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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets</title>   <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.12017v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S1" title="In Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">I Introduction: Nuclear fragmentation</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2" title="In Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">II Theoretical Formalism</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.SS1" title="In II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">II.1 Primary fragmentation or abrasion stage</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.SS2" title="In II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">II.2 Secondary fragmentation or ablation stage</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3" title="In Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III Results and discussions</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.SS1" title="In III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III.1 Single-nucleon density inputs</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.SS2" title="In III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III.2 Direct fragmentation or abrasion stage</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.SS3" title="In III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III.3 Proton evaporation contributions</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.SS3.SSS1" title="In III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III.3.1 Excitation energy distribution (EED)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.SS3.SSS2" title="In III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III.3.2 Proton evaporation probability</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S4" title="In Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">IV Summary and conclusions</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets</h1> <div class="ltx_authors"> Sukhendu De1 <a href="mailto:sukhendu_d@ph.iitr.ac.in">sukhendu_d@ph.iitr.ac.in</a> V. Choudhary2 <a href="mailto:vishal.choudhary@jecrcu.edu.in">vishal.choudhary@jecrcu.edu.in</a> R. Chatterjee1 <a href="mailto:rchatterjee@ph.iitr.ac.in">rchatterjee@ph.iitr.ac.in</a> 1Indian Institute of Technology Roorkee, Roorkee, 247667, Uttarakhand, India 2Department of Physics, School of Science, JECRC University, Jaipur, 303905, Rajasthan, India W. Horiuchi3,4,5,6 <a href="mailto:whoriuchi@omu.ac.jp">whoriuchi@omu.ac.jp</a> 3 Department of Physics, Osaka Metropolitan University, Osaka 558-8585, Japan 4 Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP) , Osaka Metropolitan University, Osaka 558-8585, Japan 5 RIKEN Nishina Center, Wako 351-0198, Japan 6 Department of Physics, Hokkaido University, Sapporo 060-0810, Japan </div> <div class="ltx_dates">(March 15, 2025)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> This study examines charge-changing cross sections for <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="id11.1.m1.1"><semantics id="id11.1.m1.1a"><mmultiscripts id="id11.1.m1.1.1" xref="id11.1.m1.1.1.cmml"><mtext id="id11.1.m1.1.1.2" xref="id11.1.m1.1.1.2a.cmml">C</mtext><mprescripts id="id11.1.m1.1.1a" xref="id11.1.m1.1.1.cmml"></mprescripts><mrow id="id11.1.m1.1.1b" xref="id11.1.m1.1.1.cmml"></mrow><mn id="id11.1.m1.1.1.3" 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The two-stage abrasion-ablation model is used, with the abrasion stage described via the Glauber model, incorporating validated single-nucleon density distributions from proton elastic scattering data. In the ablation stage, where particle evaporation occurs, the contribution to charge-changing cross sections is estimated using two approaches. First, a statistical decay model is employed to analyze the evaporation of protons following neutron removal in the abrasion stage. The second approach estimates evaporation contributions by subtracting the direct process component (abrasion) from the experimental charge-changing cross section data. A comparison between these estimated contributions from the experimental data and the predictions of the statistical model enables a systematic evaluation of the evaporation process. The key factors influencing evaporation, such as excitation energy distribution parameters, decay width of the emitted particles, and nucleon separation energies, are analyzed. A strong correlation is observed between the maximum excitation energy available for evaporation and the neutron separation energy of the projectiles across different targets, highlighting the role of evaporation dynamics in the charge-changing cross sections. </div> ††preprint: NITEP 241 <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> I Introduction: Nuclear fragmentation</h2> <div class="ltx_para" id="S1.p1"> The space radiation or Galactic cosmic ray (GCR) environment is a complex mixture of photons, electrons, protons, and other heavy ions with energy ranging from a few eV to several TeV per nucleon. GCR is emitted by sources beyond the solar system, principally supernovae, gamma-ray bursts, and other high-energy cosmic phenomena. As these intense particles move across space, they contact interstellar material and magnetic fields, which can trigger fragmentation and nuclear spallation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib1" title="">1</a>]</cite>, resulting in secondary particle creation. When high-energy heavy ions breach spacecraft shielding materials, they fracture into a slew of lesser ions with lower atomic numbers that can go deeper than the incoming particles. These nuclei in GCR can have enough energy to pierce any shielding device currently employed on mission spacecraft. Interactions among the ions in the space environment are vital in various areas critical for space study and exploration due to the secondary radiation fields they generate. Space radiation may infiltrate housing, ships, equipment, and spacesuits, thereby putting personnel at risk. One of the most difficult issues in keeping astronauts fit and healthy while travelling through the solar system is minimising the physiological changes produced by space radiation exposure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib2" title="">2</a>]</cite>. Due to the difficulty of reproducing GCR radiation environments on Earth, stochastic (Monte Carlo) or deterministic transport algorithms are required for risk assessment of exploratory mission scenarios and shielding design. Radiation transport algorithms employ fragmentation cross sections, also known as production cross sections, which include charge-changing cross sections, neutron removal cross sections, interaction cross sections, and total reaction cross sections to measure this phenomenon. The primary aim is to accurately model the interaction between particles and materials in space environments. The nuclear fragmentation cross sections used in space radiation transport codes at NASA are derived from experimental data and theoretical models. Experimental measurements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib7" title="">7</a>]</cite> of nuclear fragmentation cross sections involve bombarding a target material with a beam of energetic particles and analysing the resulting nuclear reactions. These experiments are typically performed using accelerators and detectors to measure the products of the interactions. Experimental data provide crucial information on the probabilities of various nuclear reactions and fragment production for different projectile energies and target materials. On the theoretical side, the required cross sections are computed using some semi-empirical models like the Silberberg-Tsao <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib8" title="">8</a>]</cite> model, and a series of codes like NUCFRAG <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib9" title="">9</a>]</cite>, NUCFRG2 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib10" title="">10</a>]</cite>, NUCFRG3 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib11" title="">11</a>]</cite> developed by NASA. There are also a few software packages that are used in cross section estimation for space radiation applications like GEANT4 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib12" title="">12</a>]</cite>, FLUKA <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib13" title="">13</a>]</cite>, PHITS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib14" title="">14</a>]</cite> and LISE++ <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib15" title="">15</a>]</cite>. </div> <div class="ltx_para" id="S1.p2"> A proper analysis of the fragmentation process and cross section measurements are also useful ingredients for modeling the transport code for the use of ion beam therapy dose calculations and treatment planning. When charged particles travel through matter, they lose energy largely through interactions with the material’s atomic electrons and nucleus. The depth of penetration of the particles affects the energy deposition. The Bragg curve describes the connection between particle energy deposition or stopping power and material penetration depth <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib17" title="">17</a>]</cite>. When the Bragg peak is considered, the interaction cross section or total reaction cross section directly affects the total energy deposition profile of the charged particles <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib19" title="">19</a>]</cite>. A higher value of cross sections provides a wider and shallower Bragg peak. This is due to the fact that particles with bigger cross sections have more frequent contact, resulting in a more progressive loss of energy across a wider depth range. The values of the cross sections vary from ion to ion depending on the nuclear interactions the ion undergoes when travelling through the material or human body. The fragments or ions like 4He and 12C become very useful for radiation therapy because of their favourable depth to dose Bragg curve. Recent results <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib21" title="">21</a>]</cite> on 3He and 12O ions have also proven to be very useful in heavy ion beam therapy. </div> <div class="ltx_para" id="S1.p3"> Nuclear fragmentation in heavy-ion collisions at intermediate energies has been a subject of extensive research over the past few decades <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib24" title="">24</a>]</cite>. Of particular interests are exotic nuclei, such as halo nuclei <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib26" title="">26</a>]</cite> with extended neutron distributions or bubble nuclei <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib28" title="">28</a>]</cite> with reduced central densities, which exhibit unique structural properties that influence reaction dynamics. These exotic features not only provide valuable insights into nuclear interactions but also play a critical role in understanding nucleosynthesis and the evolution of matter in astrophysical environments <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib30" title="">30</a>]</cite>, such as supernovae and neutron star mergers. </div> <div class="ltx_para" id="S1.p4"> The charge-changing cross sections across nuclei can be effectively described using the abrasion-ablation framework <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib34" title="">34</a>]</cite>. This process involves two principal mechanisms where proton removal plays a central role. In one scenario, abrasion of protons occurs during the initial impact, often accompanied by neutron removal, followed by neutron evaporation in the later stage. In the alternative scenario, neutrons are primarily removed during the initial phase, while subsequent evaporation leads to the loss of one or more protons along with additional neutrons. These distinct pathways collectively influence the final charge of the residual nucleus. In the abrasion step, nucleons from the overlapping zone of the projectile and target are abraded, depending on the impact parameter, resulting in the formation of excited remnants of the initial nuclei, known as prefragments. In the subsequent ablation step, these prefragments deexcite through the evaporation of light particles. The evaporation chain continues until the excitation energy of the prefragments falls below the particle emission threshold energies, resulting in the formation of the final fragments. </div> <div class="ltx_para" id="S1.p5"> In this paper, we study charge-changing cross sections using the abrasion-ablation model for <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mmultiscripts id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mtext id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2a.cmml">C</mtext><mprescripts id="S1.p5.1.m1.1.1a" xref="S1.p5.1.m1.1.1.cmml"></mprescripts><mrow id="S1.p5.1.m1.1.1b" xref="S1.p5.1.m1.1.1.cmml"></mrow><mn id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1">superscript</csymbol><ci id="S1.p5.1.m1.1.1.2a.cmml" xref="S1.p5.1.m1.1.1.2"><mtext id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">C</mtext></ci><cn id="S1.p5.1.m1.1.1.3.cmml" type="integer" xref="S1.p5.1.m1.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">{}^{12}\text{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT C</annotation></semantics></math>, <math alttext="{}^{14}\text{N}" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mmultiscripts id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mtext id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2a.cmml">N</mtext><mprescripts id="S1.p5.2.m2.1.1a" xref="S1.p5.2.m2.1.1.cmml"></mprescripts><mrow id="S1.p5.2.m2.1.1b" xref="S1.p5.2.m2.1.1.cmml"></mrow><mn id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml">14</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1">superscript</csymbol><ci id="S1.p5.2.m2.1.1.2a.cmml" xref="S1.p5.2.m2.1.1.2"><mtext id="S1.p5.2.m2.1.1.2.cmml" xref="S1.p5.2.m2.1.1.2">N</mtext></ci><cn id="S1.p5.2.m2.1.1.3.cmml" type="integer" xref="S1.p5.2.m2.1.1.3">14</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">{}^{14}\text{N}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">start_FLOATSUPERSCRIPT 14 end_FLOATSUPERSCRIPT N</annotation></semantics></math>, <math alttext="{}^{16}\text{O}" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mmultiscripts id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><mtext id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2a.cmml">O</mtext><mprescripts id="S1.p5.3.m3.1.1a" xref="S1.p5.3.m3.1.1.cmml"></mprescripts><mrow id="S1.p5.3.m3.1.1b" xref="S1.p5.3.m3.1.1.cmml"></mrow><mn id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml">16</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.1">superscript</csymbol><ci id="S1.p5.3.m3.1.1.2a.cmml" xref="S1.p5.3.m3.1.1.2"><mtext id="S1.p5.3.m3.1.1.2.cmml" xref="S1.p5.3.m3.1.1.2">O</mtext></ci><cn id="S1.p5.3.m3.1.1.3.cmml" type="integer" xref="S1.p5.3.m3.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">{}^{16}\text{O}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O</annotation></semantics></math>, and <math alttext="{}^{20}\text{Ne}" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mmultiscripts id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml"><mtext id="S1.p5.4.m4.1.1.2" xref="S1.p5.4.m4.1.1.2a.cmml">Ne</mtext><mprescripts id="S1.p5.4.m4.1.1a" xref="S1.p5.4.m4.1.1.cmml"></mprescripts><mrow id="S1.p5.4.m4.1.1b" xref="S1.p5.4.m4.1.1.cmml"></mrow><mn id="S1.p5.4.m4.1.1.3" xref="S1.p5.4.m4.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><apply id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S1.p5.4.m4.1.1.1.cmml" xref="S1.p5.4.m4.1.1">superscript</csymbol><ci id="S1.p5.4.m4.1.1.2a.cmml" xref="S1.p5.4.m4.1.1.2"><mtext id="S1.p5.4.m4.1.1.2.cmml" xref="S1.p5.4.m4.1.1.2">Ne</mtext></ci><cn id="S1.p5.4.m4.1.1.3.cmml" type="integer" xref="S1.p5.4.m4.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">{}^{20}\text{Ne}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT Ne</annotation></semantics></math> projectiles on elemental targets C, Al, and Cu at a beam energy around 290 MeV/nucleon. The abrasion stage, representing direct nucleon removal, is modelled using the Glauber formalism <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib36" title="">36</a>]</cite> with appropriate nuclear density distributions. The ablation stage is governed by statistical evaporation models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib15" title="">15</a>]</cite> based on Hauser-Feshbach theory, which calculates the light particle evaporation probability from excited prefragments. </div> <div class="ltx_para" id="S1.p6"> To assess the evaporation contributions, we compare statistical model calculations with estimates from experimental charge-changing cross section data by subtracting the calculated abrasion component. The evaporation process depends on several key factors, including excitation energy distributions (EED), nuclear level densities, decay width ratios, and neutron/proton separation energies of the prefragment nuclei. A detailed analysis is performed to examine the influence of EED parameters across different targets, revealing a linear inverse correlation between the maximum excitation energy and the projectile neutron separation energy. </div> <div class="ltx_para" id="S1.p7"> The paper is organized as follows. The theoretical formalism involving the abrasion-ablation formalism is briefly described in <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2" title="II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">II</a>, followed by the details of the results and discussions in <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3" title="III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III</a>, and the summary and conclusions of the present work in <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3" title="III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">III</a>. </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> II Theoretical Formalism</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> II.1 Primary fragmentation or abrasion stage</h3> <div class="ltx_para" id="S2.SS1.p1"> The Glauber multiple scattering approach <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib35" title="">35</a>]</cite> is a credible theoretical model for calculating projectile fragmentation in high-energy collisions. Let a projectile of mass number <math alttext="A_{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><msub id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">A</mi><mi id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2">𝐴</ci><ci id="S2.SS1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">A_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> that consists of <math alttext="N_{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><msub id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">N</mi><mi id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">𝑁</ci><ci id="S2.SS1.p1.2.m2.1.1.3.cmml" xref="S2.SS1.p1.2.m2.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">N_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> neutrons and <math alttext="Z_{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><msub id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml">Z</mi><mi id="S2.SS1.p1.3.m3.1.1.3" xref="S2.SS1.p1.3.m3.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2">𝑍</ci><ci id="S2.SS1.p1.3.m3.1.1.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">Z_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> protons produce a fragment of mass number <math alttext="A_{F}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><msub id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">A</mi><mi id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝐴</ci><ci id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">A_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> with neutron number <math alttext="N_{F}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><msub id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml">N</mi><mi id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">𝑁</ci><ci id="S2.SS1.p1.5.m5.1.1.3.cmml" xref="S2.SS1.p1.5.m5.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">N_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> and proton number <math alttext="Z_{F}" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><msub id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml"><mi id="S2.SS1.p1.6.m6.1.1.2" xref="S2.SS1.p1.6.m6.1.1.2.cmml">Z</mi><mi id="S2.SS1.p1.6.m6.1.1.3" xref="S2.SS1.p1.6.m6.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><apply id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2">𝑍</ci><ci id="S2.SS1.p1.6.m6.1.1.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">Z_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.1d">italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> in the collision with a target nucleus of mass number <math alttext="A_{T}" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><msub id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.2" xref="S2.SS1.p1.7.m7.1.1.2.cmml">A</mi><mi id="S2.SS1.p1.7.m7.1.1.3" xref="S2.SS1.p1.7.m7.1.1.3.cmml">T</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><apply id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2">𝐴</ci><ci id="S2.SS1.p1.7.m7.1.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">A_{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">italic_A start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math>. The differential cross-section for the primary yield of the fragment with <math alttext="N_{F}" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><msub id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml"><mi id="S2.SS1.p1.8.m8.1.1.2" xref="S2.SS1.p1.8.m8.1.1.2.cmml">N</mi><mi id="S2.SS1.p1.8.m8.1.1.3" xref="S2.SS1.p1.8.m8.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><apply id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m8.1.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.p1.8.m8.1.1.2.cmml" xref="S2.SS1.p1.8.m8.1.1.2">𝑁</ci><ci id="S2.SS1.p1.8.m8.1.1.3.cmml" xref="S2.SS1.p1.8.m8.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">N_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> neutrons and <math alttext="Z_{F}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.1"><semantics id="S2.SS1.p1.9.m9.1a"><msub id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml"><mi id="S2.SS1.p1.9.m9.1.1.2" xref="S2.SS1.p1.9.m9.1.1.2.cmml">Z</mi><mi id="S2.SS1.p1.9.m9.1.1.3" xref="S2.SS1.p1.9.m9.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.1b"><apply id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.1.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1">subscript</csymbol><ci id="S2.SS1.p1.9.m9.1.1.2.cmml" xref="S2.SS1.p1.9.m9.1.1.2">𝑍</ci><ci id="S2.SS1.p1.9.m9.1.1.3.cmml" xref="S2.SS1.p1.9.m9.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.1c">Z_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.1d">italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> protons are written as the product of the density of states <math alttext="\omega(E,Z_{F},A_{F})" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m10.3"><semantics id="S2.SS1.p1.10.m10.3a"><mrow id="S2.SS1.p1.10.m10.3.3" xref="S2.SS1.p1.10.m10.3.3.cmml"><mi id="S2.SS1.p1.10.m10.3.3.4" xref="S2.SS1.p1.10.m10.3.3.4.cmml">ω</mi><mo id="S2.SS1.p1.10.m10.3.3.3" xref="S2.SS1.p1.10.m10.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p1.10.m10.3.3.2.2" xref="S2.SS1.p1.10.m10.3.3.2.3.cmml"><mo id="S2.SS1.p1.10.m10.3.3.2.2.3" stretchy="false" xref="S2.SS1.p1.10.m10.3.3.2.3.cmml">(</mo><mi id="S2.SS1.p1.10.m10.1.1" xref="S2.SS1.p1.10.m10.1.1.cmml">E</mi><mo id="S2.SS1.p1.10.m10.3.3.2.2.4" xref="S2.SS1.p1.10.m10.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p1.10.m10.2.2.1.1.1" xref="S2.SS1.p1.10.m10.2.2.1.1.1.cmml"><mi id="S2.SS1.p1.10.m10.2.2.1.1.1.2" xref="S2.SS1.p1.10.m10.2.2.1.1.1.2.cmml">Z</mi><mi id="S2.SS1.p1.10.m10.2.2.1.1.1.3" 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xref="S2.SS1.p1.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.10.m10.2.2.1.1.1.1.cmml" xref="S2.SS1.p1.10.m10.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.10.m10.2.2.1.1.1.2.cmml" xref="S2.SS1.p1.10.m10.2.2.1.1.1.2">𝑍</ci><ci id="S2.SS1.p1.10.m10.2.2.1.1.1.3.cmml" xref="S2.SS1.p1.10.m10.2.2.1.1.1.3">𝐹</ci></apply><apply id="S2.SS1.p1.10.m10.3.3.2.2.2.cmml" xref="S2.SS1.p1.10.m10.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.10.m10.3.3.2.2.2.1.cmml" xref="S2.SS1.p1.10.m10.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p1.10.m10.3.3.2.2.2.2.cmml" xref="S2.SS1.p1.10.m10.3.3.2.2.2.2">𝐴</ci><ci id="S2.SS1.p1.10.m10.3.3.2.2.2.3.cmml" xref="S2.SS1.p1.10.m10.3.3.2.2.2.3">𝐹</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m10.3c">\omega(E,Z_{F},A_{F})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m10.3d">italic_ω ( italic_E , italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT )</annotation></semantics></math> and an integral over impact parameter <math alttext="b" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m11.1"><semantics id="S2.SS1.p1.11.m11.1a"><mi id="S2.SS1.p1.11.m11.1.1" xref="S2.SS1.p1.11.m11.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m11.1b"><ci id="S2.SS1.p1.11.m11.1.1.cmml" xref="S2.SS1.p1.11.m11.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m11.1c">b</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m11.1d">italic_b</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib36" title="">36</a>]</cite>, <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{d\sigma}{dE}(E,Z_{F},A_{F})=\omega(E,Z_{F},A_{F})\int d^{2}b\,\left[P_{p% }(\mathbf{b})\right]^{Z_{F}}\left[P_{n}(\mathbf{b})\right]^{N_{F}}\left[1-P_{n% }(\mathbf{b})\right]^{N_{P}-N_{F}}\left[1-P_{p}(\mathbf{b})\right]^{Z_{P}-Z_{F% }}." class="ltx_Math" display="block" id="S2.E1.m1.7"><semantics id="S2.E1.m1.7a"><mrow id="S2.E1.m1.7.7.1" xref="S2.E1.m1.7.7.1.1.cmml"><mrow id="S2.E1.m1.7.7.1.1" xref="S2.E1.m1.7.7.1.1.cmml"><mrow id="S2.E1.m1.7.7.1.1.2" xref="S2.E1.m1.7.7.1.1.2.cmml"><mfrac id="S2.E1.m1.7.7.1.1.2.4" xref="S2.E1.m1.7.7.1.1.2.4.cmml"><mrow id="S2.E1.m1.7.7.1.1.2.4.2" xref="S2.E1.m1.7.7.1.1.2.4.2.cmml"><mi id="S2.E1.m1.7.7.1.1.2.4.2.2" xref="S2.E1.m1.7.7.1.1.2.4.2.2.cmml">d</mi><mo id="S2.E1.m1.7.7.1.1.2.4.2.1" xref="S2.E1.m1.7.7.1.1.2.4.2.1.cmml">⁢</mo><mi id="S2.E1.m1.7.7.1.1.2.4.2.3" xref="S2.E1.m1.7.7.1.1.2.4.2.3.cmml">σ</mi></mrow><mrow id="S2.E1.m1.7.7.1.1.2.4.3" xref="S2.E1.m1.7.7.1.1.2.4.3.cmml"><mi id="S2.E1.m1.7.7.1.1.2.4.3.2" xref="S2.E1.m1.7.7.1.1.2.4.3.2.cmml">d</mi><mo id="S2.E1.m1.7.7.1.1.2.4.3.1" xref="S2.E1.m1.7.7.1.1.2.4.3.1.cmml">⁢</mo><mi id="S2.E1.m1.7.7.1.1.2.4.3.3" xref="S2.E1.m1.7.7.1.1.2.4.3.3.cmml">E</mi></mrow></mfrac><mo id="S2.E1.m1.7.7.1.1.2.3" xref="S2.E1.m1.7.7.1.1.2.3.cmml">⁢</mo><mrow id="S2.E1.m1.7.7.1.1.2.2.2" xref="S2.E1.m1.7.7.1.1.2.2.3.cmml"><mo id="S2.E1.m1.7.7.1.1.2.2.2.3" stretchy="false" xref="S2.E1.m1.7.7.1.1.2.2.3.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">E</mi><mo id="S2.E1.m1.7.7.1.1.2.2.2.4" xref="S2.E1.m1.7.7.1.1.2.2.3.cmml">,</mo><msub id="S2.E1.m1.7.7.1.1.1.1.1.1" xref="S2.E1.