M15"><mrow><mi>i</mi><mo>=</mo><mtext>n</mtext></mrow></math> ) or protons ( <math id="M16"><mrow><mi>i</mi><mo>=</mo><mtext>p</mtext></mrow></math> ), t i is the diffuseness parameter, and C i is half the density radius of the neutron or proton density distribution. Configurational Information Entropy Method 2 To determine the quantity of CIE incorporated in the fragment distributions, definitions of CIE were introduced. For a system with spatially localized clusters, when performing the CIE analysis, a set of functions <math id="M17"><mrow><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></mrow></math> and their Fourier transforms F ( k ) obey Plancherel's theorem [ 45 45 ]: <math display="block" id="M18"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><mo>|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mrow><mo stretchy="false">)</mo><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><mo>|</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><msup><mrow><mo stretchy="false">)</mo><mo>|</mo></mrow><mn>2</mn></msup><mtext>d</mtext><mi>k</mi><mn>,</mn></mrow></math> where f ( x ) is square-integrable-bounded. The model fraction f ( k ) is defined as: <math display="block" id="M19"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>|</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><msup><mrow><mo stretchy="false">)</mo><mo>|</mo></mrow><mn>2</mn></msup></mrow><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mo>|</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><msup><mrow><mo stretchy="false">)</mo><mo>|</mo></mrow><mn>2</mn></msup><msup><mtext>d</mtext><mi>d</mi></msup><mi>k</mi></mrow></mfrac><mn>,</mn></mrow></math> where the integration is over all k , F ( k ) is defined, and d is the number of spatial dimensions. The model fraction f ( k ) measures the relative weight of a given mode k . The quantity of CIE <math id="M20"><mrow><msub><mi>S</mi><mtext>C</mtext></msub><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> is defined as a summation of the Shannon information entropy of f ( k ) [ 21 21 ]: <math display="block" id="M21"><mrow><msub><mi>S</mi><mtext>C</mtext></msub><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><msub><mi>f</mi><mi>m</mi></msub><mi>ln</mi><mo stretchy="false">(</mo><msub><mi>f</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Thus, the quantity of CIE has information about configurations compatible with certain constraints of a given physical system. If all the modes k have the same mass, then f m =1/ N . The discrete configuration entropy reaches a maximum at <math id="M22"><mrow><msub><mi>S</mi><mtext>C</mtext></msub><mo>=</mo><mi>ln</mi><mi>N</mi></mrow></math> . If there is only one mode, S C = 0. Continuous CIE can also be defined for continuous distributions, such as the nuclear density distribution. For non-periodic functions in the interval ( a, b ), <math display="block" id="M23"><mrow><msub><mi>S</mi><mtext>C</mtext></msub><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>ln</mi><mo stretchy="false">[</mo><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msup><mtext>d</mtext><mi>d</mi></msup><mi>k</mi><mn>,</mn></mrow></math> where <math id="M24"><mrow><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>/</mo><mi>f</mi><msub><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>max</mtext></mrow></msub></mrow></math> ［ f ( k ) max is the maximum fraction］. The normalized function <math id="M25"><mrow><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></math> guarantees that <math id="M26"><mrow><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn></mrow></math> for all k modes, and <math id="M27"><mrow><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>ln</mi><mover accent="true"><mi>f</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></math> denotes the CIE density. Results and discussion 1 The 350 MeV/u A p Ca + 9 Be reactions were calculated using the modified SAA model ( A p refers to even mass numbers from 40 to 60). The cross sections of fragments with Z ranging from 3 to 20 were obtained. For the sake of clarity, only part of the calculated results are shown in the figures. Figure 1 Figure 1 is a plot of the Fermi-type nuclear density distributions and their fast Fourier transformation (FFT) spectra. An obvious increase in ρ n is observed from 40 Ca to 60 Ca, whereas the opposite trend is observed for ρ p . A two-peak structure is evident in the FFT spectra, where the second peak is lower than the first. The difference between the neutron and proton density distributions Δ ρ = ρ n - ρ p is also shown. For 40 Ca, Δρ is very small, while Δρ increases as the neutrons in the projectile increase. The peaks in the FFT spectra of ρ n and ρ p are not clearly shown. Based on the FFT spectra f ( k ), the CIE of ρ n , ρ p , and Δρ can be determined using Eq. (7), which are denoted by <math id="M28"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>n</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , <math id="M29"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>p</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , and <math id="M30"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><mi>Δ</mi><mi>ρ</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , respectively. The isotopic cross section ( <math id="M31"><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></math> ) distributions produced in the 350 MeV/u A p Ca + 9 Be reactions are plotted in Fig. 2 Fig. 2 . In panels (ai), from Z fr = 7 to 20, the isotopic cross-section distributions in the 40-60 Ca reactions are similar for fragments with small Z fr , while a shift to the neutron-rich side is observed for larger Z fr . The symmetric Guassian-like shape of the isotopic distribution is altered by the enhanced cross sections of neutron-rich fragments in neutron-rich reaction systems, showing the isospin effect in fragment production induced by the increased neutron density on the surface of neutron-rich nuclei [ 40 40 ]. The FFT spectra of the isotopic distributions are shown in Fig. 2 Fig. 2 (b j ). In each FFT spectrum, only one peak is observed. The amplitudes of the FFT spectra of different isotopic distributions decrease as the projectile becomes more neutron-rich, except for Z = 20. Based on the FFT spectra of <math id="M32"><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></math> distributions, the quantities of CIE of the isotopic distributions are determined according to Eq. (7), which is denoted by <math id="M33"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> . The correlation between the <math id="M34"><mrow><msub><mi>S</mi><mtext>C</mtext></msub><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> of the density distribution and δ np of the projectile nucleus is shown in Fig. 3(a) Fig. 3(a) . Both <math id="M35"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>n</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> and <math id="M36"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>p</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> decrease linearly with an increase in δ np from 40 Ca to 60 Ca. <math id="M37"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><mi>Δ</mi><mi>ρ</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> also decreases linearly with increasing δ np , except for 40 Ca. The correlation between the <math id="M38"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> of different Z fr and δ np of the projectile nuclei are plotted in panel (b). The <math id="M39"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> of