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.7.7.1.1.1.1.1.1.2" xref="S2.E1.m1.7.7.1.1.1.1.1.1.2.cmml">Z</mi><mi id="S2.E1.m1.7.7.1.1.1.1.1.1.3" xref="S2.E1.m1.7.7.1.1.1.1.1.1.3.cmml">F</mi></msub><mo 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encoding="application/x-tex" id="S2.E1.m1.7c">\frac{d\sigma}{dE}(E,Z_{F},A_{F})=\omega(E,Z_{F},A_{F})\int d^{2}b\,\left[P_{p% }(\mathbf{b})\right]^{Z_{F}}\left[P_{n}(\mathbf{b})\right]^{N_{F}}\left[1-P_{n% }(\mathbf{b})\right]^{N_{P}-N_{F}}\left[1-P_{p}(\mathbf{b})\right]^{Z_{P}-Z_{F% }}.</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.7d">divide start_ARG italic_d italic_σ end_ARG start_ARG italic_d italic_E end_ARG ( italic_E , italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) = italic_ω ( italic_E , italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b [ italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ 1 - italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ 1 - italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(1)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p2"> The integral represents the cross section for each primary fragment state as a sum over impact parameters, considering the probability that <math alttext="Z_{F}" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><msub id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><mi id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml">Z</mi><mi id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p2.1.m1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2">𝑍</ci><ci id="S2.SS1.p2.1.m1.1.1.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">Z_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> protons and <math alttext="N_{F}=A_{F}-Z_{F}" 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id="S2.SS1.p2.2.m2.1.1.3.2.3.cmml" xref="S2.SS1.p2.2.m2.1.1.3.2.3">𝐹</ci></apply><apply id="S2.SS1.p2.2.m2.1.1.3.3.cmml" xref="S2.SS1.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S2.SS1.p2.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S2.SS1.p2.2.m2.1.1.3.3.2">𝑍</ci><ci id="S2.SS1.p2.2.m2.1.1.3.3.3.cmml" xref="S2.SS1.p2.2.m2.1.1.3.3.3">𝐹</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">N_{F}=A_{F}-Z_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT - italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> neutrons from the projectile do not scatter, while the remaining ones do. The terms <math alttext="P_{p}(\mathbf{b})" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mrow id="S2.SS1.p2.3.m3.1.2" xref="S2.SS1.p2.3.m3.1.2.cmml"><msub id="S2.SS1.p2.3.m3.1.2.2" xref="S2.SS1.p2.3.m3.1.2.2.cmml"><mi id="S2.SS1.p2.3.m3.1.2.2.2" xref="S2.SS1.p2.3.m3.1.2.2.2.cmml">P</mi><mi id="S2.SS1.p2.3.m3.1.2.2.3" xref="S2.SS1.p2.3.m3.1.2.2.3.cmml">p</mi></msub><mo id="S2.SS1.p2.3.m3.1.2.1" xref="S2.SS1.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.3.m3.1.2.3.2" xref="S2.SS1.p2.3.m3.1.2.cmml"><mo id="S2.SS1.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.3.m3.1.2.cmml">(</mo><mi id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">𝐛</mi><mo id="S2.SS1.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.2.cmml" xref="S2.SS1.p2.3.m3.1.2"><times id="S2.SS1.p2.3.m3.1.2.1.cmml" xref="S2.SS1.p2.3.m3.1.2.1"></times><apply id="S2.SS1.p2.3.m3.1.2.2.cmml" xref="S2.SS1.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.2.2.1.cmml" xref="S2.SS1.p2.3.m3.1.2.2">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.2.2.2.cmml" xref="S2.SS1.p2.3.m3.1.2.2.2">𝑃</ci><ci id="S2.SS1.p2.3.m3.1.2.2.3.cmml" xref="S2.SS1.p2.3.m3.1.2.2.3">𝑝</ci></apply><ci id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">𝐛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">P_{p}(\mathbf{b})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b )</annotation></semantics></math> and <math alttext="P_{n}(\mathbf{b})" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mrow id="S2.SS1.p2.4.m4.1.2" xref="S2.SS1.p2.4.m4.1.2.cmml"><msub id="S2.SS1.p2.4.m4.1.2.2" xref="S2.SS1.p2.4.m4.1.2.2.cmml"><mi id="S2.SS1.p2.4.m4.1.2.2.2" xref="S2.SS1.p2.4.m4.1.2.2.2.cmml">P</mi><mi id="S2.SS1.p2.4.m4.1.2.2.3" xref="S2.SS1.p2.4.m4.1.2.2.3.cmml">n</mi></msub><mo id="S2.SS1.p2.4.m4.1.2.1" xref="S2.SS1.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m4.1.2.3.2" xref="S2.SS1.p2.4.m4.1.2.cmml"><mo id="S2.SS1.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m4.1.2.cmml">(</mo><mi id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">𝐛</mi><mo id="S2.SS1.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.2.cmml" xref="S2.SS1.p2.4.m4.1.2"><times id="S2.SS1.p2.4.m4.1.2.1.cmml" xref="S2.SS1.p2.4.m4.1.2.1"></times><apply id="S2.SS1.p2.4.m4.1.2.2.cmml" xref="S2.SS1.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.2.2.1.cmml" xref="S2.SS1.p2.4.m4.1.2.2">subscript</csymbol><ci id="S2.SS1.p2.4.m4.1.2.2.2.cmml" xref="S2.SS1.p2.4.m4.1.2.2.2">𝑃</ci><ci id="S2.SS1.p2.4.m4.1.2.2.3.cmml" xref="S2.SS1.p2.4.m4.1.2.2.3">𝑛</ci></apply><ci id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">𝐛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">P_{n}(\mathbf{b})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b )</annotation></semantics></math> denote the probabilities that a proton or neutron from the projectile, respectively, does not collide with the target. <math alttext="\omega(E,Z_{F},A_{F})" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.3"><semantics id="S2.SS1.p2.5.m5.3a"><mrow id="S2.SS1.p2.5.m5.3.3" xref="S2.SS1.p2.5.m5.3.3.cmml"><mi id="S2.SS1.p2.5.m5.3.3.4" xref="S2.SS1.p2.5.m5.3.3.4.cmml">ω</mi><mo id="S2.SS1.p2.5.m5.3.3.3" xref="S2.SS1.p2.5.m5.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p2.5.m5.3.3.2.2" xref="S2.SS1.p2.5.m5.3.3.2.3.cmml"><mo id="S2.SS1.p2.5.m5.3.3.2.2.3" stretchy="false" xref="S2.SS1.p2.5.m5.3.3.2.3.cmml">(</mo><mi id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">E</mi><mo id="S2.SS1.p2.5.m5.3.3.2.2.4" xref="S2.SS1.p2.5.m5.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p2.5.m5.2.2.1.1.1" xref="S2.SS1.p2.5.m5.2.2.1.1.1.cmml"><mi id="S2.SS1.p2.5.m5.2.2.1.1.1.2" xref="S2.SS1.p2.5.m5.2.2.1.1.1.2.cmml">Z</mi><mi id="S2.SS1.p2.5.m5.2.2.1.1.1.3" xref="S2.SS1.p2.5.m5.2.2.1.1.1.3.cmml">F</mi></msub><mo id="S2.SS1.p2.5.m5.3.3.2.2.5" xref="S2.SS1.p2.5.m5.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p2.5.m5.3.3.2.2.2" xref="S2.SS1.p2.5.m5.3.3.2.2.2.cmml"><mi id="S2.SS1.p2.5.m5.3.3.2.2.2.2" xref="S2.SS1.p2.5.m5.3.3.2.2.2.2.cmml">A</mi><mi id="S2.SS1.p2.5.m5.3.3.2.2.2.3" xref="S2.SS1.p2.5.m5.3.3.2.2.2.3.cmml">F</mi></msub><mo id="S2.SS1.p2.5.m5.3.3.2.2.6" stretchy="false" xref="S2.SS1.p2.5.m5.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.3b"><apply id="S2.SS1.p2.5.m5.3.3.cmml" xref="S2.SS1.p2.5.m5.3.3"><times id="S2.SS1.p2.5.m5.3.3.3.cmml" xref="S2.SS1.p2.5.m5.3.3.3"></times><ci id="S2.SS1.p2.5.m5.3.3.4.cmml" xref="S2.SS1.p2.5.m5.3.3.4">𝜔</ci><vector id="S2.SS1.p2.5.m5.3.3.2.3.cmml" xref="S2.SS1.p2.5.m5.3.3.2.2"><ci id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1">𝐸</ci><apply id="S2.SS1.p2.5.m5.2.2.1.1.1.cmml" xref="S2.SS1.p2.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS1.p2.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.5.m5.2.2.1.1.1.2.cmml" xref="S2.SS1.p2.5.m5.2.2.1.1.1.2">𝑍</ci><ci id="S2.SS1.p2.5.m5.2.2.1.1.1.3.cmml" xref="S2.SS1.p2.5.m5.2.2.1.1.1.3">𝐹</ci></apply><apply id="S2.SS1.p2.5.m5.3.3.2.2.2.cmml" xref="S2.SS1.p2.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.3.3.2.2.2.1.cmml" xref="S2.SS1.p2.5.m5.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.5.m5.3.3.2.2.2.2.cmml" xref="S2.SS1.p2.5.m5.3.3.2.2.2.2">𝐴</ci><ci id="S2.SS1.p2.5.m5.3.3.2.2.2.3.cmml" xref="S2.SS1.p2.5.m5.3.3.2.2.2.3">𝐹</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.3c">\omega(E,Z_{F},A_{F})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.3d">italic_ω ( italic_E , italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT )</annotation></semantics></math> is the density of states, which is calculated by counting all combinations of projectile holes consistent with the fragment’s charge and neutron numbers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib31" title="">31</a>]</cite>. Thus, the energy-integrated primary cross section <math alttext="\sigma(E,Z_{F},A_{F})" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.3"><semantics id="S2.SS1.p2.6.m6.3a"><mrow id="S2.SS1.p2.6.m6.3.3" xref="S2.SS1.p2.6.m6.3.3.cmml"><mi id="S2.SS1.p2.6.m6.3.3.4" xref="S2.SS1.p2.6.m6.3.3.4.cmml">σ</mi><mo id="S2.SS1.p2.6.m6.3.3.3" xref="S2.SS1.p2.6.m6.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p2.6.m6.3.3.2.2" xref="S2.SS1.p2.6.m6.3.3.2.3.cmml"><mo id="S2.SS1.p2.6.m6.3.3.2.2.3" stretchy="false" xref="S2.SS1.p2.6.m6.3.3.2.3.cmml">(</mo><mi id="S2.SS1.p2.6.m6.1.1" xref="S2.SS1.p2.6.m6.1.1.cmml">E</mi><mo id="S2.SS1.p2.6.m6.3.3.2.2.4" xref="S2.SS1.p2.6.m6.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p2.6.m6.2.2.1.1.1" xref="S2.SS1.p2.6.m6.2.2.1.1.1.cmml"><mi id="S2.SS1.p2.6.m6.2.2.1.1.1.2" xref="S2.SS1.p2.6.m6.2.2.1.1.1.2.cmml">Z</mi><mi id="S2.SS1.p2.6.m6.2.2.1.1.1.3" xref="S2.SS1.p2.6.m6.2.2.1.1.1.3.cmml">F</mi></msub><mo id="S2.SS1.p2.6.m6.3.3.2.2.5" xref="S2.SS1.p2.6.m6.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p2.6.m6.3.3.2.2.2" xref="S2.SS1.p2.6.m6.3.3.2.2.2.cmml"><mi id="S2.SS1.p2.6.m6.3.3.2.2.2.2" xref="S2.SS1.p2.6.m6.3.3.2.2.2.2.cmml">A</mi><mi id="S2.SS1.p2.6.m6.3.3.2.2.2.3" xref="S2.SS1.p2.6.m6.3.3.2.2.2.3.cmml">F</mi></msub><mo id="S2.SS1.p2.6.m6.3.3.2.2.6" stretchy="false" xref="S2.SS1.p2.6.m6.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m6.3b"><apply id="S2.SS1.p2.6.m6.3.3.cmml" xref="S2.SS1.p2.6.m6.3.3"><times id="S2.SS1.p2.6.m6.3.3.3.cmml" xref="S2.SS1.p2.6.m6.3.3.3"></times><ci id="S2.SS1.p2.6.m6.3.3.4.cmml" xref="S2.SS1.p2.6.m6.3.3.4">𝜎</ci><vector id="S2.SS1.p2.6.m6.3.3.2.3.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2"><ci id="S2.SS1.p2.6.m6.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1">𝐸</ci><apply id="S2.SS1.p2.6.m6.2.2.1.1.1.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.6.m6.2.2.1.1.1.2.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1.1.2">𝑍</ci><ci id="S2.SS1.p2.6.m6.2.2.1.1.1.3.cmml" xref="S2.SS1.p2.6.m6.2.2.1.1.1.3">𝐹</ci></apply><apply id="S2.SS1.p2.6.m6.3.3.2.2.2.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.3.3.2.2.2.1.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.6.m6.3.3.2.2.2.2.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2.2.2">𝐴</ci><ci id="S2.SS1.p2.6.m6.3.3.2.2.2.3.cmml" xref="S2.SS1.p2.6.m6.3.3.2.2.2.3">𝐹</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.3c">\sigma(E,Z_{F},A_{F})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.3d">italic_σ ( italic_E , italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT )</annotation></semantics></math> inherently includes the bionomial factor <math alttext="\binom{Z_{P}}{Z_{F}}\binom{N_{P}}{N_{F}}" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.4"><semantics id="S2.SS1.p2.7.m7.4a"><mrow id="S2.SS1.p2.7.m7.4.5" xref="S2.SS1.p2.7.m7.4.5.cmml"><mrow id="S2.SS1.p2.7.m7.2.2.4" xref="S2.SS1.p2.7.m7.2.2.3.cmml"><mo id="S2.SS1.p2.7.m7.2.2.4.1" xref="S2.SS1.p2.7.m7.2.2.3.1.cmml">(</mo><mfrac id="S2.SS1.p2.7.m7.2.2.2.2" linethickness="0pt" xref="S2.SS1.p2.7.m7.2.2.3.cmml"><msub id="S2.SS1.p2.7.m7.1.1.1.1.1.1" xref="S2.SS1.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.7.m7.1.1.1.1.1.1.3" xref="S2.SS1.p2.7.m7.1.1.1.1.1.1.3.cmml">Z</mi><mi id="S2.SS1.p2.7.m7.1.1.1.1.1.1.4" xref="S2.SS1.p2.7.m7.1.1.1.1.1.1.4.cmml">P</mi></msub><msub id="S2.SS1.p2.7.m7.2.2.2.2.2.1" xref="S2.SS1.p2.7.m7.2.2.2.2.2.1.cmml"><mi id="S2.SS1.p2.7.m7.2.2.2.2.2.1.3" xref="S2.SS1.p2.7.m7.2.2.2.2.2.1.3.cmml">Z</mi><mi id="S2.SS1.p2.7.m7.2.2.2.2.2.1.4" xref="S2.SS1.p2.7.m7.2.2.2.2.2.1.4.cmml">F</mi></msub></mfrac><mo id="S2.SS1.p2.7.m7.2.2.4.2" xref="S2.SS1.p2.7.m7.2.2.3.1.cmml">)</mo></mrow><mo id="S2.SS1.p2.7.m7.4.5.1" 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xref="S2.SS1.p2.7.m7.2.2.2.2.2.1.4">𝐹</ci></apply></apply><apply id="S2.SS1.p2.7.m7.4.4.3.cmml" xref="S2.SS1.p2.7.m7.4.4.4"><csymbol cd="latexml" id="S2.SS1.p2.7.m7.4.4.3.1.cmml" xref="S2.SS1.p2.7.m7.4.4.4.1">binomial</csymbol><apply id="S2.SS1.p2.7.m7.3.3.1.1.1.1.cmml" xref="S2.SS1.p2.7.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p2.7.m7.3.3.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.7.m7.3.3.1.1.1.1.3.cmml" xref="S2.SS1.p2.7.m7.3.3.1.1.1.1.3">𝑁</ci><ci id="S2.SS1.p2.7.m7.3.3.1.1.1.1.4.cmml" xref="S2.SS1.p2.7.m7.3.3.1.1.1.1.4">𝑃</ci></apply><apply id="S2.SS1.p2.7.m7.4.4.2.2.2.1.cmml" xref="S2.SS1.p2.7.m7.4.4.2.2.2.1"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.2.2.2.1.2.cmml" xref="S2.SS1.p2.7.m7.4.4.2.2.2.1">subscript</csymbol><ci id="S2.SS1.p2.7.m7.4.4.2.2.2.1.3.cmml" xref="S2.SS1.p2.7.m7.4.4.2.2.2.1.3">𝑁</ci><ci id="S2.SS1.p2.7.m7.4.4.2.2.2.1.4.cmml" xref="S2.SS1.p2.7.m7.4.4.2.2.2.1.4">𝐹</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.4c">\binom{Z_{P}}{Z_{F}}\binom{N_{P}}{N_{F}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.4d">( FRACOP start_ARG italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_ARG start_ARG italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG ) ( FRACOP start_ARG italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_ARG start_ARG italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math>, a component that has been incorporated in several previous models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib37" title="">37</a>]</cite>. <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma(E,Z_{F},A_{F})=\binom{N_{P}}{N_{F}}\binom{Z_{P}}{Z_{F}}\int d^{2}b\,% \left[P_{p}(\mathbf{b})\right]^{Z_{F}}\left[P_{n}(\mathbf{b})\right]^{N_{F}}% \left[1-P_{n}(\mathbf{b})\right]^{N_{P}-N_{F}}\left[1-P_{p}(\mathbf{b})\right]% ^{Z_{P}-Z_{F}}." class="ltx_Math" display="block" id="S2.E2.m1.10"><semantics id="S2.E2.m1.10a"><mrow id="S2.E2.m1.10.10.1" xref="S2.E2.m1.10.10.1.1.cmml"><mrow id="S2.E2.m1.10.10.1.1" xref="S2.E2.m1.10.10.1.1.cmml"><mrow id="S2.E2.m1.10.10.1.1.2" xref="S2.E2.m1.10.10.1.1.2.cmml"><mi id="S2.E2.m1.10.10.1.1.2.4" xref="S2.E2.m1.10.10.1.1.2.4.cmml">σ</mi><mo id="S2.E2.m1.10.10.1.1.2.3" xref="S2.E2.m1.10.10.1.1.2.3.cmml">⁢</mo><mrow id="S2.E2.m1.10.10.1.1.2.2.2" xref="S2.E2.m1.10.10.1.1.2.2.3.cmml"><mo id="S2.E2.m1.10.10.1.1.2.2.2.3" stretchy="false" xref="S2.E2.m1.10.10.1.1.2.2.3.cmml">(</mo><mi id="S2.E2.m1.5.5" xref="S2.E2.m1.5.5.cmml">E</mi><mo id="S2.E2.m1.10.10.1.1.2.2.2.4" xref="S2.E2.m1.10.10.1.1.2.2.3.cmml">,</mo><msub id="S2.E2.m1.10.10.1.1.1.1.1.1" xref="S2.E2.m1.10.10.1.1.1.1.1.1.cmml"><mi id="S2.E2.m1.10.10.1.1.1.1.1.1.2" xref="S2.E2.m1.10.10.1.1.1.1.1.1.2.cmml">Z</mi><mi id="S2.E2.m1.10.10.1.1.1.1.1.1.3" xref="S2.E2.m1.10.10.1.1.1.1.1.1.3.cmml">F</mi></msub><mo id="S2.E2.m1.10.10.1.1.2.2.2.5" xref="S2.E2.m1.10.10.1.1.2.2.3.cmml">,</mo><msub id="S2.E2.m1.10.10.1.1.2.2.2.2" xref="S2.E2.m1.10.10.1.1.2.2.2.2.cmml"><mi id="S2.E2.m1.10.10.1.1.2.2.2.2.2" xref="S2.E2.m1.10.10.1.1.2.2.2.2.2.cmml">A</mi><mi id="S2.E2.m1.10.10.1.1.2.2.2.2.3" xref="S2.E2.m1.10.10.1.1.2.2.2.2.3.cmml">F</mi></msub><mo id="S2.E2.m1.10.10.1.1.2.2.2.6" stretchy="false" xref="S2.E2.m1.10.10.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.10.10.1.1.7" xref="S2.E2.m1.10.10.1.1.7.cmml">=</mo><mrow id="S2.E2.m1.10.10.1.1.6" xref="S2.E2.m1.10.10.1.1.6.cmml"><mrow id="S2.E2.m1.2.2.4" xref="S2.E2.m1.2.2.3.cmml"><mo 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italic_P end_POSTSUBSCRIPT end_ARG start_ARG italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG ) ( FRACOP start_ARG italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_ARG start_ARG italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG ) ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b [ italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ 1 - italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ 1 - italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(2)</td> </tr></tbody> </table> Now following the work in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib38" title="">38</a>]</cite> we can calculate the required cross sections as follows. </div> <div class="ltx_para" id="S2.SS1.p3"> The total reaction cross section is obtained by taking the sum of cross sections in all events where at least one nucleon is removed, which is given as <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx1"> <tbody id="S2.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td 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encoding="application/x-tex" id="S2.E3.m1.3c">\displaystyle\sigma_{R}=\int d^{2}b\left[1-[P_{p}(\textbf{b})]^{Z_{P}}[P_{n}(% \textbf{b})]^{N_{P}}\right].</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.3d">italic_σ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT = ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b [ 1 - [ italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( b ) ] start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(3)</td> </tr></tbody> </table> The partial neutron removal cross section (<math alttext="\sigma_{-xn}" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><msub id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">σ</mi><mrow id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mo id="S2.SS1.p3.1.m1.1.1.3a" xref="S2.SS1.p3.1.m1.1.1.3.cmml">−</mo><mrow id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2.2" xref="S2.SS1.p3.1.m1.1.1.3.2.2.cmml">x</mi><mo id="S2.SS1.p3.1.m1.1.1.3.2.1" xref="S2.SS1.p3.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS1.p3.1.m1.1.1.3.2.3" xref="S2.SS1.p3.1.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">𝜎</ci><apply id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3"><minus id="S2.SS1.p3.1.m1.1.1.3.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3"></minus><apply id="S2.SS1.p3.1.m1.1.1.3.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2"><times id="S2.SS1.p3.1.m1.1.1.3.2.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2.1"></times><ci id="S2.SS1.p3.1.m1.1.1.3.2.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2.2">𝑥</ci><ci id="S2.SS1.p3.1.m1.1.1.3.2.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\sigma_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) can be calculated from Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E2" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">2</a>) by setting <math alttext="Z_{F}=Z_{P}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mrow id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml"><msub id="S2.SS1.p3.2.m2.1.1.2" xref="S2.SS1.p3.2.m2.1.1.2.cmml"><mi id="S2.SS1.p3.2.m2.1.1.2.2" xref="S2.SS1.p3.2.m2.1.1.2.2.cmml">Z</mi><mi id="S2.SS1.p3.2.m2.1.1.2.3" xref="S2.SS1.p3.2.m2.1.1.2.3.cmml">F</mi></msub><mo id="S2.SS1.p3.2.m2.1.1.1" xref="S2.SS1.p3.2.m2.1.1.1.cmml">=</mo><msub id="S2.SS1.p3.2.m2.1.1.3" xref="S2.SS1.p3.2.m2.1.1.3.cmml"><mi id="S2.SS1.p3.2.m2.1.1.3.2" xref="S2.SS1.p3.2.m2.1.1.3.2.cmml">Z</mi><mi id="S2.SS1.p3.2.m2.1.1.3.3" xref="S2.SS1.p3.2.m2.1.1.3.3.cmml">P</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><apply id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1"><eq id="S2.SS1.p3.2.m2.1.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1.1"></eq><apply id="S2.SS1.p3.2.m2.1.1.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m2.1.1.2.1.cmml" xref="S2.SS1.p3.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.2.m2.1.1.2.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2.2">𝑍</ci><ci id="S2.SS1.p3.2.m2.1.1.2.3.cmml" xref="S2.SS1.p3.2.m2.1.1.2.3">𝐹</ci></apply><apply id="S2.SS1.p3.2.m2.1.1.3.cmml" xref="S2.SS1.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m2.1.1.3.1.cmml" xref="S2.SS1.p3.2.m2.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.2.m2.1.1.3.2.cmml" xref="S2.SS1.p3.2.m2.1.1.3.2">𝑍</ci><ci id="S2.SS1.p3.2.m2.1.1.3.3.cmml" xref="S2.SS1.p3.2.m2.1.1.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">Z_{F}=Z_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_Z start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>. The cross section for removal of <math alttext="x" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><mi id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><ci id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">italic_x</annotation></semantics></math> neutrons from the projectile <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{-xn}=\binom{N_{P}}{x}\int d^{2}b\,[P_{p}(\mathbf{b})]^{Z_{P}}[P_{n}(% \mathbf{b})]^{N_{P}-x}[1-P_{n}(\mathbf{b})]^{x}." class="ltx_Math" display="block" id="S2.E4.m1.6"><semantics id="S2.E4.m1.6a"><mrow id="S2.E4.m1.6.6.1" xref="S2.E4.m1.6.6.1.1.cmml"><mrow id="S2.E4.m1.6.6.1.1" xref="S2.E4.m1.6.6.1.1.cmml"><msub id="S2.E4.m1.6.6.1.1.5" xref="S2.E4.m1.6.6.1.1.5.cmml"><mi id="S2.E4.m1.6.6.1.1.5.2" xref="S2.E4.m1.6.6.1.1.5.2.cmml">σ</mi><mrow id="S2.E4.m1.6.6.1.1.5.3" xref="S2.E4.m1.6.6.1.1.5.3.cmml"><mo id="S2.E4.m1.6.6.1.1.5.3a" xref="S2.E4.m1.6.6.1.1.5.3.cmml">−</mo><mrow id="S2.E4.m1.6.6.1.1.5.3.2" xref="S2.E4.m1.6.6.1.1.5.3.2.cmml"><mi id="S2.E4.m1.6.6.1.1.5.3.2.2" xref="S2.E4.m1.6.6.1.1.5.3.2.2.cmml">x</mi><mo id="S2.E4.m1.6.6.1.1.5.3.2.1" xref="S2.E4.m1.6.6.1.1.5.3.2.1.cmml">⁢</mo><mi id="S2.E4.m1.6.6.1.1.5.3.2.3" xref="S2.E4.m1.6.6.1.1.5.3.2.3.cmml">n</mi></mrow></mrow></msub><mo id="S2.E4.m1.6.6.1.1.4" xref="S2.E4.m1.6.6.1.1.4.cmml">=</mo><mrow id="S2.E4.m1.6.6.1.1.3" xref="S2.E4.m1.6.6.1.1.3.cmml"><mrow id="S2.E4.m1.2.2.4" xref="S2.E4.m1.2.2.3.cmml"><mo id="S2.E4.m1.2.2.4.1" xref="S2.E4.m1.2.2.3.1.cmml">(</mo><mfrac id="S2.E4.m1.2.2.2.2" linethickness="0pt" xref="S2.E4.m1.2.2.3.cmml"><msub id="S2.E4.m1.1.1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.1.1.cmml"><mi id="S2.E4.m1.1.1.1.1.1.1.3" xref="S2.E4.m1.1.1.1.1.1.1.3.cmml">N</mi><mi id="S2.E4.m1.1.1.1.1.1.1.4" xref="S2.E4.m1.1.1.1.1.1.1.4.cmml">P</mi></msub><mi id="S2.E4.m1.2.2.2.2.2.1" xref="S2.E4.m1.2.2.2.2.2.1.cmml">x</mi></mfrac><mo id="S2.E4.m1.2.2.4.2" xref="S2.E4.m1.2.2.3.1.cmml">)</mo></mrow><mo id="S2.E4.m1.6.6.1.1.3.4" xref="S2.E4.m1.6.6.1.1.3.4.cmml">⁢</mo><mrow id="S2.E4.m1.6.6.1.1.3.3" xref="S2.E4.m1.6.6.1.1.3.3.cmml"><mo id="S2.E4.m1.6.6.1.1.3.3.4" xref="S2.E4.m1.6.6.1.1.3.3.4.cmml">∫</mo><mrow id="S2.E4.m1.6.6.1.1.3.3.3" xref="S2.E4.m1.6.6.1.1.3.3.3.cmml"><msup id="S2.E4.m1.6.6.1.1.3.3.3.5" xref="S2.E4.m1.6.6.1.1.3.3.3.5.cmml"><mi id="S2.E4.m1.6.6.1.1.3.3.3.5.2" xref="S2.E4.m1.6.6.1.1.3.3.3.5.2.cmml">d</mi><mn id="S2.E4.m1.6.6.1.1.3.3.3.5.3" xref="S2.E4.m1.6.6.1.1.3.3.3.5.3.cmml">2</mn></msup><mo id="S2.E4.m1.6.6.1.1.3.3.3.4" xref="S2.E4.m1.6.6.1.1.3.3.3.4.cmml">⁢</mo><mi id="S2.E4.m1.6.6.1.1.3.3.3.6" xref="S2.E4.m1.6.6.1.1.3.3.3.6.cmml">b</mi><mo id="S2.E4.m1.6.6.1.1.3.3.3.4a" lspace="0.170em" xref="S2.E4.m1.6.6.1.1.3.3.3.4.cmml">⁢</mo><msup id="S2.E4.m1.6.6.1.1.1.1.1.1" xref="S2.E4.m1.6.6.1.1.1.1.1.1.cmml"><mrow id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><msub id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.2" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.2.2" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.2.2.cmml">P</mi><mi id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.2.3" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.cmml">p</mi></msub><mo id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.3.2" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S2.E4.m1.3.3" xref="S2.E4.m1.3.3.cmml">𝐛</mi><mo id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E4.m1.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E4.m1.6.6.1.1.1.1.1.1.1.2.1.cmml">]</mo></mrow><msub id="S2.E4.m1.6.6.1.1.1.1.1.1.3" xref="S2.E4.m1.6.6.1.1.1.1.1.1.3.cmml"><mi id="S2.E4.m1.6.6.1.1.1.1.1.1.3.2" xref="S2.E4.m1.6.6.1.1.1.1.1.1.3.2.cmml">Z</mi><mi id="S2.E4.m1.6.6.1.1.1.1.1.1.3.3" xref="S2.E4.m1.6.6.1.1.1.1.1.1.3.3.cmml">P</mi></msub></msup><mo id="S2.E4.m1.6.6.1.1.3.3.3.4b" xref="S2.E4.m1.6.6.1.1.3.3.3.4.cmml">⁢</mo><msup id="S2.E4.m1.6.6.1.1.2.2.2.2" xref="S2.E4.m1.6.6.1.1.2.2.2.2.cmml"><mrow id="S2.E4.m1.6.6.1.1.2.2.2.2.1.1" xref="S2.E4.m1.6.6.1.1.2.2.2.2.1.2.cmml"><mo id="S2.E4.m1.6.6.1.1.2.2.2.2.1.1.2" stretchy="false" 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start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_ARG start_ARG italic_x end_ARG ) ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b [ italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_x end_POSTSUPERSCRIPT [ 1 - italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_b ) ] start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(4)</td> </tr></tbody> </table> The total neutron removal cross section is calculated by adding all events with at least one neutron removed without disturbing the proton number in the projectile <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx2"> <tbody id="S2.E5"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sigma_{\Delta N}" class="ltx_Math" display="inline" id="S2.Ex1.m1.1"><semantics id="S2.Ex1.m1.1a"><msub id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.2" xref="S2.Ex1.m1.1.1.2.cmml">σ</mi><mrow id="S2.Ex1.m1.1.1.3" xref="S2.Ex1.m1.1.1.3.cmml"><mi id="S2.Ex1.m1.1.1.3.2" mathvariant="normal" xref="S2.Ex1.m1.1.1.3.2.cmml">Δ</mi><mo id="S2.Ex1.m1.1.1.3.1" xref="S2.Ex1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex1.m1.1.1.3.3" xref="S2.Ex1.m1.1.1.3.3.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.1b"><apply id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.1.cmml" xref="S2.Ex1.m1.1.1">subscript</csymbol><ci id="S2.Ex1.m1.1.1.2.cmml" xref="S2.Ex1.m1.1.1.2">𝜎</ci><apply id="S2.Ex1.m1.1.1.3.cmml" xref="S2.Ex1.m1.1.1.3"><times id="S2.Ex1.m1.1.1.3.1.cmml" xref="S2.Ex1.m1.1.1.3.1"></times><ci id="S2.Ex1.m1.1.1.3.2.cmml" xref="S2.Ex1.m1.1.1.3.2">Δ</ci><ci id="S2.Ex1.m1.1.1.3.3.cmml" xref="S2.Ex1.m1.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.1c">\displaystyle\sigma_{\Delta N}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_Δ italic_N end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex1.m2.1"><semantics id="S2.Ex1.m2.1a"><mo id="S2.Ex1.m2.1.1" xref="S2.Ex1.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex1.m2.1b"><eq id="S2.Ex1.m2.1.1.cmml" xref="S2.Ex1.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{N_{F}=0}^{N_{P}-1}\sigma(N_{F},Z_{P})" class="ltx_Math" display="inline" id="S2.Ex1.m3.2"><semantics id="S2.Ex1.m3.2a"><mrow id="S2.Ex1.m3.2.2" xref="S2.Ex1.m3.2.2.cmml"><mstyle displaystyle="true" id="S2.Ex1.m3.2.2.3" xref="S2.Ex1.m3.2.2.3.cmml"><munderover id="S2.Ex1.m3.2.2.3a" xref="S2.Ex1.m3.2.2.3.cmml"><mo id="S2.Ex1.m3.2.2.3.2.2" movablelimits="false" xref="S2.Ex1.m3.2.2.3.2.2.cmml">∑</mo><mrow id="S2.Ex1.m3.2.2.3.2.3" xref="S2.Ex1.m3.2.2.3.2.3.cmml"><msub id="S2.Ex1.m3.2.2.3.2.3.2" xref="S2.Ex1.m3.2.2.3.2.3.2.cmml"><mi id="S2.Ex1.m3.2.2.3.2.3.2.2" xref="S2.Ex1.m3.2.2.3.2.3.2.2.cmml">N</mi><mi id="S2.Ex1.m3.2.2.3.2.3.2.3" xref="S2.Ex1.m3.2.2.3.2.3.2.3.cmml">F</mi></msub><mo id="S2.Ex1.m3.2.2.3.2.3.1" xref="S2.Ex1.m3.2.2.3.2.3.1.cmml">=</mo><mn id="S2.Ex1.m3.2.2.3.2.3.3" xref="S2.Ex1.m3.2.2.3.2.3.3.cmml">0</mn></mrow><mrow id="S2.Ex1.m3.2.2.3.3" xref="S2.Ex1.m3.2.2.3.3.cmml"><msub id="S2.Ex1.m3.2.2.3.3.2" xref="S2.Ex1.m3.2.2.3.3.2.cmml"><mi id="S2.Ex1.m3.2.2.3.3.2.2" xref="S2.Ex1.m3.2.2.3.3.2.2.cmml">N</mi><mi id="S2.Ex1.m3.2.2.3.3.2.3" xref="S2.Ex1.m3.2.2.3.3.2.3.cmml">P</mi></msub><mo id="S2.Ex1.m3.2.2.3.3.1" xref="S2.Ex1.m3.2.2.3.3.1.cmml">−</mo><mn id="S2.Ex1.m3.2.2.3.3.3" xref="S2.Ex1.m3.2.2.3.3.3.cmml">1</mn></mrow></munderover></mstyle><mrow id="S2.Ex1.m3.2.2.2" xref="S2.Ex1.m3.2.2.2.cmml"><mi id="S2.Ex1.m3.2.2.2.4" xref="S2.Ex1.m3.2.2.2.4.cmml">σ</mi><mo id="S2.Ex1.m3.2.2.2.3" xref="S2.Ex1.m3.2.2.2.3.cmml">⁢</mo><mrow id="S2.Ex1.m3.2.2.2.2.2" xref="S2.Ex1.m3.2.2.2.2.3.cmml"><mo id="S2.Ex1.m3.2.2.2.2.2.3" stretchy="false" xref="S2.Ex1.m3.2.2.2.2.3.cmml">(</mo><msub id="S2.Ex1.m3.1.1.1.1.1.1" xref="S2.Ex1.m3.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m3.1.1.1.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.1.1.2.cmml">N</mi><mi id="S2.Ex1.m3.1.1.1.1.1.1.3" xref="S2.Ex1.m3.1.1.1.1.1.1.3.cmml">F</mi></msub><mo id="S2.Ex1.m3.2.2.2.2.2.4" xref="S2.Ex1.m3.2.2.2.2.3.cmml">,</mo><msub id="S2.Ex1.m3.2.2.2.2.2.2" xref="S2.Ex1.m3.2.2.2.2.2.2.cmml"><mi id="S2.Ex1.m3.2.2.2.2.2.2.2" xref="S2.Ex1.m3.2.2.2.2.2.2.2.cmml">Z</mi><mi id="S2.Ex1.m3.2.2.2.2.2.2.3" xref="S2.Ex1.m3.2.2.2.2.2.2.3.cmml">P</mi></msub><mo id="S2.Ex1.m3.2.2.2.2.2.5" stretchy="false" xref="S2.Ex1.m3.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m3.2b"><apply id="S2.Ex1.m3.2.2.cmml" xref="S2.Ex1.m3.2.2"><apply id="S2.Ex1.m3.2.2.3.cmml" xref="S2.Ex1.m3.2.2.3"><csymbol cd="ambiguous" id="S2.Ex1.m3.2.2.3.1.cmml" xref="S2.Ex1.m3.2.2.3">superscript</csymbol><apply id="S2.Ex1.m3.2.2.3.2.cmml" xref="S2.Ex1.m3.2.2.3"><csymbol cd="ambiguous" id="S2.Ex1.m3.2.2.3.2.1.cmml" xref="S2.Ex1.m3.2.2.3">subscript</csymbol><sum id="S2.Ex1.m3.2.2.3.2.2.cmml" xref="S2.Ex1.m3.2.2.3.2.2"></sum><apply id="S2.Ex1.m3.2.2.3.2.3.cmml" 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xref="S2.Ex1.m3.2.2.2.2.2.2.3">𝑃</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m3.2c">\displaystyle\sum_{N_{F}=0}^{N_{P}-1}\sigma(N_{F},Z_{P})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m3.2d">∑ start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_σ ( italic_N start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(5)</td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math 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ltx_eqn_center_padright"></td> </tr> </tbody> </table> The charge-changing cross section in the abrasion is measured by taking all the events in which at least one proton is removed and is written by subtracting the total neutron removal cross section from the total reaction cross section, <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx3"> <tbody id="S2.E6"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sigma_{\rm cc}" class="ltx_Math" display="inline" id="S2.Ex2.m1.1"><semantics id="S2.Ex2.m1.1a"><msub id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml"><mi id="S2.Ex2.m1.1.1.2" xref="S2.Ex2.m1.1.1.2.cmml">σ</mi><mi id="S2.Ex2.m1.1.1.3" xref="S2.Ex2.m1.1.1.3.cmml">cc</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.1b"><apply id="S2.Ex2.m1.1.1.cmml" xref="S2.Ex2.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex2.m1.1.1.1.cmml" xref="S2.Ex2.m1.1.1">subscript</csymbol><ci id="S2.Ex2.m1.1.1.2.cmml" xref="S2.Ex2.m1.1.1.2">𝜎</ci><ci id="S2.Ex2.m1.1.1.3.cmml" xref="S2.Ex2.m1.1.1.3">cc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.1c">\displaystyle\sigma_{\rm cc}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex2.m2.1"><semantics id="S2.Ex2.m2.1a"><mo id="S2.Ex2.m2.1.1" xref="S2.Ex2.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex2.m2.1b"><eq id="S2.Ex2.m2.1.1.cmml" xref="S2.Ex2.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sigma_{R}-\sigma_{\Delta N}" class="ltx_Math" display="inline" id="S2.Ex2.m3.1"><semantics id="S2.Ex2.m3.1a"><mrow id="S2.Ex2.m3.1.1" xref="S2.Ex2.m3.1.1.cmml"><msub id="S2.Ex2.m3.1.1.2" xref="S2.Ex2.m3.1.1.2.cmml"><mi id="S2.Ex2.m3.1.1.2.2" xref="S2.Ex2.m3.1.1.2.2.cmml">σ</mi><mi id="S2.Ex2.m3.1.1.2.3" xref="S2.Ex2.m3.1.1.2.3.cmml">R</mi></msub><mo id="S2.Ex2.m3.1.1.1" xref="S2.Ex2.m3.1.1.1.cmml">−</mo><msub id="S2.Ex2.m3.1.1.3" xref="S2.Ex2.m3.1.1.3.cmml"><mi id="S2.Ex2.m3.1.1.3.2" xref="S2.Ex2.m3.1.1.3.2.cmml">σ</mi><mrow id="S2.Ex2.m3.1.1.3.3" xref="S2.Ex2.m3.1.1.3.3.cmml"><mi id="S2.Ex2.m3.1.1.3.3.2" mathvariant="normal" xref="S2.Ex2.m3.1.1.3.3.2.cmml">Δ</mi><mo id="S2.Ex2.m3.1.1.3.3.1" xref="S2.Ex2.m3.1.1.3.3.1.cmml">⁢</mo><mi id="S2.Ex2.m3.1.1.3.3.3" xref="S2.Ex2.m3.1.1.3.3.3.cmml">N</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m3.1b"><apply id="S2.Ex2.m3.1.1.cmml" xref="S2.Ex2.m3.1.1"><minus id="S2.Ex2.m3.1.1.1.cmml" xref="S2.Ex2.m3.1.1.1"></minus><apply id="S2.Ex2.m3.1.1.2.cmml" xref="S2.Ex2.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Ex2.m3.1.1.2.1.cmml" xref="S2.Ex2.m3.1.1.2">subscript</csymbol><ci id="S2.Ex2.m3.1.1.2.2.cmml" xref="S2.Ex2.m3.1.1.2.2">𝜎</ci><ci id="S2.Ex2.m3.1.1.2.3.cmml" xref="S2.Ex2.m3.1.1.2.3">𝑅</ci></apply><apply id="S2.Ex2.m3.1.1.3.cmml" xref="S2.Ex2.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Ex2.m3.1.1.3.1.cmml" xref="S2.Ex2.m3.1.1.3">subscript</csymbol><ci id="S2.Ex2.m3.1.1.3.2.cmml" xref="S2.Ex2.m3.1.1.3.2">𝜎</ci><apply id="S2.Ex2.m3.1.1.3.3.cmml" xref="S2.Ex2.m3.1.1.3.3"><times id="S2.Ex2.m3.1.1.3.3.1.cmml" xref="S2.Ex2.m3.1.1.3.3.1"></times><ci id="S2.Ex2.m3.1.1.3.3.2.cmml" xref="S2.Ex2.m3.1.1.3.3.2">Δ</ci><ci id="S2.Ex2.m3.1.1.3.3.3.cmml" xref="S2.Ex2.m3.1.1.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m3.1c">\displaystyle\sigma_{R}-\sigma_{\Delta N}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m3.1d">italic_σ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT - italic_σ start_POSTSUBSCRIPT roman_Δ italic_N end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(6)</td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><mo id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E6.m1.1b"><eq id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\int d^{2}b\left\{1-[P_{p}(\textbf{b})]^{Z_{P}}\right\}." class="ltx_Math" display="inline" id="S2.E6.m2.2"><semantics id="S2.E6.m2.2a"><mrow id="S2.E6.m2.2.2.1" xref="S2.E6.m2.2.2.1.1.cmml"><mstyle displaystyle="true" id="S2.E6.m2.2.2.1.1" xref="S2.E6.m2.2.2.1.1.cmml"><mrow id="S2.E6.m2.2.2.1.1a" xref="S2.E6.m2.2.2.1.1.cmml"><mo id="S2.E6.m2.2.2.1.1.2" xref="S2.E6.m2.2.2.1.1.2.cmml">∫</mo><mrow id="S2.E6.m2.2.2.1.1.1" xref="S2.E6.m2.2.2.1.1.1.cmml"><msup id="S2.E6.m2.2.2.1.1.1.3" xref="S2.E6.m2.2.2.1.1.1.3.cmml"><mi id="S2.E6.m2.2.2.1.1.1.3.2" xref="S2.E6.m2.2.2.1.1.1.3.2.cmml">d</mi><mn id="S2.E6.m2.2.2.1.1.1.3.3" xref="S2.E6.m2.2.2.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E6.m2.2.2.1.1.1.2" xref="S2.E6.m2.2.2.1.1.1.2.cmml">⁢</mo><mi id="S2.E6.m2.2.2.1.1.1.4" 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xref="S2.E6.m2.2.2.1.1.1.1.1.1.1.3.3">𝑃</ci></apply></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m2.2c">\displaystyle\int d^{2}b\left\{1-[P_{p}(\textbf{b})]^{Z_{P}}\right\}.</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m2.2d">∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b { 1 - [ italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b ) ] start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUPERSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> The probability <math alttext="P_{p}(\textbf{b})" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m1.1"><semantics id="S2.SS1.p3.4.m1.1a"><mrow id="S2.SS1.p3.4.m1.1.2" xref="S2.SS1.p3.4.m1.1.2.cmml"><msub id="S2.SS1.p3.4.m1.1.2.2" xref="S2.SS1.p3.4.m1.1.2.2.cmml"><mi id="S2.SS1.p3.4.m1.1.2.2.2" xref="S2.SS1.p3.4.m1.1.2.2.2.cmml">P</mi><mi id="S2.SS1.p3.4.m1.1.2.2.3" xref="S2.SS1.p3.4.m1.1.2.2.3.cmml">p</mi></msub><mo id="S2.SS1.p3.4.m1.1.2.1" xref="S2.SS1.p3.4.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.4.m1.1.2.3.2" xref="S2.SS1.p3.4.m1.1.1a.cmml"><mo id="S2.SS1.p3.4.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.4.m1.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.4.m1.1.1" xref="S2.SS1.p3.4.m1.1.1.cmml">b</mtext><mo id="S2.SS1.p3.4.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.4.m1.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m1.1b"><apply id="S2.SS1.p3.4.m1.1.2.cmml" xref="S2.SS1.p3.4.m1.1.2"><times id="S2.SS1.p3.4.m1.1.2.1.cmml" xref="S2.SS1.p3.4.m1.1.2.1"></times><apply id="S2.SS1.p3.4.m1.1.2.2.cmml" xref="S2.SS1.p3.4.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m1.1.2.2.1.cmml" xref="S2.SS1.p3.4.m1.1.2.2">subscript</csymbol><ci id="S2.SS1.p3.4.m1.1.2.2.2.cmml" xref="S2.SS1.p3.4.m1.1.2.2.2">𝑃</ci><ci id="S2.SS1.p3.4.m1.1.2.2.3.cmml" xref="S2.SS1.p3.4.m1.1.2.2.3">𝑝</ci></apply><ci id="S2.SS1.p3.4.m1.1.1a.cmml" xref="S2.SS1.p3.4.m1.1.2.3.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.4.m1.1.1.cmml" xref="S2.SS1.p3.4.m1.1.1">b</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m1.1c">P_{p}(\textbf{b})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m1.1d">italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b )</annotation></semantics></math> can be determined from the S-matrix concept using the optical potential. The eikonal S-matrix is defined as <math alttext="\mathcal{S}_{p}(\textbf{b})=\exp[i\chi_{p}(\textbf{b})]" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m2.4"><semantics id="S2.SS1.p3.5.m2.4a"><mrow id="S2.SS1.p3.5.m2.4.4" xref="S2.SS1.p3.5.m2.4.4.cmml"><mrow id="S2.SS1.p3.5.m2.4.4.3" xref="S2.SS1.p3.5.m2.4.4.3.cmml"><msub id="S2.SS1.p3.5.m2.4.4.3.2" xref="S2.SS1.p3.5.m2.4.4.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.5.m2.4.4.3.2.2" xref="S2.SS1.p3.5.m2.4.4.3.2.2.cmml">𝒮</mi><mi id="S2.SS1.p3.5.m2.4.4.3.2.3" xref="S2.SS1.p3.5.m2.4.4.3.2.3.cmml">p</mi></msub><mo id="S2.SS1.p3.5.m2.4.4.3.1" xref="S2.SS1.p3.5.m2.4.4.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.5.m2.4.4.3.3.2" xref="S2.SS1.p3.5.m2.1.1a.cmml"><mo id="S2.SS1.p3.5.m2.4.4.3.3.2.1" stretchy="false" xref="S2.SS1.p3.5.m2.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.5.m2.1.1" xref="S2.SS1.p3.5.m2.1.1.cmml">b</mtext><mo id="S2.SS1.p3.5.m2.4.4.3.3.2.2" stretchy="false" 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cd="ambiguous" id="S2.SS1.p3.5.m2.4.4.1.1.1.1.3.1.cmml" xref="S2.SS1.p3.5.m2.4.4.1.1.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.5.m2.4.4.1.1.1.1.3.2.cmml" xref="S2.SS1.p3.5.m2.4.4.1.1.1.1.3.2">𝜒</ci><ci id="S2.SS1.p3.5.m2.4.4.1.1.1.1.3.3.cmml" xref="S2.SS1.p3.5.m2.4.4.1.1.1.1.3.3">𝑝</ci></apply><ci id="S2.SS1.p3.5.m2.2.2a.cmml" xref="S2.SS1.p3.5.m2.4.4.1.1.1.1.4.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.5.m2.2.2.cmml" xref="S2.SS1.p3.5.m2.2.2">b</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m2.4c">\mathcal{S}_{p}(\textbf{b})=\exp[i\chi_{p}(\textbf{b})]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m2.4d">caligraphic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b ) = roman_exp [ italic_i italic_χ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b ) ]</annotation></semantics></math>, for a projectile proton scattering from a target nucleus, where <math alttext="\chi_{p}(\textbf{b})" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m3.1"><semantics id="S2.SS1.p3.6.m3.1a"><mrow id="S2.SS1.p3.6.m3.1.2" xref="S2.SS1.p3.6.m3.1.2.cmml"><msub id="S2.SS1.p3.6.m3.1.2.2" xref="S2.SS1.p3.6.m3.1.2.2.cmml"><mi id="S2.SS1.p3.6.m3.1.2.2.2" xref="S2.SS1.p3.6.m3.1.2.2.2.cmml">χ</mi><mi id="S2.SS1.p3.6.m3.1.2.2.3" xref="S2.SS1.p3.6.m3.1.2.2.3.cmml">p</mi></msub><mo id="S2.SS1.p3.6.m3.1.2.1" xref="S2.SS1.p3.6.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.6.m3.1.2.3.2" xref="S2.SS1.p3.6.m3.1.1a.cmml"><mo id="S2.SS1.p3.6.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.6.m3.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.6.m3.1.1" xref="S2.SS1.p3.6.m3.1.1.cmml">b</mtext><mo id="S2.SS1.p3.6.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.6.m3.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m3.1b"><apply id="S2.SS1.p3.6.m3.1.2.cmml" xref="S2.SS1.p3.6.m3.1.2"><times id="S2.SS1.p3.6.m3.1.2.1.cmml" xref="S2.SS1.p3.6.m3.1.2.1"></times><apply id="S2.SS1.p3.6.m3.1.2.2.cmml" xref="S2.SS1.p3.6.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.6.m3.1.2.2.1.cmml" xref="S2.SS1.p3.6.m3.1.2.2">subscript</csymbol><ci id="S2.SS1.p3.6.m3.1.2.2.2.cmml" xref="S2.SS1.p3.6.m3.1.2.2.2">𝜒</ci><ci id="S2.SS1.p3.6.m3.1.2.2.3.cmml" xref="S2.SS1.p3.6.m3.1.2.2.3">𝑝</ci></apply><ci id="S2.SS1.p3.6.m3.1.1a.cmml" xref="S2.SS1.p3.6.m3.1.2.3.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.6.m3.1.1.cmml" xref="S2.SS1.p3.6.m3.1.1">b</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m3.1c">\chi_{p}(\textbf{b})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m3.1d">italic_χ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b )</annotation></semantics></math> is the eikonal phase shift function <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib36" title="">36</a>]</cite>. Then the eikonal phase shift function with the optical limit approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib40" title="">40</a>]</cite> <table class="ltx_equation ltx_eqn_table" id="S2.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\chi_{p}(\mathbf{b})=i\int d^{2}s\,\rho_{pz}^{P}(\mathbf{s})\int d^{2}t\,\Big{% [}Z_{T}\rho_{pz}^{T}(\mathbf{t})\Gamma_{pp}(\mathbf{b}+\mathbf{s}-\mathbf{t})+% N_{T}\rho_{nz}^{T}(\mathbf{t})\Gamma_{pn}(\mathbf{b}+\mathbf{s}-\mathbf{t})% \Big{]}." class="ltx_Math" display="block" id="S2.E7.m1.5"><semantics id="S2.E7.m1.5a"><mrow id="S2.E7.m1.5.5.1" xref="S2.E7.m1.5.5.1.1.cmml"><mrow id="S2.E7.m1.5.5.1.1" xref="S2.E7.m1.5.5.1.1.cmml"><mrow id="S2.E7.m1.5.5.1.1.3" xref="S2.E7.m1.5.5.1.1.3.cmml"><msub id="S2.E7.m1.5.5.1.1.3.2" xref="S2.E7.m1.5.5.1.1.3.2.cmml"><mi id="S2.E7.m1.5.5.1.1.3.2.2" xref="S2.E7.m1.5.5.1.1.3.2.2.cmml">χ</mi><mi id="S2.E7.m1.5.5.1.1.3.2.3" xref="S2.E7.m1.5.5.1.1.3.2.3.cmml">p</mi></msub><mo id="S2.E7.m1.5.5.1.1.3.1" xref="S2.E7.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.E7.m1.5.5.1.1.3.3.2" xref="S2.E7.m1.5.5.1.1.3.cmml"><mo id="S2.E7.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.E7.m1.5.5.1.1.3.cmml">(</mo><mi id="S2.E7.m1.1.1" xref="S2.E7.m1.1.1.cmml">𝐛</mi><mo id="S2.E7.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S2.E7.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E7.m1.5.5.1.1.2" xref="S2.E7.m1.5.5.1.1.2.cmml">=</mo><mrow id="S2.E7.m1.5.5.1.1.1" xref="S2.E7.m1.5.5.1.1.1.cmml"><mi id="S2.E7.m1.5.5.1.1.1.3" xref="S2.E7.m1.5.5.1.1.1.3.cmml">i</mi><mo id="S2.E7.m1.5.5.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S2.E7.m1.5.5.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.cmml"><mo id="S2.E7.m1.5.5.1.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.1.2.cmml">∫</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.cmml"><msup id="S2.E7.m1.5.5.1.1.1.1.1.3" xref="S2.E7.m1.5.5.1.1.1.1.1.3.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.3.2" xref="S2.E7.m1.5.5.1.1.1.1.1.3.2.cmml">d</mi><mn id="S2.E7.m1.5.5.1.1.1.1.1.3.3" xref="S2.E7.m1.5.5.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E7.m1.5.5.1.1.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E7.m1.5.5.1.1.1.1.1.4" xref="S2.E7.m1.5.5.1.1.1.1.1.4.cmml">s</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.2a" lspace="0.170em" xref="S2.E7.m1.5.5.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S2.E7.m1.5.5.1.1.1.1.1.5" xref="S2.E7.m1.5.5.1.1.1.1.1.5.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.5.2.2" xref="S2.E7.m1.5.5.1.1.1.1.1.5.2.2.cmml">ρ</mi><mrow id="S2.E7.m1.5.5.1.1.1.1.1.5.2.3" xref="S2.E7.m1.5.5.1.1.1.1.1.5.2.3.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.5.2.3.2" xref="S2.E7.m1.5.5.1.1.1.1.1.5.2.3.2.cmml">p</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.5.2.3.1" 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xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.3.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.3.2.cmml">d</mi><mn id="S2.E7.m1.5.5.1.1.1.1.1.1.1.3.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.4" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.4.cmml">t</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.2a" lspace="0.170em" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.2" maxsize="160%" minsize="160%" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">Z</mi><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">T</mi></msub><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml">ρ</mi><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.2.cmml">p</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.1.cmml">⁢</mo><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.2.3.3.cmml">z</mi></mrow><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.4.3.cmml">T</mi></msubsup><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2a" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.5.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.5.2.1" stretchy="false" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S2.E7.m1.3.3" xref="S2.E7.m1.3.3.cmml">𝐭</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.5.2.2" stretchy="false" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2b" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.2" mathvariant="normal" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.2.cmml">Γ</mi><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.2.cmml">p</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.1.cmml">⁢</mo><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.6.3.3.cmml">p</mi></mrow></msub><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2c" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐛</mi><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">+</mo><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">𝐬</mi></mrow><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">𝐭</mi></mrow><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.cmml">+</mo><mrow id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml"><msub id="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.3" xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi 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xref="S2.E7.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.1.1.1.3">𝐭</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.5c">\chi_{p}(\mathbf{b})=i\int d^{2}s\,\rho_{pz}^{P}(\mathbf{s})\int d^{2}t\,\Big{% [}Z_{T}\rho_{pz}^{T}(\mathbf{t})\Gamma_{pp}(\mathbf{b}+\mathbf{s}-\mathbf{t})+% N_{T}\rho_{nz}^{T}(\mathbf{t})\Gamma_{pn}(\mathbf{b}+\mathbf{s}-\mathbf{t})% \Big{]}.</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.5d">italic_χ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) = italic_i ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s italic_ρ start_POSTSUBSCRIPT italic_p italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( bold_s ) ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t [ italic_Z start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_p italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_t ) roman_Γ start_POSTSUBSCRIPT italic_p italic_p end_POSTSUBSCRIPT ( bold_b + bold_s - bold_t ) + italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_n italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_t ) roman_Γ start_POSTSUBSCRIPT italic_p italic_n end_POSTSUBSCRIPT ( bold_b + bold_s - bold_t ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(7)</td> </tr></tbody> </table> with the thickness function <math alttext="\rho_{Nz}^{P(T)}" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m1.1"><semantics id="S2.SS1.p3.7.m1.1a"><msubsup id="S2.SS1.p3.7.m1.1.2" xref="S2.SS1.p3.7.m1.1.2.cmml"><mi id="S2.SS1.p3.7.m1.1.2.2.2" xref="S2.SS1.p3.7.m1.1.2.2.2.cmml">ρ</mi><mrow id="S2.SS1.p3.7.m1.1.2.2.3" 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encoding="application/x-tex" id="S2.SS1.p3.7.m1.1c">\rho_{Nz}^{P(T)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m1.1d">italic_ρ start_POSTSUBSCRIPT italic_N italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P ( italic_T ) end_POSTSUPERSCRIPT</annotation></semantics></math> related to the density by, e.g., <table class="ltx_equation ltx_eqn_table" id="S2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho_{Nz}^{P(T)}=\int\rho_{N}^{P(T)}(\sqrt{\textbf{s}^{2}+z^{2}})dz,~{}~{}N=p,n" class="ltx_Math" display="block" id="S2.E8.m1.7"><semantics id="S2.E8.m1.7a"><mrow id="S2.E8.m1.7.7.2" xref="S2.E8.m1.7.7.3.cmml"><mrow id="S2.E8.m1.6.6.1.1" xref="S2.E8.m1.6.6.1.1.cmml"><msubsup id="S2.E8.m1.6.6.1.1.2" xref="S2.E8.m1.6.6.1.1.2.cmml"><mi id="S2.E8.m1.6.6.1.1.2.2.2" xref="S2.E8.m1.6.6.1.1.2.2.2.cmml">ρ</mi><mrow 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xref="S2.E8.m1.6.6.1.1.3.2.4.2.cmml">z</mi></mrow></mrow></mrow></mrow><mo id="S2.E8.m1.7.7.2.3" rspace="0.827em" xref="S2.E8.m1.7.7.3a.cmml">,</mo><mrow id="S2.E8.m1.7.7.2.2" xref="S2.E8.m1.7.7.2.2.cmml"><mi id="S2.E8.m1.7.7.2.2.2" xref="S2.E8.m1.7.7.2.2.2.cmml">N</mi><mo id="S2.E8.m1.7.7.2.2.1" xref="S2.E8.m1.7.7.2.2.1.cmml">=</mo><mrow id="S2.E8.m1.7.7.2.2.3.2" xref="S2.E8.m1.7.7.2.2.3.1.cmml"><mi id="S2.E8.m1.4.4" xref="S2.E8.m1.4.4.cmml">p</mi><mo id="S2.E8.m1.7.7.2.2.3.2.1" xref="S2.E8.m1.7.7.2.2.3.1.cmml">,</mo><mi id="S2.E8.m1.5.5" xref="S2.E8.m1.5.5.cmml">n</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.m1.7b"><apply id="S2.E8.m1.7.7.3.cmml" xref="S2.E8.m1.7.7.2"><csymbol cd="ambiguous" id="S2.E8.m1.7.7.3a.cmml" xref="S2.E8.m1.7.7.2.3">formulae-sequence</csymbol><apply id="S2.E8.m1.6.6.1.1.cmml" xref="S2.E8.m1.6.6.1.1"><eq id="S2.E8.m1.6.6.1.1.1.cmml" xref="S2.E8.m1.6.6.1.1.1"></eq><apply id="S2.E8.m1.6.6.1.1.2.cmml" 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xref="S2.E8.m1.3.3.2.3.3">2</cn></apply></apply></apply><apply id="S2.E8.m1.6.6.1.1.3.2.4.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4"><csymbol cd="latexml" id="S2.E8.m1.6.6.1.1.3.2.4.1.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4.1">differential-d</csymbol><ci id="S2.E8.m1.6.6.1.1.3.2.4.2.cmml" xref="S2.E8.m1.6.6.1.1.3.2.4.2">𝑧</ci></apply></apply></apply></apply><apply id="S2.E8.m1.7.7.2.2.cmml" xref="S2.E8.m1.7.7.2.2"><eq id="S2.E8.m1.7.7.2.2.1.cmml" xref="S2.E8.m1.7.7.2.2.1"></eq><ci id="S2.E8.m1.7.7.2.2.2.cmml" xref="S2.E8.m1.7.7.2.2.2">𝑁</ci><list id="S2.E8.m1.7.7.2.2.3.1.cmml" xref="S2.E8.m1.7.7.2.2.3.2"><ci id="S2.E8.m1.4.4.cmml" xref="S2.E8.m1.4.4">𝑝</ci><ci id="S2.E8.m1.5.5.cmml" xref="S2.E8.m1.5.5">𝑛</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.7c">\rho_{Nz}^{P(T)}=\int\rho_{N}^{P(T)}(\sqrt{\textbf{s}^{2}+z^{2}})dz,~{}~{}N=p,n</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.7d">italic_ρ start_POSTSUBSCRIPT italic_N italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P ( italic_T ) end_POSTSUPERSCRIPT = ∫ italic_ρ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P ( italic_T ) end_POSTSUPERSCRIPT ( square-root start_ARG s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) italic_d italic_z , italic_N = italic_p , italic_n</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(8)</td> </tr></tbody> </table> where <math alttext="z" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m1.1"><semantics id="S2.SS1.p3.8.m1.1a"><mi id="S2.SS1.p3.8.m1.1.1" xref="S2.SS1.p3.8.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m1.1b"><ci id="S2.SS1.p3.8.m1.1.1.cmml" xref="S2.SS1.p3.8.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m1.1d">italic_z</annotation></semantics></math> is the coordinate along the direction of the incident beam. The densities of the projectile(target) - proton(neutron) are <math alttext="\rho_{p(n)}^{P(T)}(r)" class="ltx_Math" display="inline" id="S2.SS1.p3.9.m2.3"><semantics id="S2.SS1.p3.9.m2.3a"><mrow id="S2.SS1.p3.9.m2.3.4" xref="S2.SS1.p3.9.m2.3.4.cmml"><msubsup id="S2.SS1.p3.9.m2.3.4.2" xref="S2.SS1.p3.9.m2.3.4.2.cmml"><mi id="S2.SS1.p3.9.m2.3.4.2.2.2" xref="S2.SS1.p3.9.m2.3.4.2.2.2.cmml">ρ</mi><mrow id="S2.SS1.p3.9.m2.1.1.1" xref="S2.SS1.p3.9.m2.1.1.1.cmml"><mi id="S2.SS1.p3.9.m2.1.1.1.3" xref="S2.SS1.p3.9.m2.1.1.1.3.cmml">p</mi><mo id="S2.SS1.p3.9.m2.1.1.1.2" xref="S2.SS1.p3.9.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.9.m2.1.1.1.4.2" xref="S2.SS1.p3.9.m2.1.1.1.cmml"><mo id="S2.SS1.p3.9.m2.1.1.1.4.2.1" stretchy="false" xref="S2.SS1.p3.9.m2.1.1.1.cmml">(</mo><mi id="S2.SS1.p3.9.m2.1.1.1.1" xref="S2.SS1.p3.9.m2.1.1.1.1.cmml">n</mi><mo id="S2.SS1.p3.9.m2.1.1.1.4.2.2" stretchy="false" xref="S2.SS1.p3.9.m2.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.SS1.p3.9.m2.2.2.1" xref="S2.SS1.p3.9.m2.2.2.1.cmml"><mi id="S2.SS1.p3.9.m2.2.2.1.3" xref="S2.SS1.p3.9.m2.2.2.1.3.cmml">P</mi><mo id="S2.SS1.p3.9.m2.2.2.1.2" xref="S2.SS1.p3.9.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.9.m2.2.2.1.4.2" xref="S2.SS1.p3.9.m2.2.2.1.cmml"><mo id="S2.SS1.p3.9.m2.2.2.1.4.2.1" stretchy="false" xref="S2.SS1.p3.9.m2.2.2.1.cmml">(</mo><mi id="S2.SS1.p3.9.m2.2.2.1.1" xref="S2.SS1.p3.9.m2.2.2.1.1.cmml">T</mi><mo id="S2.SS1.p3.9.m2.2.2.1.4.2.2" stretchy="false" xref="S2.SS1.p3.9.m2.2.2.1.cmml">)</mo></mrow></mrow></msubsup><mo id="S2.SS1.p3.9.m2.3.4.1" xref="S2.SS1.p3.9.m2.3.4.1.cmml">⁢</mo><mrow id="S2.SS1.p3.9.m2.3.4.3.2" xref="S2.SS1.p3.9.m2.3.4.cmml"><mo id="S2.SS1.p3.9.m2.3.4.3.2.1" stretchy="false" xref="S2.SS1.p3.9.m2.3.4.cmml">(</mo><mi id="S2.SS1.p3.9.m2.3.3" xref="S2.SS1.p3.9.m2.3.3.cmml">r</mi><mo id="S2.SS1.p3.9.m2.3.4.3.2.2" stretchy="false" xref="S2.SS1.p3.9.m2.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.9.m2.3b"><apply id="S2.SS1.p3.9.m2.3.4.cmml" xref="S2.SS1.p3.9.m2.3.4"><times id="S2.SS1.p3.9.m2.3.4.1.cmml" xref="S2.SS1.p3.9.m2.3.4.1"></times><apply id="S2.SS1.p3.9.m2.3.4.2.cmml" xref="S2.SS1.p3.9.m2.3.4.2"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m2.3.4.2.1.cmml" xref="S2.SS1.p3.9.m2.3.4.2">superscript</csymbol><apply id="S2.SS1.p3.9.m2.3.4.2.2.cmml" xref="S2.SS1.p3.9.m2.3.4.2"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m2.3.4.2.2.1.cmml" xref="S2.SS1.p3.9.m2.3.4.2">subscript</csymbol><ci id="S2.SS1.p3.9.m2.3.4.2.2.2.cmml" xref="S2.SS1.p3.9.m2.3.4.2.2.2">𝜌</ci><apply id="S2.SS1.p3.9.m2.1.1.1.cmml" xref="S2.SS1.p3.9.m2.1.1.1"><times id="S2.SS1.p3.9.m2.1.1.1.2.cmml" xref="S2.SS1.p3.9.m2.1.1.1.2"></times><ci id="S2.SS1.p3.9.m2.1.1.1.3.cmml" xref="S2.SS1.p3.9.m2.1.1.1.3">𝑝</ci><ci id="S2.SS1.p3.9.m2.1.1.1.1.cmml" xref="S2.SS1.p3.9.m2.1.1.1.1">𝑛</ci></apply></apply><apply id="S2.SS1.p3.9.m2.2.2.1.cmml" xref="S2.SS1.p3.9.m2.2.2.1"><times id="S2.SS1.p3.9.m2.2.2.1.2.cmml" xref="S2.SS1.p3.9.m2.2.2.1.2"></times><ci id="S2.SS1.p3.9.m2.2.2.1.3.cmml" xref="S2.SS1.p3.9.m2.2.2.1.3">𝑃</ci><ci id="S2.SS1.p3.9.m2.2.2.1.1.cmml" xref="S2.SS1.p3.9.m2.2.2.1.1">𝑇</ci></apply></apply><ci id="S2.SS1.p3.9.m2.3.3.cmml" xref="S2.SS1.p3.9.m2.3.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m2.3c">\rho_{p(n)}^{P(T)}(r)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.9.m2.3d">italic_ρ start_POSTSUBSCRIPT italic_p ( italic_n ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P ( italic_T ) end_POSTSUPERSCRIPT ( italic_r )</annotation></semantics></math> and are normalised so that <math alttext="\int\rho_{p(n)}^{P(T)}(r)d^{3}{r}=1" class="ltx_Math" display="inline" id="S2.SS1.p3.10.m3.3"><semantics id="S2.SS1.p3.10.m3.3a"><mrow id="S2.SS1.p3.10.m3.3.4" xref="S2.SS1.p3.10.m3.3.4.cmml"><mrow id="S2.SS1.p3.10.m3.3.4.2" xref="S2.SS1.p3.10.m3.3.4.2.cmml"><mo id="S2.SS1.p3.10.m3.3.4.2.1" xref="S2.SS1.p3.10.m3.3.4.2.1.cmml">∫</mo><mrow id="S2.SS1.p3.10.m3.3.4.2.2" xref="S2.SS1.p3.10.m3.3.4.2.2.cmml"><msubsup id="S2.SS1.p3.10.m3.3.4.2.2.2" xref="S2.SS1.p3.10.m3.3.4.2.2.2.cmml"><mi id="S2.SS1.p3.10.m3.3.4.2.2.2.2.2" xref="S2.SS1.p3.10.m3.3.4.2.2.2.2.2.cmml">ρ</mi><mrow id="S2.SS1.p3.10.m3.1.1.1" xref="S2.SS1.p3.10.m3.1.1.1.cmml"><mi id="S2.SS1.p3.10.m3.1.1.1.3" xref="S2.SS1.p3.10.m3.1.1.1.3.cmml">p</mi><mo id="S2.SS1.p3.10.m3.1.1.1.2" xref="S2.SS1.p3.10.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.10.m3.1.1.1.4.2" xref="S2.SS1.p3.10.m3.1.1.1.cmml"><mo id="S2.SS1.p3.10.m3.1.1.1.4.2.1" stretchy="false" xref="S2.SS1.p3.10.m3.1.1.1.cmml">(</mo><mi id="S2.SS1.p3.10.m3.1.1.1.1" xref="S2.SS1.p3.10.m3.1.1.1.1.cmml">n</mi><mo id="S2.SS1.p3.10.m3.1.1.1.4.2.2" stretchy="false" xref="S2.SS1.p3.10.m3.