isotopes from Z fr = 10 to 18 are also found to decrease with the increasing δ np of the projectile nucleus. The <math id="M40"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> of the fragment near the projectile nucleus was more sensitive to the change in δ np . The mass yield ( <math id="M42"><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></math> ) distributions in the 350 A MeV A p Ca + 9 Be reactions are shown in Fig. 4 Fig. 4 . In each reaction, the mass yield increases with the A fr of the fragment until it is close to the projectile nucleus. In different reactions, a very similar trend of mass distribution was observed, which decreased with the increasing mass number of the projectile nucleus. The corresponding quantities of CIE were determined from <math id="M43"><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></math> distributions, which are labeled as <math id="M44"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> . The correlation between <math id="M45"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> and δ np for projectile nuclei is shown in Fig. 4 (b) Fig. 4 (b) . Except for the bend point formed at δ np for 42 Ca owing to the transition of proton-skin to neutron skin, the <math id="M46"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> correlation was found to be linear for reactions of A p ≥ 44. The charge cross section is defined as the summation of the isotopic cross sections <math id="M47"><mrow><msub><mi>σ</mi><mtext>C</mtext></msub><mo>=</mo><mstyle displaystyle="true"><msub><mo>∑</mo><mi>o</mi></msub></mstyle><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>o</mi></msub><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo></mrow></math> . The charge cross section distributions in the 350 A MeV A p Ca + 9 Be reactions are shown in Fig. 5 Fig. 5 . Similar trends of <math id="M48"><mrow><msub><mi>σ</mi><mtext>C</mtext></msub></mrow></math> distribution trends similar to those of <math id="M49"><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></math> were observed. The determined CIE of <math id="M50"><mrow><msub><mi>σ</mi><mtext>C</mtext></msub></mrow></math> distributions, labeled as <math id="M51"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mtext>C</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , were linearly correlated to the neutron skin thickness of the projectile nuclei. The CIE approach transfers the experimental distributions to quantified parameters and provides information probes for determining the properties of a system. From the <math id="M53"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , <math id="M54"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , and <math id="M55"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mtext>C</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> correlations, it was observed that the CIE determined from isotopic, mass, and charge distributions decrease with increasing neutron skin thickness, respectively, and they had good linear correlations. The determination of neutron skin thickness, in particular, nuclei near the neutron drip line, is limited by the unavailability of effective probes. The linear correlation between the CIE and neutron-skin thickness of neutron-rich nuclei provides new approaches to determine the neutron skin thickness of the projectile nucleus by measuring the fragment distributions in projectile fragmentation reactions. summary 1 With the vast opportunities for very asymmetric nuclei available in the new era of radioactive ion beam facilities, neutron skin thickness is one of the most important questions in nuclear physics. In this study, CIE theory is adopted to quantify the information entropy incorporated in nuclear density distributions and fragment cross-section distributions in 350 MeV/u 40-60 Ca + 9 Be projectile fragmentation reactions calculated using the modified SAA model. CIE quantities of nuclear density distributions ( <math id="M56"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mrow><mtext>n,p</mtext></mrow></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> and <math id="M57"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><mi>Δ</mi><mi>ρ</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> ), isotopic cross-section distributions ( <math id="M58"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> ), mass cross-section distributions ( <math id="M59"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> ), and charge cross-section distributions ( <math id="M60"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mtext>C</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> ) were determined. The correlations between <math id="M61"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>p</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , <math id="M62"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>n</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , <math id="M63"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><mi>Δ</mi><mi>ρ</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , and <math id="M64"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , <math id="M65"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> , and <math id="M66"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>C</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>∼</mo><msub><mi>δ</mi><mrow><mtext>np</mtext></mrow></msub></mrow></math> were also investigated. For neutron-rich calcium projectiles, obvious linear dependences of <math id="M67"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>ρ</mi><mtext>n</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , <math id="M68"><mrow><msubsup><mi>S</mi><mi>C</mi><mrow><msub><mi>ρ</mi><mtext>p</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , and <math id="M69"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><mi>Δ</mi><mi>ρ</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> on δ np were observed. The <math id="M70"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> of fragments with different Z fr is shown to linearly depend on the δ np of the projectile nucleus. It is found that, if the isotopic distribution is sensitive to isospin effects in the projectiles, the extracted <math id="M71"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>Z</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> will also be sensitive to their δ np . Good linear correlations between <math id="M72"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mi>A</mi></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , <math id="M73"><mrow><msubsup><mi>S</mi><mtext>C</mtext><mrow><msub><mi>σ</mi><mtext>C</mtext></msub></mrow></msubsup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></math> , and δ np of the projectile nucleus were also observed. 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Material preparation, data collection and analysis were performed by Yi-Pu Liu, Hui-Ling Wei and Chun-Wang Ma. The first draft of the manuscript was written by Yi-Pu Liu and Chun-Wang Ma and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-022-00997-0 This work was supported by the National Natural Science Foundation of China (Nos. 11975091 and U1732135) and the Program for Innovative Research Team (in Science and Technology) in University of Henan Province, China (No. 21IRTSTHN011). 2021-11-09 2021-12-06 2021-12-07 2022-01-17 2022-02-16 © 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/s41365-022-00997-0 http://dx.doi.org/10.1007/s41365-022-00997-0 . 2022 Nuclear Science and Techniques 01 33