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.SS1.p3.10.m3.2.2.1" xref="S2.SS1.p3.10.m3.2.2.1.cmml"><mi id="S2.SS1.p3.10.m3.2.2.1.3" xref="S2.SS1.p3.10.m3.2.2.1.3.cmml">P</mi><mo id="S2.SS1.p3.10.m3.2.2.1.2" xref="S2.SS1.p3.10.m3.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.10.m3.2.2.1.4.2" xref="S2.SS1.p3.10.m3.2.2.1.cmml"><mo id="S2.SS1.p3.10.m3.2.2.1.4.2.1" stretchy="false" xref="S2.SS1.p3.10.m3.2.2.1.cmml">(</mo><mi id="S2.SS1.p3.10.m3.2.2.1.1" xref="S2.SS1.p3.10.m3.2.2.1.1.cmml">T</mi><mo id="S2.SS1.p3.10.m3.2.2.1.4.2.2" stretchy="false" xref="S2.SS1.p3.10.m3.2.2.1.cmml">)</mo></mrow></mrow></msubsup><mo id="S2.SS1.p3.10.m3.3.4.2.2.1" xref="S2.SS1.p3.10.m3.3.4.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.10.m3.3.4.2.2.3.2" xref="S2.SS1.p3.10.m3.3.4.2.2.cmml"><mo id="S2.SS1.p3.10.m3.3.4.2.2.3.2.1" stretchy="false" xref="S2.SS1.p3.10.m3.3.4.2.2.cmml">(</mo><mi id="S2.SS1.p3.10.m3.3.3" xref="S2.SS1.p3.10.m3.3.3.cmml">r</mi><mo id="S2.SS1.p3.10.m3.3.4.2.2.3.2.2" stretchy="false" xref="S2.SS1.p3.10.m3.3.4.2.2.cmml">)</mo></mrow><mo id="S2.SS1.p3.10.m3.3.4.2.2.1a" xref="S2.SS1.p3.10.m3.3.4.2.2.1.cmml">⁢</mo><msup id="S2.SS1.p3.10.m3.3.4.2.2.4" xref="S2.SS1.p3.10.m3.3.4.2.2.4.cmml"><mi id="S2.SS1.p3.10.m3.3.4.2.2.4.2" xref="S2.SS1.p3.10.m3.3.4.2.2.4.2.cmml">d</mi><mn id="S2.SS1.p3.10.m3.3.4.2.2.4.3" xref="S2.SS1.p3.10.m3.3.4.2.2.4.3.cmml">3</mn></msup><mo id="S2.SS1.p3.10.m3.3.4.2.2.1b" xref="S2.SS1.p3.10.m3.3.4.2.2.1.cmml">⁢</mo><mi id="S2.SS1.p3.10.m3.3.4.2.2.5" xref="S2.SS1.p3.10.m3.3.4.2.2.5.cmml">r</mi></mrow></mrow><mo id="S2.SS1.p3.10.m3.3.4.1" xref="S2.SS1.p3.10.m3.3.4.1.cmml">=</mo><mn id="S2.SS1.p3.10.m3.3.4.3" xref="S2.SS1.p3.10.m3.3.4.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.10.m3.3b"><apply id="S2.SS1.p3.10.m3.3.4.cmml" xref="S2.SS1.p3.10.m3.3.4"><eq id="S2.SS1.p3.10.m3.3.4.1.cmml" xref="S2.SS1.p3.10.m3.3.4.1"></eq><apply id="S2.SS1.p3.10.m3.3.4.2.cmml" xref="S2.SS1.p3.10.m3.3.4.2"><int id="S2.SS1.p3.10.m3.3.4.2.1.cmml" xref="S2.SS1.p3.10.m3.3.4.2.1"></int><apply id="S2.SS1.p3.10.m3.3.4.2.2.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2"><times id="S2.SS1.p3.10.m3.3.4.2.2.1.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.1"></times><apply id="S2.SS1.p3.10.m3.3.4.2.2.2.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m3.3.4.2.2.2.1.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.2">superscript</csymbol><apply id="S2.SS1.p3.10.m3.3.4.2.2.2.2.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m3.3.4.2.2.2.2.1.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.2">subscript</csymbol><ci id="S2.SS1.p3.10.m3.3.4.2.2.2.2.2.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.2.2.2">𝜌</ci><apply id="S2.SS1.p3.10.m3.1.1.1.cmml" xref="S2.SS1.p3.10.m3.1.1.1"><times id="S2.SS1.p3.10.m3.1.1.1.2.cmml" xref="S2.SS1.p3.10.m3.1.1.1.2"></times><ci id="S2.SS1.p3.10.m3.1.1.1.3.cmml" xref="S2.SS1.p3.10.m3.1.1.1.3">𝑝</ci><ci id="S2.SS1.p3.10.m3.1.1.1.1.cmml" xref="S2.SS1.p3.10.m3.1.1.1.1">𝑛</ci></apply></apply><apply id="S2.SS1.p3.10.m3.2.2.1.cmml" xref="S2.SS1.p3.10.m3.2.2.1"><times id="S2.SS1.p3.10.m3.2.2.1.2.cmml" xref="S2.SS1.p3.10.m3.2.2.1.2"></times><ci id="S2.SS1.p3.10.m3.2.2.1.3.cmml" xref="S2.SS1.p3.10.m3.2.2.1.3">𝑃</ci><ci id="S2.SS1.p3.10.m3.2.2.1.1.cmml" xref="S2.SS1.p3.10.m3.2.2.1.1">𝑇</ci></apply></apply><ci id="S2.SS1.p3.10.m3.3.3.cmml" xref="S2.SS1.p3.10.m3.3.3">𝑟</ci><apply id="S2.SS1.p3.10.m3.3.4.2.2.4.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.4"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m3.3.4.2.2.4.1.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.4">superscript</csymbol><ci id="S2.SS1.p3.10.m3.3.4.2.2.4.2.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.4.2">𝑑</ci><cn id="S2.SS1.p3.10.m3.3.4.2.2.4.3.cmml" type="integer" xref="S2.SS1.p3.10.m3.3.4.2.2.4.3">3</cn></apply><ci id="S2.SS1.p3.10.m3.3.4.2.2.5.cmml" xref="S2.SS1.p3.10.m3.3.4.2.2.5">𝑟</ci></apply></apply><cn id="S2.SS1.p3.10.m3.3.4.3.cmml" type="integer" xref="S2.SS1.p3.10.m3.3.4.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.10.m3.3c">\int\rho_{p(n)}^{P(T)}(r)d^{3}{r}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.10.m3.3d">∫ italic_ρ start_POSTSUBSCRIPT italic_p ( italic_n ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P ( italic_T ) end_POSTSUPERSCRIPT ( italic_r ) italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r = 1</annotation></semantics></math>. Then the S-matrix <table class="ltx_equation ltx_eqn_table" id="S2.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{S}_{p}(\mathbf{b})=\exp\left(-\int d^{2}s\rho_{pz}^{P}(\mathbf{s})% \int d^{2}t\left[Z_{T}\rho_{pz}^{T}(\mathbf{t})\Gamma_{pp}(\mathbf{b}+\mathbf{% s}-\mathbf{t})+N_{T}\rho_{nz}^{T}(\mathbf{t})\Gamma_{pn}(\mathbf{b}+\mathbf{s}% -\mathbf{t})\right]\right)." class="ltx_Math" display="block" id="S2.E9.m1.6"><semantics id="S2.E9.m1.6a"><mrow id="S2.E9.m1.6.6.1" xref="S2.E9.m1.6.6.1.1.cmml"><mrow id="S2.E9.m1.6.6.1.1" xref="S2.E9.m1.6.6.1.1.cmml"><mrow id="S2.E9.m1.6.6.1.1.3" xref="S2.E9.m1.6.6.1.1.3.cmml"><msub id="S2.E9.m1.6.6.1.1.3.2" xref="S2.E9.m1.6.6.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.6.6.1.1.3.2.2" xref="S2.E9.m1.6.6.1.1.3.2.2.cmml">𝒮</mi><mi id="S2.E9.m1.6.6.1.1.3.2.3" xref="S2.E9.m1.6.6.1.1.3.2.3.cmml">p</mi></msub><mo id="S2.E9.m1.6.6.1.1.3.1" xref="S2.E9.m1.6.6.1.1.3.1.cmml">⁢</mo><mrow id="S2.E9.m1.6.6.1.1.3.3.2" xref="S2.E9.m1.6.6.1.1.3.cmml"><mo id="S2.E9.m1.6.6.1.1.3.3.2.1" stretchy="false" xref="S2.E9.m1.6.6.1.1.3.cmml">(</mo><mi id="S2.E9.m1.1.1" xref="S2.E9.m1.1.1.cmml">𝐛</mi><mo id="S2.E9.m1.6.6.1.1.3.3.2.2" stretchy="false" xref="S2.E9.m1.6.6.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E9.m1.6.6.1.1.2" xref="S2.E9.m1.6.6.1.1.2.cmml">=</mo><mrow id="S2.E9.m1.6.6.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.2.cmml"><mi id="S2.E9.m1.5.5" xref="S2.E9.m1.5.5.cmml">exp</mi><mo id="S2.E9.m1.6.6.1.1.1.1a" xref="S2.E9.m1.6.6.1.1.1.2.cmml">⁡</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.2.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.2.cmml">(</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.1a" xref="S2.E9.m1.6.6.1.1.1.1.1.1.cmml">−</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.2.cmml">∫</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.cmml"><msup id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.3.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.3.2.cmml">d</mi><mn id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.3.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.4" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.4.cmml">s</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2a" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.2.cmml">ρ</mi><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.2.cmml">p</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.1.cmml">⁢</mo><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.2.3.3.cmml">z</mi></mrow><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.5.3.cmml">P</mi></msubsup><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2b" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.6.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.6.2.1" stretchy="false" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S2.E9.m1.2.2" xref="S2.E9.m1.2.2.cmml">𝐬</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.6.2.2" stretchy="false" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2c" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml">∫</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.2.cmml">d</mi><mn id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.4" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.4.cmml">t</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.2a" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">Z</mi><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">T</mi></msub><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml">ρ</mi><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.2.cmml">p</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.1.cmml">⁢</mo><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.3.cmml">z</mi></mrow><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.4.3.cmml">T</mi></msubsup><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2a" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.5.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.5.2.1" stretchy="false" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S2.E9.m1.3.3" xref="S2.E9.m1.3.3.cmml">𝐭</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.5.2.2" stretchy="false" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2b" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.2" mathvariant="normal" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.2.cmml">Γ</mi><mrow id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.3" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.3.cmml"><mi id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.3.2" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.3.2.cmml">p</mi><mo id="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.3.1" xref="S2.E9.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.6.3.1.cmml">⁢</mo><mi 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d^{2}t\left[Z_{T}\rho_{pz}^{T}(\mathbf{t})\Gamma_{pp}(\mathbf{b}+\mathbf{% s}-\mathbf{t})+N_{T}\rho_{nz}^{T}(\mathbf{t})\Gamma_{pn}(\mathbf{b}+\mathbf{s}% -\mathbf{t})\right]\right).</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.6d">caligraphic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_b ) = roman_exp ( - ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s italic_ρ start_POSTSUBSCRIPT italic_p italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT ( bold_s ) ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t [ italic_Z start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_p italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_t ) roman_Γ start_POSTSUBSCRIPT italic_p italic_p end_POSTSUBSCRIPT ( bold_b + bold_s - bold_t ) + italic_N start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_n italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_t ) roman_Γ start_POSTSUBSCRIPT italic_p italic_n end_POSTSUBSCRIPT ( bold_b + bold_s - bold_t ) ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(9)</td> </tr></tbody> </table> <math alttext="\textbf{s}~{}(\textbf{t})" class="ltx_Math" display="inline" id="S2.SS1.p3.11.m1.1"><semantics id="S2.SS1.p3.11.m1.1a"><mrow id="S2.SS1.p3.11.m1.1.2" xref="S2.SS1.p3.11.m1.1.2.cmml"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.11.m1.1.2.2" xref="S2.SS1.p3.11.m1.1.2.2a.cmml">s</mtext><mo id="S2.SS1.p3.11.m1.1.2.1" lspace="0.330em" xref="S2.SS1.p3.11.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.11.m1.1.2.3.2" xref="S2.SS1.p3.11.m1.1.1a.cmml"><mo id="S2.SS1.p3.11.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.11.m1.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.11.m1.1.1" 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projectile (target) nucleon relative to the center of mass of the projectile (target) nucleus, which lies on the plane perpendicular to the incident momentum of the projectile. The profile function <math alttext="\Gamma_{pN}(\textbf{b})" class="ltx_Math" display="inline" id="S2.SS1.p3.12.m2.1"><semantics id="S2.SS1.p3.12.m2.1a"><mrow id="S2.SS1.p3.12.m2.1.2" xref="S2.SS1.p3.12.m2.1.2.cmml"><msub id="S2.SS1.p3.12.m2.1.2.2" xref="S2.SS1.p3.12.m2.1.2.2.cmml"><mi id="S2.SS1.p3.12.m2.1.2.2.2" mathvariant="normal" xref="S2.SS1.p3.12.m2.1.2.2.2.cmml">Γ</mi><mrow id="S2.SS1.p3.12.m2.1.2.2.3" xref="S2.SS1.p3.12.m2.1.2.2.3.cmml"><mi id="S2.SS1.p3.12.m2.1.2.2.3.2" xref="S2.SS1.p3.12.m2.1.2.2.3.2.cmml">p</mi><mo id="S2.SS1.p3.12.m2.1.2.2.3.1" xref="S2.SS1.p3.12.m2.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.12.m2.1.2.2.3.3" xref="S2.SS1.p3.12.m2.1.2.2.3.3.cmml">N</mi></mrow></msub><mo id="S2.SS1.p3.12.m2.1.2.1" xref="S2.SS1.p3.12.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.12.m2.1.2.3.2" xref="S2.SS1.p3.12.m2.1.1a.cmml"><mo id="S2.SS1.p3.12.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.12.m2.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.12.m2.1.1" 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xref="S2.SS1.p3.12.m2.1.2.3.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.12.m2.1.1.cmml" xref="S2.SS1.p3.12.m2.1.1">b</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.12.m2.1c">\Gamma_{pN}(\textbf{b})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.12.m2.1d">roman_Γ start_POSTSUBSCRIPT italic_p italic_N end_POSTSUBSCRIPT ( b )</annotation></semantics></math> <math alttext="(N=p,n)" class="ltx_Math" display="inline" id="S2.SS1.p3.13.m3.3"><semantics id="S2.SS1.p3.13.m3.3a"><mrow id="S2.SS1.p3.13.m3.3.3.1" xref="S2.SS1.p3.13.m3.3.3.1.1.cmml"><mo id="S2.SS1.p3.13.m3.3.3.1.2" stretchy="false" xref="S2.SS1.p3.13.m3.3.3.1.1.cmml">(</mo><mrow id="S2.SS1.p3.13.m3.3.3.1.1" xref="S2.SS1.p3.13.m3.3.3.1.1.cmml"><mi id="S2.SS1.p3.13.m3.3.3.1.1.2" xref="S2.SS1.p3.13.m3.3.3.1.1.2.cmml">N</mi><mo id="S2.SS1.p3.13.m3.3.3.1.1.1" xref="S2.SS1.p3.13.m3.3.3.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.13.m3.3.3.1.1.3.2" xref="S2.SS1.p3.13.m3.3.3.1.1.3.1.cmml"><mi id="S2.SS1.p3.13.m3.1.1" xref="S2.SS1.p3.13.m3.1.1.cmml">p</mi><mo id="S2.SS1.p3.13.m3.3.3.1.1.3.2.1" xref="S2.SS1.p3.13.m3.3.3.1.1.3.1.cmml">,</mo><mi id="S2.SS1.p3.13.m3.2.2" xref="S2.SS1.p3.13.m3.2.2.cmml">n</mi></mrow></mrow><mo id="S2.SS1.p3.13.m3.3.3.1.3" stretchy="false" xref="S2.SS1.p3.13.m3.3.3.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.13.m3.3b"><apply id="S2.SS1.p3.13.m3.3.3.1.1.cmml" xref="S2.SS1.p3.13.m3.3.3.1"><eq id="S2.SS1.p3.13.m3.3.3.1.1.1.cmml" xref="S2.SS1.p3.13.m3.3.3.1.1.1"></eq><ci id="S2.SS1.p3.13.m3.3.3.1.1.2.cmml" xref="S2.SS1.p3.13.m3.3.3.1.1.2">𝑁</ci><list id="S2.SS1.p3.13.m3.3.3.1.1.3.1.cmml" xref="S2.SS1.p3.13.m3.3.3.1.1.3.2"><ci id="S2.SS1.p3.13.m3.1.1.cmml" xref="S2.SS1.p3.13.m3.1.1">𝑝</ci><ci id="S2.SS1.p3.13.m3.2.2.cmml" xref="S2.SS1.p3.13.m3.2.2">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.13.m3.3c">(N=p,n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.13.m3.3d">( italic_N = italic_p , italic_n )</annotation></semantics></math> describes the scattering of projectile protons (neutrons) with the target protons (neutrons), which is often parametrized in a finite range <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib41" title="">41</a>]</cite> as <table class="ltx_equation ltx_eqn_table" id="S2.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Gamma_{pN}(\mathbf{b})=\frac{1-i\alpha_{pN}}{4\pi\beta_{pN}}\,\sigma_{pN}^{% \rm tot}\,\exp\left(-\frac{\mathbf{b}^{2}}{2\beta_{pN}}\right)." class="ltx_Math" display="block" id="S2.E10.m1.3"><semantics id="S2.E10.m1.3a"><mrow id="S2.E10.m1.3.3.1" xref="S2.E10.m1.3.3.1.1.cmml"><mrow id="S2.E10.m1.3.3.1.1" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_tot end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG bold_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_β start_POSTSUBSCRIPT italic_p italic_N end_POSTSUBSCRIPT end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(10)</td> </tr></tbody> </table> The parameters used in the proton-nucleon <math alttext="(pN)" class="ltx_Math" display="inline" id="S2.SS1.p3.14.m1.1"><semantics id="S2.SS1.p3.14.m1.1a"><mrow id="S2.SS1.p3.14.m1.1.1.1" xref="S2.SS1.p3.14.m1.1.1.1.1.cmml"><mo id="S2.SS1.p3.14.m1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.14.m1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.14.m1.1.1.1.1" xref="S2.SS1.p3.14.m1.1.1.1.1.cmml"><mi id="S2.SS1.p3.14.m1.1.1.1.1.2" xref="S2.SS1.p3.14.m1.1.1.1.1.2.cmml">p</mi><mo id="S2.SS1.p3.14.m1.1.1.1.1.1" xref="S2.SS1.p3.14.m1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p3.14.m1.1.1.1.1.3" xref="S2.SS1.p3.14.m1.1.1.1.1.3.cmml">N</mi></mrow><mo id="S2.SS1.p3.14.m1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.14.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.14.m1.1b"><apply id="S2.SS1.p3.14.m1.1.1.1.1.cmml" xref="S2.SS1.p3.14.m1.1.1.1"><times id="S2.SS1.p3.14.m1.1.1.1.1.1.cmml" xref="S2.SS1.p3.14.m1.1.1.1.1.1"></times><ci id="S2.SS1.p3.14.m1.1.1.1.1.2.cmml" xref="S2.SS1.p3.14.m1.1.1.1.1.2">𝑝</ci><ci id="S2.SS1.p3.14.m1.1.1.1.1.3.cmml" xref="S2.SS1.p3.14.m1.1.1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.14.m1.1c">(pN)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.14.m1.1d">( italic_p italic_N )</annotation></semantics></math> scattering model are defined as follows: <math alttext="\alpha_{pN}" class="ltx_Math" display="inline" id="S2.SS1.p3.15.m2.1"><semantics id="S2.SS1.p3.15.m2.1a"><msub id="S2.SS1.p3.15.m2.1.1" xref="S2.SS1.p3.15.m2.1.1.cmml"><mi id="S2.SS1.p3.15.m2.1.1.2" xref="S2.SS1.p3.15.m2.1.1.2.cmml">α</mi><mrow id="S2.SS1.p3.15.m2.1.1.3" xref="S2.SS1.p3.15.m2.1.1.3.cmml"><mi id="S2.SS1.p3.15.m2.1.1.3.2" xref="S2.SS1.p3.15.m2.1.1.3.2.cmml">p</mi><mo id="S2.SS1.p3.15.m2.1.1.3.1" xref="S2.SS1.p3.15.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.15.m2.1.1.3.3" xref="S2.SS1.p3.15.m2.1.1.3.3.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.15.m2.1b"><apply id="S2.SS1.p3.15.m2.1.1.cmml" xref="S2.SS1.p3.15.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.15.m2.1.1.1.cmml" xref="S2.SS1.p3.15.m2.1.1">subscript</csymbol><ci id="S2.SS1.p3.15.m2.1.1.2.cmml" xref="S2.SS1.p3.15.m2.1.1.2">𝛼</ci><apply id="S2.SS1.p3.15.m2.1.1.3.cmml" xref="S2.SS1.p3.15.m2.1.1.3"><times id="S2.SS1.p3.15.m2.1.1.3.1.cmml" xref="S2.SS1.p3.15.m2.1.1.3.1"></times><ci id="S2.SS1.p3.15.m2.1.1.3.2.cmml" xref="S2.SS1.p3.15.m2.1.1.3.2">𝑝</ci><ci id="S2.SS1.p3.15.m2.1.1.3.3.cmml" xref="S2.SS1.p3.15.m2.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.15.m2.1c">\alpha_{pN}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.15.m2.1d">italic_α start_POSTSUBSCRIPT italic_p italic_N end_POSTSUBSCRIPT</annotation></semantics></math> is the ratio of the real to the imaginary part of the <math alttext="pN" class="ltx_Math" display="inline" id="S2.SS1.p3.16.m3.1"><semantics id="S2.SS1.p3.16.m3.1a"><mrow id="S2.SS1.p3.16.m3.1.1" xref="S2.SS1.p3.16.m3.1.1.cmml"><mi id="S2.SS1.p3.16.m3.1.1.2" xref="S2.SS1.p3.16.m3.1.1.2.cmml">p</mi><mo id="S2.SS1.p3.16.m3.1.1.1" xref="S2.SS1.p3.16.m3.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p3.16.m3.1.1.3" xref="S2.SS1.p3.16.m3.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.16.m3.1b"><apply id="S2.SS1.p3.16.m3.1.1.cmml" xref="S2.SS1.p3.16.m3.1.1"><times id="S2.SS1.p3.16.m3.1.1.1.cmml" xref="S2.SS1.p3.16.m3.1.1.1"></times><ci id="S2.SS1.p3.16.m3.1.1.2.cmml" xref="S2.SS1.p3.16.m3.1.1.2">𝑝</ci><ci id="S2.SS1.p3.16.m3.1.1.3.cmml" xref="S2.SS1.p3.16.m3.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.16.m3.1c">pN</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.16.m3.1d">italic_p italic_N</annotation></semantics></math> scattering amplitude in the forward direction,<math alttext="\sigma_{pN}^{\text{tot}}" class="ltx_Math" display="inline" id="S2.SS1.p3.17.m4.1"><semantics id="S2.SS1.p3.17.m4.1a"><msubsup id="S2.SS1.p3.17.m4.1.1" xref="S2.SS1.p3.17.m4.1.1.cmml"><mi id="S2.SS1.p3.17.m4.1.1.2.2" xref="S2.SS1.p3.17.m4.1.1.2.2.cmml">σ</mi><mrow id="S2.SS1.p3.17.m4.1.1.2.3" xref="S2.SS1.p3.17.m4.1.1.2.3.cmml"><mi id="S2.SS1.p3.17.m4.1.1.2.3.2" xref="S2.SS1.p3.17.m4.1.1.2.3.2.cmml">p</mi><mo id="S2.SS1.p3.17.m4.1.1.2.3.1" xref="S2.SS1.p3.17.m4.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.17.m4.1.1.2.3.3" xref="S2.SS1.p3.17.m4.1.1.2.3.3.cmml">N</mi></mrow><mtext id="S2.SS1.p3.17.m4.1.1.3" xref="S2.SS1.p3.17.m4.1.1.3a.cmml">tot</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.17.m4.1b"><apply id="S2.SS1.p3.17.m4.1.1.cmml" xref="S2.SS1.p3.17.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.17.m4.1.1.1.cmml" xref="S2.SS1.p3.17.m4.1.1">superscript</csymbol><apply id="S2.SS1.p3.17.m4.1.1.2.cmml" xref="S2.SS1.p3.17.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.17.m4.1.1.2.1.cmml" xref="S2.SS1.p3.17.m4.1.1">subscript</csymbol><ci id="S2.SS1.p3.17.m4.1.1.2.2.cmml" xref="S2.SS1.p3.17.m4.1.1.2.2">𝜎</ci><apply id="S2.SS1.p3.17.m4.1.1.2.3.cmml" xref="S2.SS1.p3.17.m4.1.1.2.3"><times id="S2.SS1.p3.17.m4.1.1.2.3.1.cmml" xref="S2.SS1.p3.17.m4.1.1.2.3.1"></times><ci id="S2.SS1.p3.17.m4.1.1.2.3.2.cmml" xref="S2.SS1.p3.17.m4.1.1.2.3.2">𝑝</ci><ci id="S2.SS1.p3.17.m4.1.1.2.3.3.cmml" xref="S2.SS1.p3.17.m4.1.1.2.3.3">𝑁</ci></apply></apply><ci id="S2.SS1.p3.17.m4.1.1.3a.cmml" 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xref="S2.SS1.p3.18.m5.1.1"><times id="S2.SS1.p3.18.m5.1.1.1.cmml" xref="S2.SS1.p3.18.m5.1.1.1"></times><ci id="S2.SS1.p3.18.m5.1.1.2.cmml" xref="S2.SS1.p3.18.m5.1.1.2">𝑝</ci><ci id="S2.SS1.p3.18.m5.1.1.3.cmml" xref="S2.SS1.p3.18.m5.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.18.m5.1c">pN</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.18.m5.1d">italic_p italic_N</annotation></semantics></math> cross section, <math alttext="\beta_{pN}" class="ltx_Math" display="inline" id="S2.SS1.p3.19.m6.1"><semantics id="S2.SS1.p3.19.m6.1a"><msub id="S2.SS1.p3.19.m6.1.1" xref="S2.SS1.p3.19.m6.1.1.cmml"><mi id="S2.SS1.p3.19.m6.1.1.2" xref="S2.SS1.p3.19.m6.1.1.2.cmml">β</mi><mrow id="S2.SS1.p3.19.m6.1.1.3" xref="S2.SS1.p3.19.m6.1.1.3.cmml"><mi id="S2.SS1.p3.19.m6.1.1.3.2" xref="S2.SS1.p3.19.m6.1.1.3.2.cmml">p</mi><mo id="S2.SS1.p3.19.m6.1.1.3.1" xref="S2.SS1.p3.19.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.19.m6.1.1.3.3" xref="S2.SS1.p3.19.m6.1.1.3.3.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.19.m6.1b"><apply id="S2.SS1.p3.19.m6.1.1.cmml" xref="S2.SS1.p3.19.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.19.m6.1.1.1.cmml" xref="S2.SS1.p3.19.m6.1.1">subscript</csymbol><ci id="S2.SS1.p3.19.m6.1.1.2.cmml" xref="S2.SS1.p3.19.m6.1.1.2">𝛽</ci><apply id="S2.SS1.p3.19.m6.1.1.3.cmml" xref="S2.SS1.p3.19.m6.1.1.3"><times id="S2.SS1.p3.19.m6.1.1.3.1.cmml" xref="S2.SS1.p3.19.m6.1.1.3.1"></times><ci id="S2.SS1.p3.19.m6.1.1.3.2.cmml" xref="S2.SS1.p3.19.m6.1.1.3.2">𝑝</ci><ci id="S2.SS1.p3.19.m6.1.1.3.3.cmml" xref="S2.SS1.p3.19.m6.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.19.m6.1c">\beta_{pN}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.19.m6.1d">italic_β start_POSTSUBSCRIPT italic_p italic_N end_POSTSUBSCRIPT</annotation></semantics></math> is the slope parameter of the <math alttext="pN" class="ltx_Math" display="inline" id="S2.SS1.p3.20.m7.1"><semantics id="S2.SS1.p3.20.m7.1a"><mrow id="S2.SS1.p3.20.m7.1.1" xref="S2.SS1.p3.20.m7.1.1.cmml"><mi id="S2.SS1.p3.20.m7.1.1.2" xref="S2.SS1.p3.20.m7.1.1.2.cmml">p</mi><mo id="S2.SS1.p3.20.m7.1.1.1" xref="S2.SS1.p3.20.m7.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p3.20.m7.1.1.3" xref="S2.SS1.p3.20.m7.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.20.m7.1b"><apply id="S2.SS1.p3.20.m7.1.1.cmml" xref="S2.SS1.p3.20.m7.1.1"><times id="S2.SS1.p3.20.m7.1.1.1.cmml" xref="S2.SS1.p3.20.m7.1.1.1"></times><ci id="S2.SS1.p3.20.m7.1.1.2.cmml" xref="S2.SS1.p3.20.m7.1.1.2">𝑝</ci><ci id="S2.SS1.p3.20.m7.1.1.3.cmml" xref="S2.SS1.p3.20.m7.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.20.m7.1c">pN</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.20.m7.1d">italic_p italic_N</annotation></semantics></math> differential cross-section, characterizing the spatial distribution of the scattering amplitude. The proton-proton (Coulomb removed) and proton-neutron total cross sections (<math alttext="\sigma_{pN}^{\rm tot}" class="ltx_Math" display="inline" id="S2.SS1.p3.21.m8.1"><semantics id="S2.SS1.p3.21.m8.1a"><msubsup id="S2.SS1.p3.21.m8.1.1" xref="S2.SS1.p3.21.m8.1.1.cmml"><mi id="S2.SS1.p3.21.m8.1.1.2.2" xref="S2.SS1.p3.21.m8.1.1.2.2.cmml">σ</mi><mrow id="S2.SS1.p3.21.m8.1.1.2.3" xref="S2.SS1.p3.21.m8.1.1.2.3.cmml"><mi id="S2.SS1.p3.21.m8.1.1.2.3.2" xref="S2.SS1.p3.21.m8.1.1.2.3.2.cmml">p</mi><mo id="S2.SS1.p3.21.m8.1.1.2.3.1" xref="S2.SS1.p3.21.m8.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.21.m8.1.1.2.3.3" xref="S2.SS1.p3.21.m8.1.1.2.3.3.cmml">N</mi></mrow><mi id="S2.SS1.p3.21.m8.1.1.3" xref="S2.SS1.p3.21.m8.1.1.3.cmml">tot</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.21.m8.1b"><apply id="S2.SS1.p3.21.m8.1.1.cmml" xref="S2.SS1.p3.21.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.21.m8.1.1.1.cmml" xref="S2.SS1.p3.21.m8.1.1">superscript</csymbol><apply id="S2.SS1.p3.21.m8.1.1.2.cmml" xref="S2.SS1.p3.21.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.21.m8.1.1.2.1.cmml" xref="S2.SS1.p3.21.m8.1.1">subscript</csymbol><ci id="S2.SS1.p3.21.m8.1.1.2.2.cmml" xref="S2.SS1.p3.21.m8.1.1.2.2">𝜎</ci><apply id="S2.SS1.p3.21.m8.1.1.2.3.cmml" xref="S2.SS1.p3.21.m8.1.1.2.3"><times id="S2.SS1.p3.21.m8.1.1.2.3.1.cmml" xref="S2.SS1.p3.21.m8.1.1.2.3.1"></times><ci id="S2.SS1.p3.21.m8.1.1.2.3.2.cmml" xref="S2.SS1.p3.21.m8.1.1.2.3.2">𝑝</ci><ci id="S2.SS1.p3.21.m8.1.1.2.3.3.cmml" xref="S2.SS1.p3.21.m8.1.1.2.3.3">𝑁</ci></apply></apply><ci id="S2.SS1.p3.21.m8.1.1.3.cmml" xref="S2.SS1.p3.21.m8.1.1.3">tot</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.21.m8.1c">\sigma_{pN}^{\rm tot}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.21.m8.1d">italic_σ start_POSTSUBSCRIPT italic_p italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_tot end_POSTSUPERSCRIPT</annotation></semantics></math>) parameters are generally obtained from a fit of experimental data. The standard parameter set for proton-neutron and proton-proton scattering can be found in tabulated form in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib41" title="">41</a>]</cite>. The probability in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E1" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">1</a>) is then calculated <math alttext="P_{p}(\textbf{b})=\left|\mathcal{S}_{p}(\textbf{b})\right|^{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.22.m9.3"><semantics id="S2.SS1.p3.22.m9.3a"><mrow id="S2.SS1.p3.22.m9.3.3" xref="S2.SS1.p3.22.m9.3.3.cmml"><mrow id="S2.SS1.p3.22.m9.3.3.3" xref="S2.SS1.p3.22.m9.3.3.3.cmml"><msub id="S2.SS1.p3.22.m9.3.3.3.2" xref="S2.SS1.p3.22.m9.3.3.3.2.cmml"><mi id="S2.SS1.p3.22.m9.3.3.3.2.2" xref="S2.SS1.p3.22.m9.3.3.3.2.2.cmml">P</mi><mi id="S2.SS1.p3.22.m9.3.3.3.2.3" xref="S2.SS1.p3.22.m9.3.3.3.2.3.cmml">p</mi></msub><mo id="S2.SS1.p3.22.m9.3.3.3.1" xref="S2.SS1.p3.22.m9.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.22.m9.3.3.3.3.2" xref="S2.SS1.p3.22.m9.1.1a.cmml"><mo id="S2.SS1.p3.22.m9.3.3.3.3.2.1" stretchy="false" xref="S2.SS1.p3.22.m9.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.22.m9.1.1" xref="S2.SS1.p3.22.m9.1.1.cmml">b</mtext><mo id="S2.SS1.p3.22.m9.3.3.3.3.2.2" stretchy="false" xref="S2.SS1.p3.22.m9.1.1a.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.22.m9.3.3.2" xref="S2.SS1.p3.22.m9.3.3.2.cmml">=</mo><msup id="S2.SS1.p3.22.m9.3.3.1" xref="S2.SS1.p3.22.m9.3.3.1.cmml"><mrow id="S2.SS1.p3.22.m9.3.3.1.1.1" xref="S2.SS1.p3.22.m9.3.3.1.1.2.cmml"><mo id="S2.SS1.p3.22.m9.3.3.1.1.1.2" xref="S2.SS1.p3.22.m9.3.3.1.1.2.1.cmml">|</mo><mrow id="S2.SS1.p3.22.m9.3.3.1.1.1.1" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.cmml"><msub id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.2" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.2.cmml">𝒮</mi><mi id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.3" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.3.cmml">p</mi></msub><mo id="S2.SS1.p3.22.m9.3.3.1.1.1.1.1" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p3.22.m9.3.3.1.1.1.1.3.2" xref="S2.SS1.p3.22.m9.2.2a.cmml"><mo id="S2.SS1.p3.22.m9.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p3.22.m9.2.2a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.22.m9.2.2" xref="S2.SS1.p3.22.m9.2.2.cmml">b</mtext><mo id="S2.SS1.p3.22.m9.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p3.22.m9.2.2a.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.22.m9.3.3.1.1.1.3" xref="S2.SS1.p3.22.m9.3.3.1.1.2.1.cmml">|</mo></mrow><mn id="S2.SS1.p3.22.m9.3.3.1.3" xref="S2.SS1.p3.22.m9.3.3.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.22.m9.3b"><apply id="S2.SS1.p3.22.m9.3.3.cmml" xref="S2.SS1.p3.22.m9.3.3"><eq id="S2.SS1.p3.22.m9.3.3.2.cmml" xref="S2.SS1.p3.22.m9.3.3.2"></eq><apply id="S2.SS1.p3.22.m9.3.3.3.cmml" xref="S2.SS1.p3.22.m9.3.3.3"><times id="S2.SS1.p3.22.m9.3.3.3.1.cmml" 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id="S2.SS1.p3.22.m9.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.1"></times><apply id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.1.cmml" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.2.cmml" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.2">𝒮</ci><ci id="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.3.cmml" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.2.3">𝑝</ci></apply><ci id="S2.SS1.p3.22.m9.2.2a.cmml" xref="S2.SS1.p3.22.m9.3.3.1.1.1.1.3.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.22.m9.2.2.cmml" xref="S2.SS1.p3.22.m9.2.2">b</mtext></ci></apply></apply><cn id="S2.SS1.p3.22.m9.3.3.1.3.cmml" type="integer" xref="S2.SS1.p3.22.m9.3.3.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.22.m9.3c">P_{p}(\textbf{b})=\left|\mathcal{S}_{p}(\textbf{b})\right|^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.22.m9.3d">italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b ) = | caligraphic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( b ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Similarly, the probability <math alttext="P_{n}(\textbf{b})=\left|\mathcal{S}_{n}(\textbf{b})\right|^{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.23.m10.3"><semantics id="S2.SS1.p3.23.m10.3a"><mrow id="S2.SS1.p3.23.m10.3.3" xref="S2.SS1.p3.23.m10.3.3.cmml"><mrow id="S2.SS1.p3.23.m10.3.3.3" xref="S2.SS1.p3.23.m10.3.3.3.cmml"><msub id="S2.SS1.p3.23.m10.3.3.3.2" xref="S2.SS1.p3.23.m10.3.3.3.2.cmml"><mi id="S2.SS1.p3.23.m10.3.3.3.2.2" xref="S2.SS1.p3.23.m10.3.3.3.2.2.cmml">P</mi><mi id="S2.SS1.p3.23.m10.3.3.3.2.3" xref="S2.SS1.p3.23.m10.3.3.3.2.3.cmml">n</mi></msub><mo id="S2.SS1.p3.23.m10.3.3.3.1" xref="S2.SS1.p3.23.m10.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.23.m10.3.3.3.3.2" xref="S2.SS1.p3.23.m10.1.1a.cmml"><mo id="S2.SS1.p3.23.m10.3.3.3.3.2.1" stretchy="false" xref="S2.SS1.p3.23.m10.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.23.m10.1.1" xref="S2.SS1.p3.23.m10.1.1.cmml">b</mtext><mo id="S2.SS1.p3.23.m10.3.3.3.3.2.2" stretchy="false" xref="S2.SS1.p3.23.m10.1.1a.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.23.m10.3.3.2" xref="S2.SS1.p3.23.m10.3.3.2.cmml">=</mo><msup id="S2.SS1.p3.23.m10.3.3.1" xref="S2.SS1.p3.23.m10.3.3.1.cmml"><mrow id="S2.SS1.p3.23.m10.3.3.1.1.1" xref="S2.SS1.p3.23.m10.3.3.1.1.2.cmml"><mo id="S2.SS1.p3.23.m10.3.3.1.1.1.2" xref="S2.SS1.p3.23.m10.3.3.1.1.2.1.cmml">|</mo><mrow id="S2.SS1.p3.23.m10.3.3.1.1.1.1" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.cmml"><msub id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.2" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.2.cmml">𝒮</mi><mi id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.3" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.SS1.p3.23.m10.3.3.1.1.1.1.1" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p3.23.m10.3.3.1.1.1.1.3.2" xref="S2.SS1.p3.23.m10.2.2a.cmml"><mo id="S2.SS1.p3.23.m10.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p3.23.m10.2.2a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.23.m10.2.2" xref="S2.SS1.p3.23.m10.2.2.cmml">b</mtext><mo id="S2.SS1.p3.23.m10.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p3.23.m10.2.2a.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.23.m10.3.3.1.1.1.3" xref="S2.SS1.p3.23.m10.3.3.1.1.2.1.cmml">|</mo></mrow><mn id="S2.SS1.p3.23.m10.3.3.1.3" xref="S2.SS1.p3.23.m10.3.3.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.23.m10.3b"><apply id="S2.SS1.p3.23.m10.3.3.cmml" xref="S2.SS1.p3.23.m10.3.3"><eq id="S2.SS1.p3.23.m10.3.3.2.cmml" xref="S2.SS1.p3.23.m10.3.3.2"></eq><apply id="S2.SS1.p3.23.m10.3.3.3.cmml" xref="S2.SS1.p3.23.m10.3.3.3"><times id="S2.SS1.p3.23.m10.3.3.3.1.cmml" xref="S2.SS1.p3.23.m10.3.3.3.1"></times><apply id="S2.SS1.p3.23.m10.3.3.3.2.cmml" xref="S2.SS1.p3.23.m10.3.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.23.m10.3.3.3.2.1.cmml" xref="S2.SS1.p3.23.m10.3.3.3.2">subscript</csymbol><ci id="S2.SS1.p3.23.m10.3.3.3.2.2.cmml" xref="S2.SS1.p3.23.m10.3.3.3.2.2">𝑃</ci><ci id="S2.SS1.p3.23.m10.3.3.3.2.3.cmml" xref="S2.SS1.p3.23.m10.3.3.3.2.3">𝑛</ci></apply><ci id="S2.SS1.p3.23.m10.1.1a.cmml" xref="S2.SS1.p3.23.m10.3.3.3.3.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.23.m10.1.1.cmml" xref="S2.SS1.p3.23.m10.1.1">b</mtext></ci></apply><apply id="S2.SS1.p3.23.m10.3.3.1.cmml" xref="S2.SS1.p3.23.m10.3.3.1"><csymbol cd="ambiguous" id="S2.SS1.p3.23.m10.3.3.1.2.cmml" xref="S2.SS1.p3.23.m10.3.3.1">superscript</csymbol><apply id="S2.SS1.p3.23.m10.3.3.1.1.2.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1"><abs id="S2.SS1.p3.23.m10.3.3.1.1.2.1.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.2"></abs><apply id="S2.SS1.p3.23.m10.3.3.1.1.1.1.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1"><times id="S2.SS1.p3.23.m10.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.1"></times><apply id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.1.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.2.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.2">𝒮</ci><ci id="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.3.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.2.3">𝑛</ci></apply><ci id="S2.SS1.p3.23.m10.2.2a.cmml" xref="S2.SS1.p3.23.m10.3.3.1.1.1.1.3.2"><mtext class="ltx_mathvariant_bold" id="S2.SS1.p3.23.m10.2.2.cmml" xref="S2.SS1.p3.23.m10.2.2">b</mtext></ci></apply></apply><cn id="S2.SS1.p3.23.m10.3.3.1.3.cmml" type="integer" xref="S2.SS1.p3.23.m10.3.3.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.23.m10.3c">P_{n}(\textbf{b})=\left|\mathcal{S}_{n}(\textbf{b})\right|^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.23.m10.3d">italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( b ) = | caligraphic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( b ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, given by interchanging the role of a proton with a neutron<math alttext="(p\leftrightarrow n)" class="ltx_Math" display="inline" id="S2.SS1.p3.24.m11.1"><semantics id="S2.SS1.p3.24.m11.1a"><mrow id="S2.SS1.p3.24.m11.1.1.1" xref="S2.SS1.p3.24.m11.1.1.1.1.cmml"><mo id="S2.SS1.p3.24.m11.1.1.1.2" stretchy="false" xref="S2.SS1.p3.24.m11.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.24.m11.1.1.1.1" xref="S2.SS1.p3.24.m11.1.1.1.1.cmml"><mi id="S2.SS1.p3.24.m11.1.1.1.1.2" xref="S2.SS1.p3.24.m11.1.1.1.1.2.cmml">p</mi><mo id="S2.SS1.p3.24.m11.1.1.1.1.1" stretchy="false" xref="S2.SS1.p3.24.m11.1.1.1.1.1.cmml">↔</mo><mi id="S2.SS1.p3.24.m11.1.1.1.1.3" xref="S2.SS1.p3.24.m11.1.1.1.1.3.cmml">n</mi></mrow><mo id="S2.SS1.p3.24.m11.1.1.1.3" stretchy="false" xref="S2.SS1.p3.24.m11.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.24.m11.1b"><apply id="S2.SS1.p3.24.m11.1.1.1.1.cmml" xref="S2.SS1.p3.24.m11.1.1.1"><ci id="S2.SS1.p3.24.m11.1.1.1.1.1.cmml" xref="S2.SS1.p3.24.m11.1.1.1.1.1">↔</ci><ci id="S2.SS1.p3.24.m11.1.1.1.1.2.cmml" xref="S2.SS1.p3.24.m11.1.1.1.1.2">𝑝</ci><ci id="S2.SS1.p3.24.m11.1.1.1.1.3.cmml" xref="S2.SS1.p3.24.m11.1.1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.24.m11.1c">(p\leftrightarrow n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.24.m11.1d">( italic_p ↔ italic_n )</annotation></semantics></math> in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E9" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">9</a>) and Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E10" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">10</a>). </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> II.2 Secondary fragmentation or ablation stage</h3> <div class="ltx_para" id="S2.SS2.p1"> In the ablation stage <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>]</cite>, the final fragmentation products are formed by deexciting the residual fragment in the abrasion stage through the evaporation of light particles. Here, we are interested in the process where neutrons are removed during the initial abrasion stage, followed by the evaporation of protons during the final stage. </div> <div class="ltx_para" id="S2.SS2.p2"> The proton particle evaporation contribution is calculated following the development by Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib34" title="">34</a>]</cite> <table class="ltx_equation ltx_eqn_table" id="S2.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{\rm cc}^{{\rm evap}}=\sum_{x=1}^{N_{P}}\sigma_{-xn}P_{-xn}," class="ltx_Math" display="block" id="S2.E11.m1.1"><semantics id="S2.E11.m1.1a"><mrow id="S2.E11.m1.1.1.1" xref="S2.E11.m1.1.1.1.1.cmml"><mrow id="S2.E11.m1.1.1.1.1" xref="S2.E11.m1.1.1.1.1.cmml"><msubsup id="S2.E11.m1.1.1.1.1.2" xref="S2.E11.m1.1.1.1.1.2.cmml"><mi id="S2.E11.m1.1.1.1.1.2.2.2" xref="S2.E11.m1.1.1.1.1.2.2.2.cmml">σ</mi><mi id="S2.E11.m1.1.1.1.1.2.2.3" xref="S2.E11.m1.1.1.1.1.2.2.3.cmml">cc</mi><mi id="S2.E11.m1.1.1.1.1.2.3" xref="S2.E11.m1.1.1.1.1.2.3.cmml">evap</mi></msubsup><mo id="S2.E11.m1.1.1.1.1.1" rspace="0.111em" xref="S2.E11.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E11.m1.1.1.1.1.3" xref="S2.E11.m1.1.1.1.1.3.cmml"><munderover id="S2.E11.m1.1.1.1.1.3.1" xref="S2.E11.m1.1.1.1.1.3.1.cmml"><mo id="S2.E11.m1.1.1.1.1.3.1.2.2" movablelimits="false" xref="S2.E11.m1.1.1.1.1.3.1.2.2.cmml">∑</mo><mrow id="S2.E11.m1.1.1.1.1.3.1.2.3" xref="S2.E11.m1.1.1.1.1.3.1.2.3.cmml"><mi id="S2.E11.m1.1.1.1.1.3.1.2.3.2" xref="S2.E11.m1.1.1.1.1.3.1.2.3.2.cmml">x</mi><mo id="S2.E11.m1.1.1.1.1.3.1.2.3.1" xref="S2.E11.m1.1.1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S2.E11.m1.1.1.1.1.3.1.2.3.3" xref="S2.E11.m1.1.1.1.1.3.1.2.3.3.cmml">1</mn></mrow><msub id="S2.E11.m1.1.1.1.1.3.1.3" xref="S2.E11.m1.1.1.1.1.3.1.3.cmml"><mi id="S2.E11.m1.1.1.1.1.3.1.3.2" xref="S2.E11.m1.1.1.1.1.3.1.3.2.cmml">N</mi><mi id="S2.E11.m1.1.1.1.1.3.1.3.3" xref="S2.E11.m1.1.1.1.1.3.1.3.3.cmml">P</mi></msub></munderover><mrow id="S2.E11.m1.1.1.1.1.3.2" xref="S2.E11.m1.1.1.1.1.3.2.cmml"><msub id="S2.E11.m1.1.1.1.1.3.2.2" xref="S2.E11.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E11.m1.1.1.1.1.3.2.2.2" xref="S2.E11.m1.1.1.1.1.3.2.2.2.cmml">σ</mi><mrow id="S2.E11.m1.1.1.1.1.3.2.2.3" xref="S2.E11.m1.1.1.1.1.3.2.2.3.cmml"><mo id="S2.E11.m1.1.1.1.1.3.2.2.3a" xref="S2.E11.m1.1.1.1.1.3.2.2.3.cmml">−</mo><mrow id="S2.E11.m1.1.1.1.1.3.2.2.3.2" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.cmml"><mi id="S2.E11.m1.1.1.1.1.3.2.2.3.2.2" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.2.cmml">x</mi><mo id="S2.E11.m1.1.1.1.1.3.2.2.3.2.1" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.1.cmml">⁢</mo><mi id="S2.E11.m1.1.1.1.1.3.2.2.3.2.3" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.3.cmml">n</mi></mrow></mrow></msub><mo id="S2.E11.m1.1.1.1.1.3.2.1" xref="S2.E11.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.E11.m1.1.1.1.1.3.2.3" xref="S2.E11.m1.1.1.1.1.3.2.3.cmml"><mi id="S2.E11.m1.1.1.1.1.3.2.3.2" xref="S2.E11.m1.1.1.1.1.3.2.3.2.cmml">P</mi><mrow id="S2.E11.m1.1.1.1.1.3.2.3.3" xref="S2.E11.m1.1.1.1.1.3.2.3.3.cmml"><mo id="S2.E11.m1.1.1.1.1.3.2.3.3a" xref="S2.E11.m1.1.1.1.1.3.2.3.3.cmml">−</mo><mrow id="S2.E11.m1.1.1.1.1.3.2.3.3.2" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.cmml"><mi id="S2.E11.m1.1.1.1.1.3.2.3.3.2.2" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.2.cmml">x</mi><mo id="S2.E11.m1.1.1.1.1.3.2.3.3.2.1" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.1.cmml">⁢</mo><mi id="S2.E11.m1.1.1.1.1.3.2.3.3.2.3" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.3.cmml">n</mi></mrow></mrow></msub></mrow></mrow></mrow><mo id="S2.E11.m1.1.1.1.2" xref="S2.E11.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E11.m1.1b"><apply id="S2.E11.m1.1.1.1.1.cmml" xref="S2.E11.m1.1.1.1"><eq id="S2.E11.m1.1.1.1.1.1.cmml" xref="S2.E11.m1.1.1.1.1.1"></eq><apply id="S2.E11.m1.1.1.1.1.2.cmml" xref="S2.E11.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E11.m1.1.1.1.1.2.1.cmml" xref="S2.E11.m1.1.1.1.1.2">superscript</csymbol><apply id="S2.E11.m1.1.1.1.1.2.2.cmml" xref="S2.E11.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E11.m1.1.1.1.1.2.2.1.cmml" xref="S2.E11.m1.1.1.1.1.2">subscript</csymbol><ci id="S2.E11.m1.1.1.1.1.2.2.2.cmml" xref="S2.E11.m1.1.1.1.1.2.2.2">𝜎</ci><ci id="S2.E11.m1.1.1.1.1.2.2.3.cmml" xref="S2.E11.m1.1.1.1.1.2.2.3">cc</ci></apply><ci id="S2.E11.m1.1.1.1.1.2.3.cmml" xref="S2.E11.m1.1.1.1.1.2.3">evap</ci></apply><apply id="S2.E11.m1.1.1.1.1.3.cmml" xref="S2.E11.m1.1.1.1.1.3"><apply id="S2.E11.m1.1.1.1.1.3.1.cmml" xref="S2.E11.m1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.E11.m1.1.1.1.1.3.1.1.cmml" xref="S2.E11.m1.1.1.1.1.3.1">superscript</csymbol><apply id="S2.E11.m1.1.1.1.1.3.1.2.cmml" xref="S2.E11.m1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.E11.m1.1.1.1.1.3.1.2.1.cmml" 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id="S2.E11.m1.1.1.1.1.3.2.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.E11.m1.1.1.1.1.3.2.2.1.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="S2.E11.m1.1.1.1.1.3.2.2.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.2">𝜎</ci><apply id="S2.E11.m1.1.1.1.1.3.2.2.3.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.3"><minus id="S2.E11.m1.1.1.1.1.3.2.2.3.1.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.3"></minus><apply id="S2.E11.m1.1.1.1.1.3.2.2.3.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2"><times id="S2.E11.m1.1.1.1.1.3.2.2.3.2.1.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.1"></times><ci id="S2.E11.m1.1.1.1.1.3.2.2.3.2.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.2">𝑥</ci><ci id="S2.E11.m1.1.1.1.1.3.2.2.3.2.3.cmml" xref="S2.E11.m1.1.1.1.1.3.2.2.3.2.3">𝑛</ci></apply></apply></apply><apply id="S2.E11.m1.1.1.1.1.3.2.3.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.E11.m1.1.1.1.1.3.2.3.1.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3">subscript</csymbol><ci id="S2.E11.m1.1.1.1.1.3.2.3.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.2">𝑃</ci><apply id="S2.E11.m1.1.1.1.1.3.2.3.3.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.3"><minus id="S2.E11.m1.1.1.1.1.3.2.3.3.1.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.3"></minus><apply id="S2.E11.m1.1.1.1.1.3.2.3.3.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2"><times id="S2.E11.m1.1.1.1.1.3.2.3.3.2.1.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.1"></times><ci id="S2.E11.m1.1.1.1.1.3.2.3.3.2.2.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.2">𝑥</ci><ci id="S2.E11.m1.1.1.1.1.3.2.3.3.2.3.cmml" xref="S2.E11.m1.1.1.1.1.3.2.3.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m1.1c">\sigma_{\rm cc}^{{\rm evap}}=\sum_{x=1}^{N_{P}}\sigma_{-xn}P_{-xn},</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_evap end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_x = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(11)</td> </tr></tbody> </table> with <math alttext="\sigma_{-xn}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><msub id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml">σ</mi><mrow id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml"><mo id="S2.SS2.p2.1.m1.1.1.3a" xref="S2.SS2.p2.1.m1.1.1.3.cmml">−</mo><mrow id="S2.SS2.p2.1.m1.1.1.3.2" xref="S2.SS2.p2.1.m1.1.1.3.2.cmml"><mi id="S2.SS2.p2.1.m1.1.1.3.2.2" xref="S2.SS2.p2.1.m1.1.1.3.2.2.cmml">x</mi><mo id="S2.SS2.p2.1.m1.1.1.3.2.1" xref="S2.SS2.p2.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS2.p2.1.m1.1.1.3.2.3" xref="S2.SS2.p2.1.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">𝜎</ci><apply id="S2.SS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3"><minus id="S2.SS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3"></minus><apply id="S2.SS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2"><times id="S2.SS2.p2.1.m1.1.1.3.2.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2.1"></times><ci id="S2.SS2.p2.1.m1.1.1.3.2.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2.2">𝑥</ci><ci id="S2.SS2.p2.1.m1.1.1.3.2.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\sigma_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as the partial neutron removal cross section in the abrasion stage as formulated in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E4" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">4</a>). <math alttext="P_{-xn}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><msub id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml">P</mi><mrow id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml"><mo id="S2.SS2.p2.2.m2.1.1.3a" xref="S2.SS2.p2.2.m2.1.1.3.cmml">−</mo><mrow id="S2.SS2.p2.2.m2.1.1.3.2" xref="S2.SS2.p2.2.m2.1.1.3.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.3.2.2" xref="S2.SS2.p2.2.m2.1.1.3.2.2.cmml">x</mi><mo id="S2.SS2.p2.2.m2.1.1.3.2.1" xref="S2.SS2.p2.2.m2.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS2.p2.2.m2.1.1.3.2.3" xref="S2.SS2.p2.2.m2.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><apply id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2">𝑃</ci><apply id="S2.SS2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3"><minus id="S2.SS2.p2.2.m2.1.1.3.1.cmml" xref="S2.SS2.p2.2.m2.1.1.3"></minus><apply id="S2.SS2.p2.2.m2.1.1.3.2.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2"><times id="S2.SS2.p2.2.m2.1.1.3.2.1.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2.1"></times><ci id="S2.SS2.p2.2.m2.1.1.3.2.2.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2.2">𝑥</ci><ci id="S2.SS2.p2.2.m2.1.1.3.2.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">P_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">italic_P start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as the corresponding charged particle evaporation probability after initial neutron removal, that depends on the excitation energy distribution (EED) of the prefragment and the probability to evaporate at least one proton at a certain excitation energy as, <table class="ltx_equation ltx_eqn_table" id="S2.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{-xn}=\int_{0}^{\infty}w_{-xn}(E_{\text{ex}})f(E_{\text{ex}},A_{P}-x,Z_{P})% \,dE_{\text{ex}}" class="ltx_Math" display="block" id="S2.E12.m1.4"><semantics id="S2.E12.m1.4a"><mrow id="S2.E12.m1.4.4" xref="S2.E12.m1.4.4.cmml"><msub id="S2.E12.m1.4.4.6" xref="S2.E12.m1.4.4.6.cmml"><mi id="S2.E12.m1.4.4.6.2" xref="S2.E12.m1.4.4.6.2.cmml">P</mi><mrow id="S2.E12.m1.4.4.6.3" xref="S2.E12.m1.4.4.6.3.cmml"><mo id="S2.E12.m1.4.4.6.3a" xref="S2.E12.m1.4.4.6.3.cmml">−</mo><mrow id="S2.E12.m1.4.4.6.3.2" xref="S2.E12.m1.4.4.6.3.2.cmml"><mi id="S2.E12.m1.4.4.6.3.2.2" xref="S2.E12.m1.4.4.6.3.2.2.cmml">x</mi><mo id="S2.E12.m1.4.4.6.3.2.1" xref="S2.E12.m1.4.4.6.3.2.1.cmml">⁢</mo><mi id="S2.E12.m1.4.4.6.3.2.3" xref="S2.E12.m1.4.4.6.3.2.3.cmml">n</mi></mrow></mrow></msub><mo id="S2.E12.m1.4.4.5" rspace="0.111em" xref="S2.E12.m1.4.4.5.cmml">=</mo><mrow id="S2.E12.m1.4.4.4" xref="S2.E12.m1.4.4.4.cmml"><msubsup id="S2.E12.m1.4.4.4.5" xref="S2.E12.m1.4.4.4.5.cmml"><mo id="S2.E12.m1.4.4.4.5.2.2" xref="S2.E12.m1.4.4.4.5.2.2.cmml">∫</mo><mn id="S2.E12.m1.4.4.4.5.2.3" xref="S2.E12.m1.4.4.4.5.2.3.cmml">0</mn><mi id="S2.E12.m1.4.4.4.5.3" mathvariant="normal" xref="S2.E12.m1.4.4.4.5.3.cmml">∞</mi></msubsup><mrow id="S2.E12.m1.4.4.4.4" xref="S2.E12.m1.4.4.4.4.cmml"><msub id="S2.E12.m1.4.4.4.4.6" xref="S2.E12.m1.4.4.4.4.6.cmml"><mi id="S2.E12.m1.4.4.4.4.6.2" xref="S2.E12.m1.4.4.4.4.6.2.cmml">w</mi><mrow id="S2.E12.m1.4.4.4.4.6.3" 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<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(12)</td> </tr></tbody> </table> <math alttext="E_{\mathrm{ex}}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m1.1"><semantics id="S2.SS2.p2.3.m1.1a"><msub id="S2.SS2.p2.3.m1.1.1" xref="S2.SS2.p2.3.m1.1.1.cmml"><mi id="S2.SS2.p2.3.m1.1.1.2" xref="S2.SS2.p2.3.m1.1.1.2.cmml">E</mi><mi id="S2.SS2.p2.3.m1.1.1.3" xref="S2.SS2.p2.3.m1.1.1.3.cmml">ex</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m1.1b"><apply id="S2.SS2.p2.3.m1.1.1.cmml" xref="S2.SS2.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m1.1.1.1.cmml" xref="S2.SS2.p2.3.m1.1.1">subscript</csymbol><ci id="S2.SS2.p2.3.m1.1.1.2.cmml" xref="S2.SS2.p2.3.m1.1.1.2">𝐸</ci><ci id="S2.SS2.p2.3.m1.1.1.3.cmml" xref="S2.SS2.p2.3.m1.1.1.3">ex</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m1.1c">E_{\mathrm{ex}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m1.1d">italic_E start_POSTSUBSCRIPT roman_ex end_POSTSUBSCRIPT</annotation></semantics></math> is the excitation energy of the residual fragment after <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m2.1"><semantics id="S2.SS2.p2.4.m2.1a"><mi id="S2.SS2.p2.4.m2.1.1" xref="S2.SS2.p2.4.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m2.1b"><ci id="S2.SS2.p2.4.m2.1.1.cmml" xref="S2.SS2.p2.4.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m2.1d">italic_x</annotation></semantics></math> neutron removal, <math alttext="f(E_{\rm ex},A_{p}-x,Z_{p})" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m3.3"><semantics id="S2.SS2.p2.5.m3.3a"><mrow id="S2.SS2.p2.5.m3.3.3" xref="S2.SS2.p2.5.m3.3.3.cmml"><mi id="S2.SS2.p2.5.m3.3.3.5" xref="S2.SS2.p2.5.m3.3.3.5.cmml">f</mi><mo id="S2.SS2.p2.5.m3.3.3.4" 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xref="S2.SS2.p2.5.m3.2.2.2.2.2.3.cmml">x</mi></mrow><mo id="S2.SS2.p2.5.m3.3.3.3.3.6" xref="S2.SS2.p2.5.m3.3.3.3.4.cmml">,</mo><msub id="S2.SS2.p2.5.m3.3.3.3.3.3" xref="S2.SS2.p2.5.m3.3.3.3.3.3.cmml"><mi id="S2.SS2.p2.5.m3.3.3.3.3.3.2" xref="S2.SS2.p2.5.m3.3.3.3.3.3.2.cmml">Z</mi><mi id="S2.SS2.p2.5.m3.3.3.3.3.3.3" xref="S2.SS2.p2.5.m3.3.3.3.3.3.3.cmml">p</mi></msub><mo id="S2.SS2.p2.5.m3.3.3.3.3.7" stretchy="false" xref="S2.SS2.p2.5.m3.3.3.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m3.3b"><apply id="S2.SS2.p2.5.m3.3.3.cmml" xref="S2.SS2.p2.5.m3.3.3"><times id="S2.SS2.p2.5.m3.3.3.4.cmml" xref="S2.SS2.p2.5.m3.3.3.4"></times><ci id="S2.SS2.p2.5.m3.3.3.5.cmml" xref="S2.SS2.p2.5.m3.3.3.5">𝑓</ci><vector id="S2.SS2.p2.5.m3.3.3.3.4.cmml" xref="S2.SS2.p2.5.m3.3.3.3.3"><apply id="S2.SS2.p2.5.m3.1.1.1.1.1.cmml" xref="S2.SS2.p2.5.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m3.1.1.1.1.1.1.cmml" 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xref="S2.SS2.p2.5.m3.3.3.3.3.3">subscript</csymbol><ci id="S2.SS2.p2.5.m3.3.3.3.3.3.2.cmml" xref="S2.SS2.p2.5.m3.3.3.3.3.3.2">𝑍</ci><ci id="S2.SS2.p2.5.m3.3.3.3.3.3.3.cmml" xref="S2.SS2.p2.5.m3.3.3.3.3.3.3">𝑝</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m3.3c">f(E_{\rm ex},A_{p}-x,Z_{p})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m3.3d">italic_f ( italic_E start_POSTSUBSCRIPT roman_ex end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - italic_x , italic_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math> is the proton evaporation probability (same as the branching ratio) at certain excitation energy, <math alttext="w_{-xn}(E_{\rm ex})" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m4.1"><semantics id="S2.SS2.p2.6.m4.1a"><mrow id="S2.SS2.p2.6.m4.1.1" xref="S2.SS2.p2.6.m4.1.1.cmml"><msub id="S2.SS2.p2.6.m4.1.1.3" xref="S2.SS2.p2.6.m4.1.1.3.cmml"><mi id="S2.SS2.p2.6.m4.1.1.3.2" xref="S2.SS2.p2.6.m4.1.1.3.2.cmml">w</mi><mrow id="S2.SS2.p2.6.m4.1.1.3.3" xref="S2.SS2.p2.6.m4.1.1.3.3.cmml"><mo id="S2.SS2.p2.6.m4.1.1.3.3a" xref="S2.SS2.p2.6.m4.1.1.3.3.cmml">−</mo><mrow id="S2.SS2.p2.6.m4.1.1.3.3.2" xref="S2.SS2.p2.6.m4.1.1.3.3.2.cmml"><mi id="S2.SS2.p2.6.m4.1.1.3.3.2.2" xref="S2.SS2.p2.6.m4.1.1.3.3.2.2.cmml">x</mi><mo id="S2.SS2.p2.6.m4.1.1.3.3.2.1" xref="S2.SS2.p2.6.m4.1.1.3.3.2.1.cmml">⁢</mo><mi id="S2.SS2.p2.6.m4.1.1.3.3.2.3" xref="S2.SS2.p2.6.m4.1.1.3.3.2.3.cmml">n</mi></mrow></mrow></msub><mo id="S2.SS2.p2.6.m4.1.1.2" xref="S2.SS2.p2.6.m4.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.6.m4.1.1.1.1" xref="S2.SS2.p2.6.m4.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.6.m4.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.6.m4.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p2.6.m4.1.1.1.1.1" xref="S2.SS2.p2.6.m4.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.6.m4.1.1.1.1.1.2" xref="S2.SS2.p2.6.m4.1.1.1.1.1.2.cmml">E</mi><mi id="S2.SS2.p2.6.m4.1.1.1.1.1.3" xref="S2.SS2.p2.6.m4.1.1.1.1.1.3.cmml">ex</mi></msub><mo id="S2.SS2.p2.6.m4.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.6.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m4.1b"><apply id="S2.SS2.p2.6.m4.1.1.cmml" xref="S2.SS2.p2.6.m4.1.1"><times id="S2.SS2.p2.6.m4.1.1.2.cmml" xref="S2.SS2.p2.6.m4.1.1.2"></times><apply id="S2.SS2.p2.6.m4.1.1.3.cmml" xref="S2.SS2.p2.6.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.6.m4.1.1.3.1.cmml" xref="S2.SS2.p2.6.m4.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.6.m4.1.1.3.2.cmml" xref="S2.SS2.p2.6.m4.1.1.3.2">𝑤</ci><apply id="S2.SS2.p2.6.m4.1.1.3.3.cmml" xref="S2.SS2.p2.6.m4.1.1.3.3"><minus id="S2.SS2.p2.6.m4.1.1.3.3.1.cmml" xref="S2.SS2.p2.6.m4.1.1.3.3"></minus><apply id="S2.SS2.p2.6.m4.1.1.3.3.2.cmml" xref="S2.SS2.p2.6.m4.1.1.3.3.2"><times id="S2.SS2.p2.6.m4.1.1.3.3.2.1.cmml" xref="S2.SS2.p2.6.m4.1.1.3.3.2.1"></times><ci id="S2.SS2.p2.6.m4.1.1.3.3.2.2.cmml" 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The Gaimard-Schmidt(GS) approach <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib42" title="">42</a>]</cite> is used to calculate the EED of the residual fragment as in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>]</cite>. 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(<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E6" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">6</a>)) calculated in the abrasion stage that uses the Glauber model with realistic density inputs. We then calculate proton evaporation contributions in the ablation stage and analyze their correlation with key influencing factors related to the evaporation process. </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> III.1 Single-nucleon density inputs</h3> <figure class="ltx_figure" id="S3.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="607" id="S3.F1.g1" src="x1.png" width="498"/> <figcaption class="ltx_caption ltx_centering">Figure 1: Single-nucleon density distributions for <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.F1.8.m1.1"><semantics id="S3.F1.8.m1.1b"><mmultiscripts id="S3.F1.8.m1.1.1" xref="S3.F1.8.m1.1.1.cmml"><mtext id="S3.F1.8.m1.1.1.2" xref="S3.F1.8.m1.1.1.2a.cmml">C</mtext><mprescripts id="S3.F1.8.m1.1.1b" xref="S3.F1.8.m1.1.1.cmml"></mprescripts><mrow id="S3.F1.8.m1.1.1c" xref="S3.F1.8.m1.1.1.cmml"></mrow><mn id="S3.F1.8.m1.1.1.3" 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id="S3.F1.10.m3.1e">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O</annotation></semantics></math>, and <math alttext="{}^{20}\text{Ne}" class="ltx_Math" display="inline" id="S3.F1.11.m4.1"><semantics id="S3.F1.11.m4.1b"><mmultiscripts id="S3.F1.11.m4.1.1" xref="S3.F1.11.m4.1.1.cmml"><mtext id="S3.F1.11.m4.1.1.2" xref="S3.F1.11.m4.1.1.2a.cmml">Ne</mtext><mprescripts id="S3.F1.11.m4.1.1b" xref="S3.F1.11.m4.1.1.cmml"></mprescripts><mrow id="S3.F1.11.m4.1.1c" xref="S3.F1.11.m4.1.1.cmml"></mrow><mn id="S3.F1.11.m4.1.1.3" xref="S3.F1.11.m4.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F1.11.m4.1c"><apply id="S3.F1.11.m4.1.1.cmml" xref="S3.F1.11.m4.1.1"><csymbol cd="ambiguous" id="S3.F1.11.m4.1.1.1.cmml" xref="S3.F1.11.m4.1.1">superscript</csymbol><ci id="S3.F1.11.m4.1.1.2a.cmml" xref="S3.F1.11.m4.1.1.2"><mtext id="S3.F1.11.m4.1.1.2.cmml" xref="S3.F1.11.m4.1.1.2">Ne</mtext></ci><cn id="S3.F1.11.m4.1.1.3.cmml" type="integer" xref="S3.F1.11.m4.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.11.m4.1d">{}^{20}\text{Ne}</annotation><annotation encoding="application/x-llamapun" id="S3.F1.11.m4.1e">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT Ne</annotation></semantics></math>, presented from top to bottom, respectively. Different nuclear density configurations taken from Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib45" title="">45</a>]</cite> for 12C and 16O and Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib46" title="">46</a>]</cite> for 20Ne are considered, as discussed in the text.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p1"> We now show a few input density configurations for 12C, 14N, 16O, and 20Ne in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F1" title="Figure 1 ‣ III.1 Single-nucleon density inputs ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">1</a>, considered in the literature. For <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mmultiscripts id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml"><mtext id="S3.SS1.p1.5.m5.1.1.2" xref="S3.SS1.p1.5.m5.1.1.2a.cmml">C</mtext><mprescripts id="S3.SS1.p1.5.m5.1.1a" xref="S3.SS1.p1.5.m5.1.1.cmml"></mprescripts><mrow id="S3.SS1.p1.5.m5.1.1b" xref="S3.SS1.p1.5.m5.1.1.cmml"></mrow><mn id="S3.SS1.p1.5.m5.1.1.3" xref="S3.SS1.p1.5.m5.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><apply id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">superscript</csymbol><ci id="S3.SS1.p1.5.m5.1.1.2a.cmml" xref="S3.SS1.p1.5.m5.1.1.2"><mtext id="S3.SS1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.p1.5.m5.1.1.2">C</mtext></ci><cn id="S3.SS1.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.p1.5.m5.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">{}^{12}\text{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT C</annotation></semantics></math> and <math alttext="{}^{16}\text{O}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><mmultiscripts id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml"><mtext id="S3.SS1.p1.6.m6.1.1.2" xref="S3.SS1.p1.6.m6.1.1.2a.cmml">O</mtext><mprescripts id="S3.SS1.p1.6.m6.1.1a" xref="S3.SS1.p1.6.m6.1.1.cmml"></mprescripts><mrow id="S3.SS1.p1.6.m6.1.1b" xref="S3.SS1.p1.6.m6.1.1.cmml"></mrow><mn id="S3.SS1.p1.6.m6.1.1.3" xref="S3.SS1.p1.6.m6.1.1.3.cmml">16</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><apply id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1">superscript</csymbol><ci id="S3.SS1.p1.6.m6.1.1.2a.cmml" xref="S3.SS1.p1.6.m6.1.1.2"><mtext id="S3.SS1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2">O</mtext></ci><cn id="S3.SS1.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.SS1.p1.6.m6.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">{}^{16}\text{O}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O</annotation></semantics></math>, the single-nucleon densities are calculated based on the 3<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.1"><semantics id="S3.SS1.p1.7.m7.1a"><mi id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><ci id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.1d">italic_α</annotation></semantics></math> and 4<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m8.1"><semantics id="S3.SS1.p1.8.m8.1a"><mi id="S3.SS1.p1.8.m8.1.1" xref="S3.SS1.p1.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m8.1b"><ci id="S3.SS1.p1.8.m8.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m8.1d">italic_α</annotation></semantics></math> cluster configurations (C-type), respectively. A shell model type (S-type) density is also considered for both nuclei. It is also well known that <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m9.1"><semantics id="S3.SS1.p1.9.m9.1a"><mmultiscripts id="S3.SS1.p1.9.m9.1.1" xref="S3.SS1.p1.9.m9.1.1.cmml"><mtext id="S3.SS1.p1.9.m9.1.1.2" xref="S3.SS1.p1.9.m9.1.1.2a.cmml">C</mtext><mprescripts id="S3.SS1.p1.9.m9.1.1a" xref="S3.SS1.p1.9.m9.1.1.cmml"></mprescripts><mrow id="S3.SS1.p1.9.m9.1.1b" xref="S3.SS1.p1.9.m9.1.1.cmml"></mrow><mn id="S3.SS1.p1.9.m9.1.1.3" xref="S3.SS1.p1.9.m9.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m9.1b"><apply id="S3.SS1.p1.9.m9.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m9.1.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1">superscript</csymbol><ci id="S3.SS1.p1.9.m9.1.1.2a.cmml" xref="S3.SS1.p1.9.m9.1.1.2"><mtext id="S3.SS1.p1.9.m9.1.1.2.cmml" xref="S3.SS1.p1.9.m9.1.1.2">C</mtext></ci><cn id="S3.SS1.p1.9.m9.1.1.3.cmml" type="integer" xref="S3.SS1.p1.9.m9.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m9.1c">{}^{12}\text{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m9.1d">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT C</annotation></semantics></math> represents a mixture of shell and cluster configurations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib48" title="">48</a>]</cite>. To account for this mixture, the mixed configuration (M-type) is additionally considered for <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m10.1"><semantics id="S3.SS1.p1.10.m10.1a"><mmultiscripts id="S3.SS1.p1.10.m10.1.1" xref="S3.SS1.p1.10.m10.1.1.cmml"><mtext id="S3.SS1.p1.10.m10.1.1.2" xref="S3.SS1.p1.10.m10.1.1.2a.cmml">C</mtext><mprescripts id="S3.SS1.p1.10.m10.1.1a" xref="S3.SS1.p1.10.m10.1.1.cmml"></mprescripts><mrow id="S3.SS1.p1.10.m10.1.1b" xref="S3.SS1.p1.10.m10.1.1.cmml"></mrow><mn id="S3.SS1.p1.10.m10.1.1.3" xref="S3.SS1.p1.10.m10.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m10.1b"><apply id="S3.SS1.p1.10.m10.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1">superscript</csymbol><ci id="S3.SS1.p1.10.m10.1.1.2a.cmml" xref="S3.SS1.p1.10.m10.1.1.2"><mtext id="S3.SS1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2">C</mtext></ci><cn id="S3.SS1.p1.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p1.10.m10.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m10.1c">{}^{12}\text{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m10.1d">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT C</annotation></semantics></math>. The shell and cluster density profiles are derived by the approach outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib49" title="">49</a>]</cite> and tailoring it for the current multi-<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p1.11.m11.1"><semantics id="S3.SS1.p1.11.m11.1a"><mi id="S3.SS1.p1.11.m11.1.1" xref="S3.SS1.p1.11.m11.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m11.1b"><ci id="S3.SS1.p1.11.m11.1.1.cmml" xref="S3.SS1.p1.11.m11.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m11.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.11.m11.1d">italic_α</annotation></semantics></math> scenarios. Details of the theoretical framework underlying this calculation are provided in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib49" title="">49</a>]</cite>. The S-type and C-type configurations show significantly different density profiles, despite having the same root-mean-square radii. The density profile near the surface of the C-type configuration declines more steeply than that of the S-type configuration, which may be more effective for peripheral nucleon removal. </div> <div class="ltx_para" id="S3.SS1.p2"> The density of 14N is calculated using shell model configuration with a harmonic oscillator (HO) basis. </div> <div class="ltx_para" id="S3.SS1.p3"> For <math alttext="{}^{20}\text{Ne}" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mmultiscripts id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml"><mtext id="S3.SS1.p3.1.m1.1.1.2" xref="S3.SS1.p3.1.m1.1.1.2a.cmml">Ne</mtext><mprescripts id="S3.SS1.p3.1.m1.1.1a" xref="S3.SS1.p3.1.m1.1.1.cmml"></mprescripts><mrow id="S3.SS1.p3.1.m1.1.1b" xref="S3.SS1.p3.1.m1.1.1.cmml"></mrow><mn id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">superscript</csymbol><ci id="S3.SS1.p3.1.m1.1.1.2a.cmml" xref="S3.SS1.p3.1.m1.1.1.2"><mtext id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.2">Ne</mtext></ci><cn id="S3.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p3.1.m1.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">{}^{20}\text{Ne}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT Ne</annotation></semantics></math>, two cluster configurations (5<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mi id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">italic_α</annotation></semantics></math> and <math alttext="{}^{16}\text{O}+\alpha" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mrow id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mmultiscripts id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml"><mtext id="S3.SS1.p3.3.m3.1.1.2.2" xref="S3.SS1.p3.3.m3.1.1.2.2a.cmml">O</mtext><mprescripts id="S3.SS1.p3.3.m3.1.1.2a" xref="S3.SS1.p3.3.m3.1.1.2.cmml"></mprescripts><mrow id="S3.SS1.p3.3.m3.1.1.2b" xref="S3.SS1.p3.3.m3.1.1.2.cmml"></mrow><mn id="S3.SS1.p3.3.m3.1.1.2.3" xref="S3.SS1.p3.3.m3.1.1.2.3.cmml">16</mn></mmultiscripts><mo id="S3.SS1.p3.3.m3.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.cmml">+</mo><mi id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><plus id="S3.SS1.p3.3.m3.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1.1"></plus><apply id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.2.1.cmml" xref="S3.SS1.p3.3.m3.1.1.2">superscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.2.2a.cmml" xref="S3.SS1.p3.3.m3.1.1.2.2"><mtext id="S3.SS1.p3.3.m3.1.1.2.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2.2">O</mtext></ci><cn id="S3.SS1.p3.3.m3.1.1.2.3.cmml" type="integer" xref="S3.SS1.p3.3.m3.1.1.2.3">16</cn></apply><ci id="S3.SS1.p3.3.m3.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">{}^{16}\text{O}+\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O + italic_α</annotation></semantics></math>) and two shell-model configurations (<math alttext="j" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mi id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><ci id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">italic_j</annotation></semantics></math>-<math alttext="j" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.1"><semantics id="S3.SS1.p3.5.m5.1a"><mi id="S3.SS1.p3.5.m5.1.1" xref="S3.SS1.p3.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.1b"><ci id="S3.SS1.p3.5.m5.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.1d">italic_j</annotation></semantics></math> coupling and SU(3)) are examined. The <math alttext="j" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.1"><semantics id="S3.SS1.p3.6.m6.1a"><mi id="S3.SS1.p3.6.m6.1.1" xref="S3.SS1.p3.6.m6.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.1b"><ci id="S3.SS1.p3.6.m6.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.1d">italic_j</annotation></semantics></math>-<math alttext="j" class="ltx_Math" display="inline" id="S3.SS1.p3.7.m7.1"><semantics id="S3.SS1.p3.7.m7.1a"><mi id="S3.SS1.p3.7.m7.1.1" xref="S3.SS1.p3.7.m7.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.7.m7.1b"><ci id="S3.SS1.p3.7.m7.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.7.m7.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.7.m7.1d">italic_j</annotation></semantics></math> coupling configuration corresponds to <math alttext="(0d_{5/2})^{4}" class="ltx_Math" display="inline" id="S3.SS1.p3.8.m8.1"><semantics id="S3.SS1.p3.8.m8.1a"><msup id="S3.SS1.p3.8.m8.1.1" xref="S3.SS1.p3.8.m8.1.1.cmml"><mrow id="S3.SS1.p3.8.m8.1.1.1.1" xref="S3.SS1.p3.8.m8.1.1.1.1.1.cmml"><mo id="S3.SS1.p3.8.m8.1.1.1.1.2" stretchy="false" xref="S3.SS1.p3.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.p3.8.m8.1.1.1.1.1" xref="S3.SS1.p3.8.m8.1.1.1.1.1.cmml"><mn id="S3.SS1.p3.8.m8.1.1.1.1.1.2" xref="S3.SS1.p3.8.m8.1.1.1.1.1.2.cmml">0</mn><mo id="S3.SS1.p3.8.m8.1.1.1.1.1.1" xref="S3.SS1.p3.8.m8.1.1.1.1.1.1.cmml">⁢</mo><msub id="S3.SS1.p3.8.m8.1.1.1.1.1.3" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.cmml"><mi id="S3.SS1.p3.8.m8.1.1.1.1.1.3.2" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.2.cmml">d</mi><mrow id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.cmml"><mn id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.2" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.2.cmml">5</mn><mo id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.1" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.1.cmml">/</mo><mn id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.3" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.3.cmml">2</mn></mrow></msub></mrow><mo id="S3.SS1.p3.8.m8.1.1.1.1.3" stretchy="false" xref="S3.SS1.p3.8.m8.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.SS1.p3.8.m8.1.1.3" xref="S3.SS1.p3.8.m8.1.1.3.cmml">4</mn></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.8.m8.1b"><apply id="S3.SS1.p3.8.m8.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.8.m8.1.1.2.cmml" xref="S3.SS1.p3.8.m8.1.1">superscript</csymbol><apply id="S3.SS1.p3.8.m8.1.1.1.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1"><times id="S3.SS1.p3.8.m8.1.1.1.1.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1.1.1"></times><cn id="S3.SS1.p3.8.m8.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS1.p3.8.m8.1.1.1.1.1.2">0</cn><apply id="S3.SS1.p3.8.m8.1.1.1.1.1.3.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.8.m8.1.1.1.1.1.3.1.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.8.m8.1.1.1.1.1.3.2.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.2">𝑑</ci><apply id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3"><divide id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.1.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.1"></divide><cn id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.2">5</cn><cn id="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p3.8.m8.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply><cn id="S3.SS1.p3.8.m8.1.1.3.cmml" type="integer" xref="S3.SS1.p3.8.m8.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.8.m8.1c">(0d_{5/2})^{4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.8.m8.1d">( 0 italic_d start_POSTSUBSCRIPT 5 / 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, where four nucleons occupy the <math alttext="0d_{5/2}" class="ltx_Math" display="inline" id="S3.SS1.p3.9.m9.1"><semantics id="S3.SS1.p3.9.m9.1a"><mrow id="S3.SS1.p3.9.m9.1.1" xref="S3.SS1.p3.9.m9.1.1.cmml"><mn id="S3.SS1.p3.9.m9.1.1.2" xref="S3.SS1.p3.9.m9.1.1.2.cmml">0</mn><mo id="S3.SS1.p3.9.m9.1.1.1" xref="S3.SS1.p3.9.m9.1.1.1.cmml">⁢</mo><msub id="S3.SS1.p3.9.m9.1.1.3" xref="S3.SS1.p3.9.m9.1.1.3.cmml"><mi id="S3.SS1.p3.9.m9.1.1.3.2" xref="S3.SS1.p3.9.m9.1.1.3.2.cmml">d</mi><mrow id="S3.SS1.p3.9.m9.1.1.3.3" xref="S3.SS1.p3.9.m9.1.1.3.3.cmml"><mn id="S3.SS1.p3.9.m9.1.1.3.3.2" xref="S3.SS1.p3.9.m9.1.1.3.3.2.cmml">5</mn><mo id="S3.SS1.p3.9.m9.1.1.3.3.1" xref="S3.SS1.p3.9.m9.1.1.3.3.1.cmml">/</mo><mn id="S3.SS1.p3.9.m9.1.1.3.3.3" xref="S3.SS1.p3.9.m9.1.1.3.3.3.cmml">2</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.9.m9.1b"><apply id="S3.SS1.p3.9.m9.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1"><times id="S3.SS1.p3.9.m9.1.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1.1"></times><cn id="S3.SS1.p3.9.m9.1.1.2.cmml" type="integer" xref="S3.SS1.p3.9.m9.1.1.2">0</cn><apply id="S3.SS1.p3.9.m9.1.1.3.cmml" xref="S3.SS1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.9.m9.1.1.3.1.cmml" xref="S3.SS1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.9.m9.1.1.3.2.cmml" xref="S3.SS1.p3.9.m9.1.1.3.2">𝑑</ci><apply id="S3.SS1.p3.9.m9.1.1.3.3.cmml" xref="S3.SS1.p3.9.m9.1.1.3.3"><divide id="S3.SS1.p3.9.m9.1.1.3.3.1.cmml" xref="S3.SS1.p3.9.m9.1.1.3.3.1"></divide><cn id="S3.SS1.p3.9.m9.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p3.9.m9.1.1.3.3.2">5</cn><cn id="S3.SS1.p3.9.m9.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p3.9.m9.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.9.m9.1c">0d_{5/2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.9.m9.1d">0 italic_d start_POSTSUBSCRIPT 5 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> orbital, while the SU(3) configuration corresponds to <math alttext="(1s0d)^{4}" class="ltx_Math" display="inline" id="S3.SS1.p3.10.m10.1"><semantics id="S3.SS1.p3.10.m10.1a"><msup id="S3.SS1.p3.10.m10.1.1" xref="S3.SS1.p3.10.m10.1.1.cmml"><mrow id="S3.SS1.p3.10.m10.1.1.1.1" xref="S3.SS1.p3.10.m10.1.1.1.1.1.cmml"><mo id="S3.SS1.p3.10.m10.1.1.1.1.2" stretchy="false" xref="S3.SS1.p3.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.p3.10.m10.1.1.1.1.1" xref="S3.SS1.p3.10.m10.1.1.1.1.1.cmml"><mn id="S3.SS1.p3.10.m10.1.1.1.1.1.2" xref="S3.SS1.p3.10.m10.1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS1.p3.10.m10.1.1.1.1.1.1" xref="S3.SS1.p3.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p3.10.m10.1.1.1.1.1.3" xref="S3.SS1.p3.10.m10.1.1.1.1.1.3.cmml">s</mi><mo id="S3.SS1.p3.10.m10.1.1.1.1.1.1a" xref="S3.SS1.p3.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mn id="S3.SS1.p3.10.m10.1.1.1.1.1.4" xref="S3.SS1.p3.10.m10.1.1.1.1.1.4.cmml">0</mn><mo 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id="S3.SS1.p3.12.m12.1b"><apply id="S3.SS1.p3.12.m12.1.1.cmml" xref="S3.SS1.p3.12.m12.1.1"><times id="S3.SS1.p3.12.m12.1.1.1.cmml" xref="S3.SS1.p3.12.m12.1.1.1"></times><cn id="S3.SS1.p3.12.m12.1.1.2.cmml" type="integer" xref="S3.SS1.p3.12.m12.1.1.2">0</cn><ci id="S3.SS1.p3.12.m12.1.1.3.cmml" xref="S3.SS1.p3.12.m12.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.12.m12.1c">0d</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.12.m12.1d">0 italic_d</annotation></semantics></math> shells based on SU(3) symmetry. </div> <div class="ltx_para" id="S3.SS1.p4"> These different density configurations have been systematically studied through calculations of angular distributions in proton elastic scattering, as reported in Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib46" title="">46</a>]</cite>. Based on these studies, the most appropriate density configurations are the Mixed type (M-type) for 12C, a 4<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mi id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><ci id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">italic_α</annotation></semantics></math> cluster type (C-type) for 16O, and a 16O + <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p4.5.m5.1"><semantics id="S3.SS1.p4.5.m5.1a"><mi id="S3.SS1.p4.5.m5.1.1" xref="S3.SS1.p4.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.5.m5.1b"><ci id="S3.SS1.p4.5.m5.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.5.m5.1d">italic_α</annotation></semantics></math> (O-<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p4.6.m6.1"><semantics id="S3.SS1.p4.6.m6.1a"><mi id="S3.SS1.p4.6.m6.1.1" xref="S3.SS1.p4.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.6.m6.1b"><ci id="S3.SS1.p4.6.m6.1.1.cmml" xref="S3.SS1.p4.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.6.m6.1d">italic_α</annotation></semantics></math> type) bicluster structure for 20Ne. Our study will use these specific density distributions for cross section calculations in the abrasion step, ensuring consistency with the most appropriate structure models identified from proton elastic scattering studies. </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> III.2 Direct fragmentation or abrasion stage</h3> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption ltx_centering">Table 1: Charge-changing cross sections in the abrasion stage of projectile fragmentation for 12C, 14N, 16O, and 20Ne on elemental targets C, Al, and Cu at a beam energy of 290 MeV/nucleon. Calculations were performed using the finite-range optical limit approximation (FROLA) with the most appropriate nuclear density inputs: mixed-type (M-type) density for 12C, a shell model with a harmonic oscillator basis for 14N, C-type (4<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.T1.18.m7.1"><semantics id="S3.T1.18.m7.1b"><mi id="S3.T1.18.m7.1.1" xref="S3.T1.18.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.T1.18.m7.1c"><ci id="S3.T1.18.m7.1.1.cmml" xref="S3.T1.18.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.18.m7.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.T1.18.m7.1e">italic_α</annotation></semantics></math> cluster) density for 16O, and a 16O + <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.T1.21.m10.1"><semantics id="S3.T1.21.m10.1b"><mi id="S3.T1.21.m10.1.1" xref="S3.T1.21.m10.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.T1.21.m10.1c"><ci id="S3.T1.21.m10.1.1.cmml" xref="S3.T1.21.m10.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.21.m10.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.T1.21.m10.1e">italic_α</annotation></semantics></math> cluster configuration for 20Ne. All cross sections in this table are rounded off to the nearest integer.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T1.28"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T1.23.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T1.23.1.2" style="padding:-1pt 5.0pt;">Projectile</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T1.23.1.3" style="padding:-1pt 5.0pt;">Target</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T1.23.1.4" style="padding:-1pt 5.0pt;">Projectile</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T1.23.1.1" style="padding:-1pt 5.0pt;"> <math alttext="\sigma_{\rm cc}" class="ltx_Math" display="inline" id="S3.T1.23.1.1.m1.1"><semantics id="S3.T1.23.1.1.m1.1a"><msub id="S3.T1.23.1.1.m1.1.1" xref="S3.T1.23.1.1.m1.1.1.cmml"><mi id="S3.T1.23.1.1.m1.1.1.2" xref="S3.T1.23.1.1.m1.1.1.2.cmml">σ</mi><mi id="S3.T1.23.1.1.m1.1.1.3" xref="S3.T1.23.1.1.m1.1.1.3.cmml">cc</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.23.1.1.m1.1b"><apply id="S3.T1.23.1.1.m1.1.1.cmml" xref="S3.T1.23.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.23.1.1.m1.1.1.1.cmml" xref="S3.T1.23.1.1.m1.1.1">subscript</csymbol><ci id="S3.T1.23.1.1.m1.1.1.2.cmml" xref="S3.T1.23.1.1.m1.1.1.2">𝜎</ci><ci id="S3.T1.23.1.1.m1.1.1.3.cmml" xref="S3.T1.23.1.1.m1.1.1.3">cc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.23.1.1.m1.1c">\sigma_{\rm cc}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.23.1.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math> (mb)</th> </tr> <tr class="ltx_tr" id="S3.T1.28.7.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.28.7.1.1" style="padding:-1pt 5.0pt;">Nucleus</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.28.7.1.2" style="padding:-1pt 5.0pt;">Nucleus</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.28.7.1.3" style="padding:-1pt 5.0pt;">Density Type</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.28.7.1.4" style="padding:-1pt 5.0pt;">(Abrasion)</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.24.2"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.24.2.1" style="padding:-1pt 5.0pt;"> 12C</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.24.2.2" style="padding:-1pt 5.0pt;">C</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.24.2.3" style="padding:-1pt 5.0pt;">M-type</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.24.2.4" style="padding:-1pt 5.0pt;">650</td> </tr> <tr class="ltx_tr" id="S3.T1.28.8.1"> <td class="ltx_td" id="S3.T1.28.8.1.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.8.1.2" style="padding:-1pt 5.0pt;">Al</td> <td class="ltx_td" id="S3.T1.28.8.1.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.8.1.4" style="padding:-1pt 5.0pt;">1001</td> </tr> <tr class="ltx_tr" id="S3.T1.28.9.2"> <td class="ltx_td" id="S3.T1.28.9.2.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.9.2.2" style="padding:-1pt 5.0pt;">Cu</td> <td class="ltx_td" id="S3.T1.28.9.2.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.9.2.4" style="padding:-1pt 5.0pt;">1568</td> </tr> <tr class="ltx_tr" id="S3.T1.25.3"> <td class="ltx_td ltx_align_center" id="S3.T1.25.3.1" style="padding:-1pt 5.0pt;"> 14N</td> <td class="ltx_td ltx_align_center" id="S3.T1.25.3.2" style="padding:-1pt 5.0pt;">C</td> <td class="ltx_td ltx_align_center" id="S3.T1.25.3.3" style="padding:-1pt 5.0pt;">HO-type</td> <td class="ltx_td ltx_align_center" id="S3.T1.25.3.4" style="padding:-1pt 5.0pt;">720</td> </tr> <tr class="ltx_tr" id="S3.T1.28.10.3"> <td class="ltx_td" id="S3.T1.28.10.3.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.10.3.2" style="padding:-1pt 5.0pt;">Al</td> <td class="ltx_td" id="S3.T1.28.10.3.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.10.3.4" style="padding:-1pt 5.0pt;">1070</td> </tr> <tr class="ltx_tr" id="S3.T1.28.11.4"> <td class="ltx_td" id="S3.T1.28.11.4.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.11.4.2" style="padding:-1pt 5.0pt;">Cu</td> <td class="ltx_td" id="S3.T1.28.11.4.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.11.4.4" style="padding:-1pt 5.0pt;">1648</td> </tr> <tr class="ltx_tr" id="S3.T1.26.4"> <td class="ltx_td ltx_align_center" id="S3.T1.26.4.1" style="padding:-1pt 5.0pt;"> 16O</td> <td class="ltx_td ltx_align_center" id="S3.T1.26.4.2" style="padding:-1pt 5.0pt;">C</td> <td class="ltx_td ltx_align_center" id="S3.T1.26.4.3" style="padding:-1pt 5.0pt;">C-type</td> <td class="ltx_td ltx_align_center" id="S3.T1.26.4.4" style="padding:-1pt 5.0pt;">772</td> </tr> <tr class="ltx_tr" id="S3.T1.28.12.5"> <td class="ltx_td" id="S3.T1.28.12.5.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.12.5.2" style="padding:-1pt 5.0pt;">Al</td> <td class="ltx_td" id="S3.T1.28.12.5.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.12.5.4" style="padding:-1pt 5.0pt;">1130</td> </tr> <tr class="ltx_tr" id="S3.T1.28.13.6"> <td class="ltx_td" id="S3.T1.28.13.6.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.13.6.2" style="padding:-1pt 5.0pt;">Cu</td> <td class="ltx_td" id="S3.T1.28.13.6.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.13.6.4" style="padding:-1pt 5.0pt;">1722</td> </tr> <tr class="ltx_tr" id="S3.T1.28.6"> <td class="ltx_td ltx_align_center" id="S3.T1.27.5.1" style="padding:-1pt 5.0pt;"> 20Ne</td> <td class="ltx_td ltx_align_center" id="S3.T1.28.6.3" style="padding:-1pt 5.0pt;">C</td> <td class="ltx_td ltx_align_center" id="S3.T1.28.6.2" style="padding:-1pt 5.0pt;">O-<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.T1.28.6.2.m1.1"><semantics id="S3.T1.28.6.2.m1.1a"><mi id="S3.T1.28.6.2.m1.1.1" xref="S3.T1.28.6.2.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.T1.28.6.2.m1.1b"><ci id="S3.T1.28.6.2.m1.1.1.cmml" xref="S3.T1.28.6.2.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.28.6.2.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.T1.28.6.2.m1.1d">italic_α</annotation></semantics></math> type</td> <td class="ltx_td ltx_align_center" id="S3.T1.28.6.4" style="padding:-1pt 5.0pt;">911</td> </tr> <tr class="ltx_tr" id="S3.T1.28.14.7"> <td class="ltx_td" id="S3.T1.28.14.7.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.14.7.2" style="padding:-1pt 5.0pt;">Al</td> <td class="ltx_td" id="S3.T1.28.14.7.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center" id="S3.T1.28.14.7.4" style="padding:-1pt 5.0pt;">1308</td> </tr> <tr class="ltx_tr" id="S3.T1.28.15.8"> <td class="ltx_td ltx_border_bb" id="S3.T1.28.15.8.1" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.28.15.8.2" style="padding:-1pt 5.0pt;">Cu</td> <td class="ltx_td ltx_border_bb" id="S3.T1.28.15.8.3" style="padding:-1pt 5.0pt;"></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.28.15.8.4" style="padding:-1pt 5.0pt;">1951</td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="579" id="S3.F2.g1" src="x2.png" width="829"/> <figcaption class="ltx_caption ltx_centering">Figure 2: Charge-changing cross section as a function of incident energy of the projectile nucleus for <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.F2.13.m1.1"><semantics id="S3.F2.13.m1.1b"><mmultiscripts id="S3.F2.13.m1.1.1" xref="S3.F2.13.m1.1.1.cmml"><mtext id="S3.F2.13.m1.1.1.2" xref="S3.F2.13.m1.1.1.2a.cmml">C</mtext><mprescripts id="S3.F2.13.m1.1.1b" xref="S3.F2.13.m1.1.1.cmml"></mprescripts><mrow id="S3.F2.13.m1.1.1c" xref="S3.F2.13.m1.1.1.cmml"></mrow><mn id="S3.F2.13.m1.1.1.3" xref="S3.F2.13.m1.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.13.m1.1c"><apply id="S3.F2.13.m1.1.1.cmml" xref="S3.F2.13.m1.1.1"><csymbol cd="ambiguous" id="S3.F2.13.m1.1.1.1.cmml" xref="S3.F2.13.m1.1.1">superscript</csymbol><ci id="S3.F2.13.m1.1.1.2a.cmml" xref="S3.F2.13.m1.1.1.2"><mtext id="S3.F2.13.m1.1.1.2.cmml" xref="S3.F2.13.m1.1.1.2">C</mtext></ci><cn id="S3.F2.13.m1.1.1.3.cmml" type="integer" xref="S3.F2.13.m1.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" 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id="S3.F2.15.m3.1.1.1.cmml" xref="S3.F2.15.m3.1.1">superscript</csymbol><ci id="S3.F2.15.m3.1.1.2a.cmml" xref="S3.F2.15.m3.1.1.2"><mtext id="S3.F2.15.m3.1.1.2.cmml" xref="S3.F2.15.m3.1.1.2">O</mtext></ci><cn id="S3.F2.15.m3.1.1.3.cmml" type="integer" xref="S3.F2.15.m3.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.15.m3.1d">{}^{16}\text{O}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.15.m3.1e">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O</annotation></semantics></math>, and <math alttext="{}^{20}\text{Ne}" class="ltx_Math" display="inline" id="S3.F2.16.m4.1"><semantics id="S3.F2.16.m4.1b"><mmultiscripts id="S3.F2.16.m4.1.1" xref="S3.F2.16.m4.1.1.cmml"><mtext id="S3.F2.16.m4.1.1.2" xref="S3.F2.16.m4.1.1.2a.cmml">Ne</mtext><mprescripts id="S3.F2.16.m4.1.1b" xref="S3.F2.16.m4.1.1.cmml"></mprescripts><mrow id="S3.F2.16.m4.1.1c" xref="S3.F2.16.m4.1.1.cmml"></mrow><mn id="S3.F2.16.m4.1.1.3" xref="S3.F2.16.m4.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.16.m4.1c"><apply id="S3.F2.16.m4.1.1.cmml" xref="S3.F2.16.m4.1.1"><csymbol cd="ambiguous" id="S3.F2.16.m4.1.1.1.cmml" xref="S3.F2.16.m4.1.1">superscript</csymbol><ci id="S3.F2.16.m4.1.1.2a.cmml" xref="S3.F2.16.m4.1.1.2"><mtext id="S3.F2.16.m4.1.1.2.cmml" xref="S3.F2.16.m4.1.1.2">Ne</mtext></ci><cn id="S3.F2.16.m4.1.1.3.cmml" type="integer" xref="S3.F2.16.m4.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.16.m4.1d">{}^{20}\text{Ne}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.16.m4.1e">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT Ne</annotation></semantics></math> on elemental targets: Carbon (C), Aluminium (Al), and Copper (Cu). The plots are organized into four columns corresponding to different projectiles: (a) <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.F2.17.m5.1"><semantics id="S3.F2.17.m5.1b"><mmultiscripts id="S3.F2.17.m5.1.1" xref="S3.F2.17.m5.1.1.cmml"><mtext id="S3.F2.17.m5.1.1.2" xref="S3.F2.17.m5.1.1.2a.cmml">C</mtext><mprescripts id="S3.F2.17.m5.1.1b" xref="S3.F2.17.m5.1.1.cmml"></mprescripts><mrow id="S3.F2.17.m5.1.1c" xref="S3.F2.17.m5.1.1.cmml"></mrow><mn id="S3.F2.17.m5.1.1.3" xref="S3.F2.17.m5.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.17.m5.1c"><apply id="S3.F2.17.m5.1.1.cmml" xref="S3.F2.17.m5.1.1"><csymbol cd="ambiguous" id="S3.F2.17.m5.1.1.1.cmml" xref="S3.F2.17.m5.1.1">superscript</csymbol><ci id="S3.F2.17.m5.1.1.2a.cmml" xref="S3.F2.17.m5.1.1.2"><mtext id="S3.F2.17.m5.1.1.2.cmml" xref="S3.F2.17.m5.1.1.2">C</mtext></ci><cn id="S3.F2.17.m5.1.1.3.cmml" type="integer" xref="S3.F2.17.m5.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.17.m5.1d">{}^{12}\text{C}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.17.m5.1e">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT C</annotation></semantics></math>, (b) <math alttext="{}^{14}\text{N}" class="ltx_Math" display="inline" id="S3.F2.18.m6.1"><semantics id="S3.F2.18.m6.1b"><mmultiscripts id="S3.F2.18.m6.1.1" xref="S3.F2.18.m6.1.1.cmml"><mtext id="S3.F2.18.m6.1.1.2" xref="S3.F2.18.m6.1.1.2a.cmml">N</mtext><mprescripts id="S3.F2.18.m6.1.1b" xref="S3.F2.18.m6.1.1.cmml"></mprescripts><mrow id="S3.F2.18.m6.1.1c" xref="S3.F2.18.m6.1.1.cmml"></mrow><mn id="S3.F2.18.m6.1.1.3" xref="S3.F2.18.m6.1.1.3.cmml">14</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.18.m6.1c"><apply id="S3.F2.18.m6.1.1.cmml" xref="S3.F2.18.m6.1.1"><csymbol cd="ambiguous" id="S3.F2.18.m6.1.1.1.cmml" xref="S3.F2.18.m6.1.1">superscript</csymbol><ci id="S3.F2.18.m6.1.1.2a.cmml" xref="S3.F2.18.m6.1.1.2"><mtext id="S3.F2.18.m6.1.1.2.cmml" xref="S3.F2.18.m6.1.1.2">N</mtext></ci><cn id="S3.F2.18.m6.1.1.3.cmml" type="integer" xref="S3.F2.18.m6.1.1.3">14</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.18.m6.1d">{}^{14}\text{N}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.18.m6.1e">start_FLOATSUPERSCRIPT 14 end_FLOATSUPERSCRIPT N</annotation></semantics></math>, (c) <math alttext="{}^{16}\text{O}" class="ltx_Math" display="inline" id="S3.F2.19.m7.1"><semantics id="S3.F2.19.m7.1b"><mmultiscripts id="S3.F2.19.m7.1.1" xref="S3.F2.19.m7.1.1.cmml"><mtext id="S3.F2.19.m7.1.1.2" xref="S3.F2.19.m7.1.1.2a.cmml">O</mtext><mprescripts id="S3.F2.19.m7.1.1b" xref="S3.F2.19.m7.1.1.cmml"></mprescripts><mrow id="S3.F2.19.m7.1.1c" xref="S3.F2.19.m7.1.1.cmml"></mrow><mn id="S3.F2.19.m7.1.1.3" xref="S3.F2.19.m7.1.1.3.cmml">16</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.19.m7.1c"><apply id="S3.F2.19.m7.1.1.cmml" xref="S3.F2.19.m7.1.1"><csymbol cd="ambiguous" id="S3.F2.19.m7.1.1.1.cmml" xref="S3.F2.19.m7.1.1">superscript</csymbol><ci id="S3.F2.19.m7.1.1.2a.cmml" xref="S3.F2.19.m7.1.1.2"><mtext id="S3.F2.19.m7.1.1.2.cmml" xref="S3.F2.19.m7.1.1.2">O</mtext></ci><cn id="S3.F2.19.m7.1.1.3.cmml" type="integer" xref="S3.F2.19.m7.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.19.m7.1d">{}^{16}\text{O}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.19.m7.1e">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O</annotation></semantics></math>, and (d) <math alttext="{}^{20}\text{Ne}" class="ltx_Math" display="inline" id="S3.F2.20.m8.1"><semantics id="S3.F2.20.m8.1b"><mmultiscripts id="S3.F2.20.m8.1.1" xref="S3.F2.20.m8.1.1.cmml"><mtext id="S3.F2.20.m8.1.1.2" xref="S3.F2.20.m8.1.1.2a.cmml">Ne</mtext><mprescripts id="S3.F2.20.m8.1.1b" xref="S3.F2.20.m8.1.1.cmml"></mprescripts><mrow id="S3.F2.20.m8.1.1c" xref="S3.F2.20.m8.1.1.cmml"></mrow><mn id="S3.F2.20.m8.1.1.3" xref="S3.F2.20.m8.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.20.m8.1c"><apply id="S3.F2.20.m8.1.1.cmml" xref="S3.F2.20.m8.1.1"><csymbol cd="ambiguous" id="S3.F2.20.m8.1.1.1.cmml" xref="S3.F2.20.m8.1.1">superscript</csymbol><ci id="S3.F2.20.m8.1.1.2a.cmml" xref="S3.F2.20.m8.1.1.2"><mtext id="S3.F2.20.m8.1.1.2.cmml" xref="S3.F2.20.m8.1.1.2">Ne</mtext></ci><cn id="S3.F2.20.m8.1.1.3.cmml" type="integer" xref="S3.F2.20.m8.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.20.m8.1d">{}^{20}\text{Ne}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.20.m8.1e">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT Ne</annotation></semantics></math>. Each column presents the results for the respective projectiles on three targets: Carbon (C), Aluminium (Al), and Copper (Cu). The solid line represents theoretical calculations using the FROLA, and experimental data for the charge-changing cross sections in the energy range from 40 to 1000 MeV/nucleon are included, which are taken from Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib50" title="">50</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib51" title="">51</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib52" title="">52</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib56" title="">56</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib57" title="">57</a>]</cite> for <math alttext="{}^{12}\text{C}" class="ltx_Math" display="inline" id="S3.F2.21.m9.1"><semantics id="S3.F2.21.m9.1b"><mmultiscripts id="S3.F2.21.m9.1.1" xref="S3.F2.21.m9.1.1.cmml"><mtext id="S3.F2.21.m9.1.1.2" xref="S3.F2.21.m9.1.1.2a.cmml">C</mtext><mprescripts id="S3.F2.21.m9.1.1b" xref="S3.F2.21.m9.1.1.cmml"></mprescripts><mrow id="S3.F2.21.m9.1.1c" xref="S3.F2.21.m9.1.1.cmml"></mrow><mn id="S3.F2.21.m9.1.1.3" xref="S3.F2.21.m9.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.21.m9.1c"><apply id="S3.F2.21.m9.1.1.cmml" xref="S3.F2.21.m9.1.1"><csymbol cd="ambiguous" id="S3.F2.21.m9.1.1.1.cmml" xref="S3.F2.21.m9.1.1">superscript</csymbol><ci id="S3.F2.21.m9.1.1.2a.cmml" xref="S3.F2.21.m9.1.1.2"><mtext id="S3.F2.21.m9.1.1.2.cmml" xref="S3.F2.21.m9.1.1.2">C</mtext></ci><cn id="S3.F2.21.m9.1.1.3.cmml" type="integer" xref="S3.F2.21.m9.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.21.m9.1d">{}^{12}\text{C}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.21.m9.1e">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT C</annotation></semantics></math>, from Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib57" title="">57</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib52" title="">52</a>]</cite> for <math alttext="{}^{14}\text{N}" class="ltx_Math" display="inline" id="S3.F2.22.m10.1"><semantics id="S3.F2.22.m10.1b"><mmultiscripts id="S3.F2.22.m10.1.1" xref="S3.F2.22.m10.1.1.cmml"><mtext id="S3.F2.22.m10.1.1.2" xref="S3.F2.22.m10.1.1.2a.cmml">N</mtext><mprescripts id="S3.F2.22.m10.1.1b" xref="S3.F2.22.m10.1.1.cmml"></mprescripts><mrow id="S3.F2.22.m10.1.1c" xref="S3.F2.22.m10.1.1.cmml"></mrow><mn id="S3.F2.22.m10.1.1.3" xref="S3.F2.22.m10.1.1.3.cmml">14</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.22.m10.1c"><apply id="S3.F2.22.m10.1.1.cmml" xref="S3.F2.22.m10.1.1"><csymbol cd="ambiguous" id="S3.F2.22.m10.1.1.1.cmml" xref="S3.F2.22.m10.1.1">superscript</csymbol><ci id="S3.F2.22.m10.1.1.2a.cmml" xref="S3.F2.22.m10.1.1.2"><mtext id="S3.F2.22.m10.1.1.2.cmml" xref="S3.F2.22.m10.1.1.2">N</mtext></ci><cn id="S3.F2.22.m10.1.1.3.cmml" type="integer" xref="S3.F2.22.m10.1.1.3">14</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.22.m10.1d">{}^{14}\text{N}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.22.m10.1e">start_FLOATSUPERSCRIPT 14 end_FLOATSUPERSCRIPT N</annotation></semantics></math>, from Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib57" title="">57</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib52" title="">52</a>]</cite> for <math alttext="{}^{16}\text{O}" class="ltx_Math" display="inline" id="S3.F2.23.m11.1"><semantics id="S3.F2.23.m11.1b"><mmultiscripts id="S3.F2.23.m11.1.1" xref="S3.F2.23.m11.1.1.cmml"><mtext id="S3.F2.23.m11.1.1.2" xref="S3.F2.23.m11.1.1.2a.cmml">O</mtext><mprescripts id="S3.F2.23.m11.1.1b" xref="S3.F2.23.m11.1.1.cmml"></mprescripts><mrow id="S3.F2.23.m11.1.1c" xref="S3.F2.23.m11.1.1.cmml"></mrow><mn id="S3.F2.23.m11.1.1.3" xref="S3.F2.23.m11.1.1.3.cmml">16</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.23.m11.1c"><apply id="S3.F2.23.m11.1.1.cmml" xref="S3.F2.23.m11.1.1"><csymbol cd="ambiguous" id="S3.F2.23.m11.1.1.1.cmml" xref="S3.F2.23.m11.1.1">superscript</csymbol><ci id="S3.F2.23.m11.1.1.2a.cmml" xref="S3.F2.23.m11.1.1.2"><mtext id="S3.F2.23.m11.1.1.2.cmml" xref="S3.F2.23.m11.1.1.2">O</mtext></ci><cn id="S3.F2.23.m11.1.1.3.cmml" type="integer" xref="S3.F2.23.m11.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.23.m11.1d">{}^{16}\text{O}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.23.m11.1e">start_FLOATSUPERSCRIPT 16 end_FLOATSUPERSCRIPT O</annotation></semantics></math>, and from Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib57" title="">57</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib59" title="">59</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib60" title="">60</a>]</cite> for <math alttext="{}^{20}\text{Ne}" class="ltx_Math" display="inline" id="S3.F2.24.m12.1"><semantics id="S3.F2.24.m12.1b"><mmultiscripts id="S3.F2.24.m12.1.1" xref="S3.F2.24.m12.1.1.cmml"><mtext id="S3.F2.24.m12.1.1.2" xref="S3.F2.24.m12.1.1.2a.cmml">Ne</mtext><mprescripts id="S3.F2.24.m12.1.1b" xref="S3.F2.24.m12.1.1.cmml"></mprescripts><mrow id="S3.F2.24.m12.1.1c" xref="S3.F2.24.m12.1.1.cmml"></mrow><mn id="S3.F2.24.m12.1.1.3" xref="S3.F2.24.m12.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F2.24.m12.1c"><apply id="S3.F2.24.m12.1.1.cmml" xref="S3.F2.24.m12.1.1"><csymbol cd="ambiguous" id="S3.F2.24.m12.1.1.1.cmml" xref="S3.F2.24.m12.1.1">superscript</csymbol><ci id="S3.F2.24.m12.1.1.2a.cmml" xref="S3.F2.24.m12.1.1.2"><mtext id="S3.F2.24.m12.1.1.2.cmml" xref="S3.F2.24.m12.1.1.2">Ne</mtext></ci><cn id="S3.F2.24.m12.1.1.3.cmml" type="integer" xref="S3.F2.24.m12.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.24.m12.1d">{}^{20}\text{Ne}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.24.m12.1e">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT Ne</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p1"> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.T1" title="Table 1 ‣ III.2 Direct fragmentation or abrasion stage ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">1</a> presents the charge-changing cross section for projectile nuclei <math alttext="{}^{12}\mathrm{C}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mmultiscripts id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.p1.1.m1.1.1.2.cmml">C</mi><mprescripts id="S3.SS2.p1.1.m1.1.1a" xref="S3.SS2.p1.1.m1.1.1.cmml"></mprescripts><mrow id="S3.SS2.p1.1.m1.1.1b" xref="S3.SS2.p1.1.m1.1.1.cmml"></mrow><mn id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">12</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">superscript</csymbol><ci id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2">C</ci><cn id="S3.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">{}^{12}\mathrm{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT roman_C</annotation></semantics></math>,14N,16O, and <math alttext="{}^{20}\mathrm{Ne}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.1"><semantics id="S3.SS2.p1.4.m4.1a"><mmultiscripts id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml"><mi id="S3.SS2.p1.4.m4.1.1.2" xref="S3.SS2.p1.4.m4.1.1.2.cmml">Ne</mi><mprescripts id="S3.SS2.p1.4.m4.1.1a" xref="S3.SS2.p1.4.m4.1.1.cmml"></mprescripts><mrow id="S3.SS2.p1.4.m4.1.1b" xref="S3.SS2.p1.4.m4.1.1.cmml"></mrow><mn id="S3.SS2.p1.4.m4.1.1.3" xref="S3.SS2.p1.4.m4.1.1.3.cmml">20</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.1b"><apply id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m4.1.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1">superscript</csymbol><ci id="S3.SS2.p1.4.m4.1.1.2.cmml" xref="S3.SS2.p1.4.m4.1.1.2">Ne</ci><cn id="S3.SS2.p1.4.m4.1.1.3.cmml" type="integer" xref="S3.SS2.p1.4.m4.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.1c">{}^{20}\mathrm{Ne}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.1d">start_FLOATSUPERSCRIPT 20 end_FLOATSUPERSCRIPT roman_Ne</annotation></semantics></math> on elemental targets such as carbon (C), aluminum (Al), and copper (Cu) with beam energy 290 MeV/nucleon, during the direct fragmentation (abrasion) stage. The cross sections calculated using the finite-range optical limit approximation (FROLA) correspond to the most appropriate density configurations, as discussed in the previous section. These cross sections show slight variations when compared to other density configurations, with differences generally within <math alttext="2\%" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.1"><semantics id="S3.SS2.p1.5.m5.1a"><mrow id="S3.SS2.p1.5.m5.1.1" xref="S3.SS2.p1.5.m5.1.1.cmml"><mn id="S3.SS2.p1.5.m5.1.1.2" xref="S3.SS2.p1.5.m5.1.1.2.cmml">2</mn><mo id="S3.SS2.p1.5.m5.1.1.1" xref="S3.SS2.p1.5.m5.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.1b"><apply id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1"><csymbol cd="latexml" id="S3.SS2.p1.5.m5.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1">percent</csymbol><cn id="S3.SS2.p1.5.m5.1.1.2.cmml" type="integer" xref="S3.SS2.p1.5.m5.1.1.2">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.1c">2\%</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.1d">2 %</annotation></semantics></math>. </div> <div class="ltx_para" id="S3.SS2.p2"> Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F2" title="Figure 2 ‣ III.2 Direct fragmentation or abrasion stage ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">2</a> shows the FROLA calculation for the charge changing cross sections (<math alttext="\sigma_{\rm cc}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><msub id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">σ</mi><mi id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml">cc</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2">𝜎</ci><ci id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3">cc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">\sigma_{\rm cc}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math>) on the elemental targets, as a function of beam energy. The results in the abrasion stage (<math alttext="\sigma_{\rm cc}" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><msub id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml"><mi id="S3.SS2.p2.2.m2.1.1.2" xref="S3.SS2.p2.2.m2.1.1.2.cmml">σ</mi><mi id="S3.SS2.p2.2.m2.1.1.3" xref="S3.SS2.p2.2.m2.1.1.3.cmml">cc</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><apply id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p2.2.m2.1.1.2.cmml" xref="S3.SS2.p2.2.m2.1.1.2">𝜎</ci><ci id="S3.SS2.p2.2.m2.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.3">cc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">\sigma_{\rm cc}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math>) systematically underestimate charge-changing cross sections compared to experimental data, as also observed in previous studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib34" title="">34</a>]</cite>. Calculations of charge-changing cross sections using the Glauber model indicate that not only the protons in the projectile but also the presence of neutrons play a significant role in explaining the observed data. </div> <div class="ltx_para" id="S3.SS2.p3"> Some studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib62" title="">62</a>]</cite> propose using a phenomenological energy dependent or <math alttext="Z/N" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml">Z</mi><mo id="S3.SS2.p3.1.m1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.cmml">/</mo><mi id="S3.SS2.p3.1.m1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1"><divide id="S3.SS2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1"></divide><ci id="S3.SS2.p3.1.m1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.2">𝑍</ci><ci id="S3.SS2.p3.1.m1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">Z/N</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_Z / italic_N</annotation></semantics></math> ratio dependent correction parameter that accounts for the contribution of neutrons in the projectile, allowing for a more precise description of the data. On the other hand, the Glauber model only considers the direct removal of protons from the projectile, neglecting secondary processes. In reality, charged particles, predominantly protons, can be emitted following the direct removal of neutrons, especially when the prefragment is in a highly excited state <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib63" title="">63</a>]</cite>. When neutrons are removed from the projectile during the abrasion stage (direct fragmentation), the residual nucleus may become excited to states beyond the particle evaporation threshold. In such cases, deexcitation occurs through the removal of charged particles, such as protons or alpha particles. This process contributes to the overall charge changing cross section, increasing the likelihood of charge changing reactions beyond the initial neutron removal. This motivates us to explore the proton evaporation contribution to the charge changing process on various elemental targets (C, Al, Cu) in the next section. </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> III.3 Proton evaporation contributions</h3> <div class="ltx_para" id="S3.SS3.p1"> Estimating proton evaporation contributions in the secondary fragmentation or ablation stage is done in two ways. The first involves evaporation contributions calculated following Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E11" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">11</a>) and Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E12" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">12</a>). The second is subtracting the direct process (estimated with the FROLA) from the experimental data. A comparison of these two methods would then be made at the end. </div> <div class="ltx_para" id="S3.SS3.p2"> Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E11" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">11</a>) involves calculations of partial neutron removal cross section (<math alttext="\sigma_{-xn}" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><msub id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mi id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">σ</mi><mrow id="S3.SS3.p2.1.m1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.3.cmml"><mo id="S3.SS3.p2.1.m1.1.1.3a" xref="S3.SS3.p2.1.m1.1.1.3.cmml">−</mo><mrow id="S3.SS3.p2.1.m1.1.1.3.2" xref="S3.SS3.p2.1.m1.1.1.3.2.cmml"><mi id="S3.SS3.p2.1.m1.1.1.3.2.2" xref="S3.SS3.p2.1.m1.1.1.3.2.2.cmml">x</mi><mo id="S3.SS3.p2.1.m1.1.1.3.2.1" xref="S3.SS3.p2.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.p2.1.m1.1.1.3.2.3" xref="S3.SS3.p2.1.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.2">𝜎</ci><apply id="S3.SS3.p2.1.m1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3"><minus id="S3.SS3.p2.1.m1.1.1.3.1.cmml" xref="S3.SS3.p2.1.m1.1.1.3"></minus><apply id="S3.SS3.p2.1.m1.1.1.3.2.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2"><times id="S3.SS3.p2.1.m1.1.1.3.2.1.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2.1"></times><ci id="S3.SS3.p2.1.m1.1.1.3.2.2.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2.2">𝑥</ci><ci id="S3.SS3.p2.1.m1.1.1.3.2.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\sigma_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) and proton evaporation probability (<math alttext="P_{-xn}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><msub id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mi id="S3.SS3.p2.2.m2.1.1.2" xref="S3.SS3.p2.2.m2.1.1.2.cmml">P</mi><mrow id="S3.SS3.p2.2.m2.1.1.3" xref="S3.SS3.p2.2.m2.1.1.3.cmml"><mo id="S3.SS3.p2.2.m2.1.1.3a" xref="S3.SS3.p2.2.m2.1.1.3.cmml">−</mo><mrow id="S3.SS3.p2.2.m2.1.1.3.2" xref="S3.SS3.p2.2.m2.1.1.3.2.cmml"><mi id="S3.SS3.p2.2.m2.1.1.3.2.2" xref="S3.SS3.p2.2.m2.1.1.3.2.2.cmml">x</mi><mo id="S3.SS3.p2.2.m2.1.1.3.2.1" xref="S3.SS3.p2.2.m2.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.p2.2.m2.1.1.3.2.3" xref="S3.SS3.p2.2.m2.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p2.2.m2.1.1.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2">𝑃</ci><apply id="S3.SS3.p2.2.m2.1.1.3.cmml" xref="S3.SS3.p2.2.m2.1.1.3"><minus id="S3.SS3.p2.2.m2.1.1.3.1.cmml" xref="S3.SS3.p2.2.m2.1.1.3"></minus><apply id="S3.SS3.p2.2.m2.1.1.3.2.cmml" xref="S3.SS3.p2.2.m2.1.1.3.2"><times id="S3.SS3.p2.2.m2.1.1.3.2.1.cmml" xref="S3.SS3.p2.2.m2.1.1.3.2.1"></times><ci id="S3.SS3.p2.2.m2.1.1.3.2.2.cmml" xref="S3.SS3.p2.2.m2.1.1.3.2.2">𝑥</ci><ci id="S3.SS3.p2.2.m2.1.1.3.2.3.cmml" xref="S3.SS3.p2.2.m2.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">P_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">italic_P start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>). The probability, in turn, depends on key factors like excitation energy distributions (EED) of the prefragment and probability at a particular excitation energy. We now proceed to present the calculations of these factors in detail. </div> <section class="ltx_subsubsection" id="S3.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> III.3.1 Excitation energy distribution (EED)</h4> <div class="ltx_para" id="S3.SS3.SSS1.p1"> The excitation energy distribution (EED) <math alttext="w_{-xn}(E_{ex})" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.1.m1.1"><semantics id="S3.SS3.SSS1.p1.1.m1.1a"><mrow id="S3.SS3.SSS1.p1.1.m1.1.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.cmml"><msub id="S3.SS3.SSS1.p1.1.m1.1.1.3" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.cmml"><mi id="S3.SS3.SSS1.p1.1.m1.1.1.3.2" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.2.cmml">w</mi><mrow id="S3.SS3.SSS1.p1.1.m1.1.1.3.3" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.cmml"><mo id="S3.SS3.SSS1.p1.1.m1.1.1.3.3a" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.cmml">−</mo><mrow id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.cmml"><mi id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.2" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.2.cmml">x</mi><mo id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.3" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.3.cmml">n</mi></mrow></mrow></msub><mo id="S3.SS3.SSS1.p1.1.m1.1.1.2" xref="S3.SS3.SSS1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.SSS1.p1.1.m1.1.1.1.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.2" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.2.cmml">E</mi><mrow id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.2" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.2.cmml">e</mi><mo id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.3" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.3.cmml">x</mi></mrow></msub><mo id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.1.m1.1b"><apply id="S3.SS3.SSS1.p1.1.m1.1.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1"><times id="S3.SS3.SSS1.p1.1.m1.1.1.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.2"></times><apply id="S3.SS3.SSS1.p1.1.m1.1.1.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.2">𝑤</ci><apply id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3"><minus id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3"></minus><apply id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2"><times id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.1"></times><ci id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.2">𝑥</ci><ci id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.2.3">𝑛</ci></apply></apply></apply><apply id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.2">𝐸</ci><apply id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3"><times id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.1"></times><ci id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.2">𝑒</ci><ci id="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.1.1.1.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.1.m1.1c">w_{-xn}(E_{ex})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.1.m1.1d">italic_w start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_e italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> of the prefragment, which is an important parameter in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E12" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">12</a>) is calculated using the LISE++ simulation code <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib15" title="">15</a>]</cite>. LISE++ offers several theoretical approaches for determining the EED, with the Gaimard-Schmidt (GS) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib42" title="">42</a>]</cite> model being one of the primary options. The prefragment excitation energy is determined by summing the energies of the hole states remaining in the core of the initial nucleus after nucleon removal. GS model adopts a simple approach, that uses a functional shape corresponds to an approximation of the single hole state density in the Woods-Saxon potential, as done in Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib34" title="">34</a>]</cite> requiring only two input parameters: the number of nucleons removed and the maximum excitation energy (<math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.2.m2.1"><semantics id="S3.SS3.SSS1.p1.2.m2.1a"><msub id="S3.SS3.SSS1.p1.2.m2.1.1" xref="S3.SS3.SSS1.p1.2.m2.1.1.cmml"><mi id="S3.SS3.SSS1.p1.2.m2.1.1.2" xref="S3.SS3.SSS1.p1.2.m2.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS1.p1.2.m2.1.1.3" xref="S3.SS3.SSS1.p1.2.m2.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.2.m2.1b"><apply id="S3.SS3.SSS1.p1.2.m2.1.1.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.2.m2.1.1.1.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.SSS1.p1.2.m2.1.1.2.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1.2">𝐸</ci><ci id="S3.SS3.SSS1.p1.2.m2.1.1.3a.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1.3"><mtext id="S3.SS3.SSS1.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS1.p1.2.m2.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.2.m2.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.2.m2.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>) of the prefragment formed following a one-nucleon removal reaction. <math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.3.m3.1"><semantics id="S3.SS3.SSS1.p1.3.m3.1a"><msub id="S3.SS3.SSS1.p1.3.m3.1.1" xref="S3.SS3.SSS1.p1.3.m3.1.1.cmml"><mi id="S3.SS3.SSS1.p1.3.m3.1.1.2" xref="S3.SS3.SSS1.p1.3.m3.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS1.p1.3.m3.1.1.3" xref="S3.SS3.SSS1.p1.3.m3.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.3.m3.1b"><apply id="S3.SS3.SSS1.p1.3.m3.1.1.cmml" xref="S3.SS3.SSS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.3.m3.1.1.1.cmml" xref="S3.SS3.SSS1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.SSS1.p1.3.m3.1.1.2.cmml" xref="S3.SS3.SSS1.p1.3.m3.1.1.2">𝐸</ci><ci id="S3.SS3.SSS1.p1.3.m3.1.1.3a.cmml" xref="S3.SS3.SSS1.p1.3.m3.1.1.3"><mtext id="S3.SS3.SSS1.p1.3.m3.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS1.p1.3.m3.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.3.m3.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.3.m3.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> plays a crucial role in determining the excitation energy of prefragments in nuclear reactions. It represents the maximum energy that can be deposited by a single removed nucleon into the prefragment. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F3" title="Figure 3 ‣ III.3.1 Excitation energy distribution (EED) ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">3</a> illustrates the excitation energy distribution of the prefragment following the removal of one, two, and three nucleons from the nucleus. The distributions are characterized by the maximum excitation energy, which is set at 20 MeV, 40 MeV, and 60 MeV for the respective cases, for example. For one neutron abrasion case the distribution is normalized and decreases linearly with increasing excitation energy until <math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.4.m4.1"><semantics id="S3.SS3.SSS1.p1.4.m4.1a"><msub id="S3.SS3.SSS1.p1.4.m4.1.1" xref="S3.SS3.SSS1.p1.4.m4.1.1.cmml"><mi id="S3.SS3.SSS1.p1.4.m4.1.1.2" xref="S3.SS3.SSS1.p1.4.m4.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS1.p1.4.m4.1.1.3" xref="S3.SS3.SSS1.p1.4.m4.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.4.m4.1b"><apply id="S3.SS3.SSS1.p1.4.m4.1.1.cmml" xref="S3.SS3.SSS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.4.m4.1.1.1.cmml" xref="S3.SS3.SSS1.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.SSS1.p1.4.m4.1.1.2.cmml" xref="S3.SS3.SSS1.p1.4.m4.1.1.2">𝐸</ci><ci id="S3.SS3.SSS1.p1.4.m4.1.1.3a.cmml" xref="S3.SS3.SSS1.p1.4.m4.1.1.3"><mtext id="S3.SS3.SSS1.p1.4.m4.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS1.p1.4.m4.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.4.m4.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.4.m4.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>, where it abruptly drops to zero. This behavior reflects the simplified assumption that the probability of excitation decreases as energy approaches the maximum limit. As more neutrons are removed, the excitation energy distribution generally becomes broader, more complex, and shifted towards higher energies. The mean energy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib33" title="">33</a>]</cite> per removed nucleon <math alttext="\langle E\rangle=({x}/{x+1})E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.5.m5.2"><semantics id="S3.SS3.SSS1.p1.5.m5.2a"><mrow id="S3.SS3.SSS1.p1.5.m5.2.2" xref="S3.SS3.SSS1.p1.5.m5.2.2.cmml"><mrow id="S3.SS3.SSS1.p1.5.m5.2.2.3.2" xref="S3.SS3.SSS1.p1.5.m5.2.2.3.1.cmml"><mo id="S3.SS3.SSS1.p1.5.m5.2.2.3.2.1" stretchy="false" xref="S3.SS3.SSS1.p1.5.m5.2.2.3.1.1.cmml">⟨</mo><mi id="S3.SS3.SSS1.p1.5.m5.1.1" xref="S3.SS3.SSS1.p1.5.m5.1.1.cmml">E</mi><mo id="S3.SS3.SSS1.p1.5.m5.2.2.3.2.2" stretchy="false" xref="S3.SS3.SSS1.p1.5.m5.2.2.3.1.1.cmml">⟩</mo></mrow><mo id="S3.SS3.SSS1.p1.5.m5.2.2.2" xref="S3.SS3.SSS1.p1.5.m5.2.2.2.cmml">=</mo><mrow id="S3.SS3.SSS1.p1.5.m5.2.2.1" xref="S3.SS3.SSS1.p1.5.m5.2.2.1.cmml"><mrow id="S3.SS3.SSS1.p1.5.m5.2.2.1.1.1" 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xref="S3.SS3.SSS1.p1.5.m5.2.2.1.3.2">𝐸</ci><ci id="S3.SS3.SSS1.p1.5.m5.2.2.1.3.3a.cmml" xref="S3.SS3.SSS1.p1.5.m5.2.2.1.3.3"><mtext id="S3.SS3.SSS1.p1.5.m5.2.2.1.3.3.cmml" mathsize="70%" xref="S3.SS3.SSS1.p1.5.m5.2.2.1.3.3">max</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.5.m5.2c">\langle E\rangle=({x}/{x+1})E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.5.m5.2d">⟨ italic_E ⟩ = ( italic_x / italic_x + 1 ) italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>, which gives 10 MeV for first neutron removal for <math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.6.m6.1"><semantics id="S3.SS3.SSS1.p1.6.m6.1a"><msub id="S3.SS3.SSS1.p1.6.m6.1.1" xref="S3.SS3.SSS1.p1.6.m6.1.1.cmml"><mi id="S3.SS3.SSS1.p1.6.m6.1.1.2" xref="S3.SS3.SSS1.p1.6.m6.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS1.p1.6.m6.1.1.3" xref="S3.SS3.SSS1.p1.6.m6.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.6.m6.1b"><apply id="S3.SS3.SSS1.p1.6.m6.1.1.cmml" xref="S3.SS3.SSS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.6.m6.1.1.1.cmml" xref="S3.SS3.SSS1.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.SSS1.p1.6.m6.1.1.2.cmml" xref="S3.SS3.SSS1.p1.6.m6.1.1.2">𝐸</ci><ci id="S3.SS3.SSS1.p1.6.m6.1.1.3a.cmml" xref="S3.SS3.SSS1.p1.6.m6.1.1.3"><mtext id="S3.SS3.SSS1.p1.6.m6.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS1.p1.6.m6.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.6.m6.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.6.m6.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>= 20 MeV and consequently this value increases and become maximum for higher neutron removal. </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="614" id="S3.F3.g1" src="x3.png" width="581"/> <figcaption class="ltx_caption ltx_centering">Figure 3: Excitation energy distributions of the prefragment after the removal of <math alttext="x=1" class="ltx_Math" display="inline" id="S3.F3.3.m1.1"><semantics id="S3.F3.3.m1.1b"><mrow id="S3.F3.3.m1.1.1" xref="S3.F3.3.m1.1.1.cmml"><mi id="S3.F3.3.m1.1.1.2" xref="S3.F3.3.m1.1.1.2.cmml">x</mi><mo id="S3.F3.3.m1.1.1.1" xref="S3.F3.3.m1.1.1.1.cmml">=</mo><mn id="S3.F3.3.m1.1.1.3" xref="S3.F3.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.3.m1.1c"><apply id="S3.F3.3.m1.1.1.cmml" xref="S3.F3.3.m1.1.1"><eq id="S3.F3.3.m1.1.1.1.cmml" xref="S3.F3.3.m1.1.1.1"></eq><ci id="S3.F3.3.m1.1.1.2.cmml" xref="S3.F3.3.m1.1.1.2">𝑥</ci><cn id="S3.F3.3.m1.1.1.3.cmml" type="integer" xref="S3.F3.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.3.m1.1d">x=1</annotation><annotation encoding="application/x-llamapun" id="S3.F3.3.m1.1e">italic_x = 1</annotation></semantics></math>-3 neutrons in direct fragmentation(abrasion stage), shown for <math alttext="E_{\text{max}}=20" class="ltx_Math" display="inline" id="S3.F3.4.m2.1"><semantics id="S3.F3.4.m2.1b"><mrow id="S3.F3.4.m2.1.1" xref="S3.F3.4.m2.1.1.cmml"><msub id="S3.F3.4.m2.1.1.2" xref="S3.F3.4.m2.1.1.2.cmml"><mi id="S3.F3.4.m2.1.1.2.2" xref="S3.F3.4.m2.1.1.2.2.cmml">E</mi><mtext id="S3.F3.4.m2.1.1.2.3" xref="S3.F3.4.m2.1.1.2.3a.cmml">max</mtext></msub><mo id="S3.F3.4.m2.1.1.1" xref="S3.F3.4.m2.1.1.1.cmml">=</mo><mn id="S3.F3.4.m2.1.1.3" xref="S3.F3.4.m2.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.4.m2.1c"><apply id="S3.F3.4.m2.1.1.cmml" xref="S3.F3.4.m2.1.1"><eq id="S3.F3.4.m2.1.1.1.cmml" xref="S3.F3.4.m2.1.1.1"></eq><apply id="S3.F3.4.m2.1.1.2.cmml" xref="S3.F3.4.m2.1.1.2"><csymbol cd="ambiguous" id="S3.F3.4.m2.1.1.2.1.cmml" xref="S3.F3.4.m2.1.1.2">subscript</csymbol><ci id="S3.F3.4.m2.1.1.2.2.cmml" xref="S3.F3.4.m2.1.1.2.2">𝐸</ci><ci id="S3.F3.4.m2.1.1.2.3a.cmml" xref="S3.F3.4.m2.1.1.2.3"><mtext id="S3.F3.4.m2.1.1.2.3.cmml" mathsize="70%" xref="S3.F3.4.m2.1.1.2.3">max</mtext></ci></apply><cn id="S3.F3.4.m2.1.1.3.cmml" type="integer" xref="S3.F3.4.m2.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.4.m2.1d">E_{\text{max}}=20</annotation><annotation encoding="application/x-llamapun" id="S3.F3.4.m2.1e">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT = 20</annotation></semantics></math>, 40, and 60 MeV, respectively.</figcaption> </figure> </section> <section class="ltx_subsubsection" id="S3.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> III.3.2 Proton evaporation probability</h4> <div class="ltx_para" id="S3.SS3.SSS2.p1"> We now describe the calculation of the probability of proton evaporation (<math alttext="f(E_{\text{ex}},A_{P}-x,Z_{P})" 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encoding="application/x-tex" id="S3.SS3.SSS2.p1.1.m1.3c">f(E_{\text{ex}},A_{P}-x,Z_{P})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p1.1.m1.3d">italic_f ( italic_E start_POSTSUBSCRIPT ex end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_x , italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT )</annotation></semantics></math>) as a function of excitation energy. The calculations are performed using the LISE++ (GEMINI++) simulation code <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib44" title="">44</a>]</cite>, which incorporates the Hauser-Feshbach and Weisskopf formalisms for compound nucleus decay. This approach accounts for decays involving the emission of at least one proton during cascade deexitation. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F4" title="Figure 4 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">4</a> illustrates the variation in the charged particle evaporation probability as a function of excitation energy (<math alttext="E_{\text{ex}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p1.2.m2.1"><semantics id="S3.SS3.SSS2.p1.2.m2.1a"><msub id="S3.SS3.SSS2.p1.2.m2.1.1" xref="S3.SS3.SSS2.p1.2.m2.1.1.cmml"><mi id="S3.SS3.SSS2.p1.2.m2.1.1.2" xref="S3.SS3.SSS2.p1.2.m2.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p1.2.m2.1.1.3" xref="S3.SS3.SSS2.p1.2.m2.1.1.3a.cmml">ex</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p1.2.m2.1b"><apply id="S3.SS3.SSS2.p1.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p1.2.m2.1.1"><csymbol cd="ambiguous" 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The evaporation probability curve begins to rise at or near the energy that equals to proton separation energy plus the Coulomb barrier for the prefragments. This is the excitation energy threshold beyond which the prefragment is sufficiently excited for charged particle evaporation processes to occur. After this energy threshold, the prefragment nucleus loses at least one proton, which contributes to the charge changing cross section. This trend is also followed by the other prefragments produced after the removal of two or three neutrons in the abrasion stage. </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="618" id="S3.F4.g1" src="x4.png" width="498"/> <figcaption class="ltx_caption ltx_centering">Figure 4: Probability <math alttext="(f(E_{\text{ex}},A_{P}-x,Z_{P}))" class="ltx_Math" display="inline" id="S3.F4.8.m1.1"><semantics id="S3.F4.8.m1.1b"><mrow id="S3.F4.8.m1.1.1.1" xref="S3.F4.8.m1.1.1.1.1.cmml"><mo id="S3.F4.8.m1.1.1.1.2" stretchy="false" xref="S3.F4.8.m1.1.1.1.1.cmml">(</mo><mrow id="S3.F4.8.m1.1.1.1.1" xref="S3.F4.8.m1.1.1.1.1.cmml"><mi id="S3.F4.8.m1.1.1.1.1.5" xref="S3.F4.8.m1.1.1.1.1.5.cmml">f</mi><mo id="S3.F4.8.m1.1.1.1.1.4" xref="S3.F4.8.m1.1.1.1.1.4.cmml">⁢</mo><mrow id="S3.F4.8.m1.1.1.1.1.3.3" xref="S3.F4.8.m1.1.1.1.1.3.4.cmml"><mo id="S3.F4.8.m1.1.1.1.1.3.3.4" stretchy="false" 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xref="S3.F4.8.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S3.F4.8.m1.1.1.1.1.3.3.3.2.cmml" xref="S3.F4.8.m1.1.1.1.1.3.3.3.2">𝑍</ci><ci id="S3.F4.8.m1.1.1.1.1.3.3.3.3.cmml" xref="S3.F4.8.m1.1.1.1.1.3.3.3.3">𝑃</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.8.m1.1d">(f(E_{\text{ex}},A_{P}-x,Z_{P}))</annotation><annotation encoding="application/x-llamapun" id="S3.F4.8.m1.1e">( italic_f ( italic_E start_POSTSUBSCRIPT ex end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - italic_x , italic_Z start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ) )</annotation></semantics></math> of proton evaporation after initial removal of neutrons (<math alttext="x=1" class="ltx_Math" display="inline" id="S3.F4.9.m2.1"><semantics id="S3.F4.9.m2.1b"><mrow id="S3.F4.9.m2.1.1" xref="S3.F4.9.m2.1.1.cmml"><mi id="S3.F4.9.m2.1.1.2" xref="S3.F4.9.m2.1.1.2.cmml">x</mi><mo id="S3.F4.9.m2.1.1.1" xref="S3.F4.9.m2.1.1.1.cmml">=</mo><mn id="S3.F4.9.m2.1.1.3" xref="S3.F4.9.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.9.m2.1c"><apply id="S3.F4.9.m2.1.1.cmml" xref="S3.F4.9.m2.1.1"><eq id="S3.F4.9.m2.1.1.1.cmml" xref="S3.F4.9.m2.1.1.1"></eq><ci id="S3.F4.9.m2.1.1.2.cmml" xref="S3.F4.9.m2.1.1.2">𝑥</ci><cn id="S3.F4.9.m2.1.1.3.cmml" type="integer" xref="S3.F4.9.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.9.m2.1d">x=1</annotation><annotation encoding="application/x-llamapun" id="S3.F4.9.m2.1e">italic_x = 1</annotation></semantics></math>-3) from the projectiles 12C, 14N, 16O, and 20Ne as a function of prefragment excitation energy (<math alttext="E_{\mathrm{ex}}" class="ltx_Math" display="inline" id="S3.F4.14.m7.1"><semantics id="S3.F4.14.m7.1b"><msub id="S3.F4.14.m7.1.1" xref="S3.F4.14.m7.1.1.cmml"><mi id="S3.F4.14.m7.1.1.2" xref="S3.F4.14.m7.1.1.2.cmml">E</mi><mi id="S3.F4.14.m7.1.1.3" xref="S3.F4.14.m7.1.1.3.cmml">ex</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F4.14.m7.1c"><apply id="S3.F4.14.m7.1.1.cmml" xref="S3.F4.14.m7.1.1"><csymbol cd="ambiguous" id="S3.F4.14.m7.1.1.1.cmml" xref="S3.F4.14.m7.1.1">subscript</csymbol><ci id="S3.F4.14.m7.1.1.2.cmml" xref="S3.F4.14.m7.1.1.2">𝐸</ci><ci id="S3.F4.14.m7.1.1.3.cmml" xref="S3.F4.14.m7.1.1.3">ex</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.14.m7.1d">E_{\mathrm{ex}}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.14.m7.1e">italic_E start_POSTSUBSCRIPT roman_ex end_POSTSUBSCRIPT</annotation></semantics></math>).</figcaption> </figure> <div class="ltx_para" id="S3.SS3.SSS2.p2"> With these values of <math alttext="f(E_{\text{ex}})" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p2.1.m1.1"><semantics id="S3.SS3.SSS2.p2.1.m1.1a"><mrow id="S3.SS3.SSS2.p2.1.m1.1.1" xref="S3.SS3.SSS2.p2.1.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p2.1.m1.1.1.3" xref="S3.SS3.SSS2.p2.1.m1.1.1.3.cmml">f</mi><mo id="S3.SS3.SSS2.p2.1.m1.1.1.2" xref="S3.SS3.SSS2.p2.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.SSS2.p2.1.m1.1.1.1.1" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.cmml"><mi id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.2" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.3" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.3a.cmml">ex</mtext></msub><mo id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.3" stretchy="false" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p2.1.m1.1b"><apply id="S3.SS3.SSS2.p2.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1"><times id="S3.SS3.SSS2.p2.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1.2"></times><ci id="S3.SS3.SSS2.p2.1.m1.1.1.3.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1.3">𝑓</ci><apply id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.3a.cmml" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.3"><mtext id="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p2.1.m1.1.1.1.1.1.3">ex</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p2.1.m1.1c">f(E_{\text{ex}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p2.1.m1.1d">italic_f ( italic_E start_POSTSUBSCRIPT ex end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="{w}_{-xn}(E_{\text{ex}})" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p2.2.m2.1"><semantics id="S3.SS3.SSS2.p2.2.m2.1a"><mrow id="S3.SS3.SSS2.p2.2.m2.1.1" xref="S3.SS3.SSS2.p2.2.m2.1.1.cmml"><msub id="S3.SS3.SSS2.p2.2.m2.1.1.3" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.cmml"><mi id="S3.SS3.SSS2.p2.2.m2.1.1.3.2" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.2.cmml">w</mi><mrow id="S3.SS3.SSS2.p2.2.m2.1.1.3.3" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.cmml"><mo id="S3.SS3.SSS2.p2.2.m2.1.1.3.3a" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.cmml">−</mo><mrow id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.cmml"><mi id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.2" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.2.cmml">x</mi><mo id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.1" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.3" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.3.cmml">n</mi></mrow></mrow></msub><mo id="S3.SS3.SSS2.p2.2.m2.1.1.2" xref="S3.SS3.SSS2.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.SSS2.p2.2.m2.1.1.1.1" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.2" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.3" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.3a.cmml">ex</mtext></msub><mo id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p2.2.m2.1b"><apply id="S3.SS3.SSS2.p2.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1"><times id="S3.SS3.SSS2.p2.2.m2.1.1.2.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.2"></times><apply id="S3.SS3.SSS2.p2.2.m2.1.1.3.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p2.2.m2.1.1.3.1.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p2.2.m2.1.1.3.2.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.2">𝑤</ci><apply id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3"><minus id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.1.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3"></minus><apply id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2"><times id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.1.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.1"></times><ci id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.2.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.2">𝑥</ci><ci id="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.3.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.3.3.2.3">𝑛</ci></apply></apply></apply><apply id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.3a.cmml" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.3"><mtext id="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p2.2.m2.1.1.1.1.1.3">ex</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p2.2.m2.1c">{w}_{-xn}(E_{\text{ex}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p2.2.m2.1d">italic_w start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT ex end_POSTSUBSCRIPT )</annotation></semantics></math>, the evaporation probability (<math alttext="P_{-xn}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p2.3.m3.1"><semantics id="S3.SS3.SSS2.p2.3.m3.1a"><msub id="S3.SS3.SSS2.p2.3.m3.1.1" xref="S3.SS3.SSS2.p2.3.m3.1.1.cmml"><mi id="S3.SS3.SSS2.p2.3.m3.1.1.2" xref="S3.SS3.SSS2.p2.3.m3.1.1.2.cmml">P</mi><mrow id="S3.SS3.SSS2.p2.3.m3.1.1.3" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.cmml"><mo id="S3.SS3.SSS2.p2.3.m3.1.1.3a" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.cmml">−</mo><mrow id="S3.SS3.SSS2.p2.3.m3.1.1.3.2" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.cmml"><mi id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.2" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.2.cmml">x</mi><mo id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.1" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.3" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p2.3.m3.1b"><apply id="S3.SS3.SSS2.p2.3.m3.1.1.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p2.3.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p2.3.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.2">𝑃</ci><apply id="S3.SS3.SSS2.p2.3.m3.1.1.3.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.3"><minus id="S3.SS3.SSS2.p2.3.m3.1.1.3.1.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.3"></minus><apply id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2"><times id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.1"></times><ci id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.2.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.2">𝑥</ci><ci id="S3.SS3.SSS2.p2.3.m3.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p2.3.m3.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p2.3.m3.1c">P_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p2.3.m3.1d">italic_P start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) is calculated and shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F5" title="Figure 5 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">5</a> for up to 5 neutron removals from the projectile. The parameter <math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p2.4.m4.1"><semantics id="S3.SS3.SSS2.p2.4.m4.1a"><msub id="S3.SS3.SSS2.p2.4.m4.1.1" xref="S3.SS3.SSS2.p2.4.m4.1.1.cmml"><mi id="S3.SS3.SSS2.p2.4.m4.1.1.2" xref="S3.SS3.SSS2.p2.4.m4.1.1.2.cmml">E</mi><mi id="S3.SS3.SSS2.p2.4.m4.1.1.3" xref="S3.SS3.SSS2.p2.4.m4.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p2.4.m4.1b"><apply id="S3.SS3.SSS2.p2.4.m4.1.1.cmml" xref="S3.SS3.SSS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p2.4.m4.1.1.1.cmml" xref="S3.SS3.SSS2.p2.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p2.4.m4.1.1.2.cmml" xref="S3.SS3.SSS2.p2.4.m4.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p2.4.m4.1.1.3.cmml" xref="S3.SS3.SSS2.p2.4.m4.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p2.4.m4.1c">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p2.4.m4.1d">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> is fixed to a value that reproduces the estimated evaporation cross sections from the experimental data at beam energy 290 MeV/nucleon on carbon target. The variation of <math alttext="P_{-xn}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p2.5.m5.1"><semantics id="S3.SS3.SSS2.p2.5.m5.1a"><msub id="S3.SS3.SSS2.p2.5.m5.1.1" xref="S3.SS3.SSS2.p2.5.m5.1.1.cmml"><mi id="S3.SS3.SSS2.p2.5.m5.1.1.2" xref="S3.SS3.SSS2.p2.5.m5.1.1.2.cmml">P</mi><mrow id="S3.SS3.SSS2.p2.5.m5.1.1.3" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.cmml"><mo id="S3.SS3.SSS2.p2.5.m5.1.1.3a" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.cmml">−</mo><mrow id="S3.SS3.SSS2.p2.5.m5.1.1.3.2" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.cmml"><mi id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.2" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.2.cmml">x</mi><mo id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.1" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.3" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p2.5.m5.1b"><apply id="S3.SS3.SSS2.p2.5.m5.1.1.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p2.5.m5.1.1.1.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p2.5.m5.1.1.2.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.2">𝑃</ci><apply id="S3.SS3.SSS2.p2.5.m5.1.1.3.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.3"><minus id="S3.SS3.SSS2.p2.5.m5.1.1.3.1.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.3"></minus><apply id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2"><times id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.1"></times><ci id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.2.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.2">𝑥</ci><ci id="S3.SS3.SSS2.p2.5.m5.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p2.5.m5.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p2.5.m5.1c">P_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p2.5.m5.1d">italic_P start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math> indicates that as the number of neutrons removed from the projectile increases, the charged particle evaporation probability also increases. For all projectile nuclei, one notices this probability tends to 1 when the <math alttext="Z/N" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p2.6.m6.1"><semantics id="S3.SS3.SSS2.p2.6.m6.1a"><mrow id="S3.SS3.SSS2.p2.6.m6.1.1" xref="S3.SS3.SSS2.p2.6.m6.1.1.cmml"><mi id="S3.SS3.SSS2.p2.6.m6.1.1.2" xref="S3.SS3.SSS2.p2.6.m6.1.1.2.cmml">Z</mi><mo id="S3.SS3.SSS2.p2.6.m6.1.1.1" xref="S3.SS3.SSS2.p2.6.m6.1.1.1.cmml">/</mo><mi id="S3.SS3.SSS2.p2.6.m6.1.1.3" xref="S3.SS3.SSS2.p2.6.m6.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p2.6.m6.1b"><apply id="S3.SS3.SSS2.p2.6.m6.1.1.cmml" xref="S3.SS3.SSS2.p2.6.m6.1.1"><divide id="S3.SS3.SSS2.p2.6.m6.1.1.1.cmml" xref="S3.SS3.SSS2.p2.6.m6.1.1.1"></divide><ci id="S3.SS3.SSS2.p2.6.m6.1.1.2.cmml" xref="S3.SS3.SSS2.p2.6.m6.1.1.2">𝑍</ci><ci id="S3.SS3.SSS2.p2.6.m6.1.1.3.cmml" xref="S3.SS3.SSS2.p2.6.m6.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p2.6.m6.1c">Z/N</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p2.6.m6.1d">italic_Z / italic_N</annotation></semantics></math> ratio becomes greater than 1, reflecting a direct relationship between the degree of neutron removal and the likelihood of charged particle evaporation, contributing significantly to the charge-changing cross section. </div> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="422" id="S3.F5.g1" src="x5.png" width="581"/> <figcaption class="ltx_caption ltx_centering">Figure 5: Proton evaporation probability (<math alttext="P_{-xn}" class="ltx_Math" display="inline" id="S3.F5.3.m1.1"><semantics id="S3.F5.3.m1.1b"><msub id="S3.F5.3.m1.1.1" xref="S3.F5.3.m1.1.1.cmml"><mi id="S3.F5.3.m1.1.1.2" xref="S3.F5.3.m1.1.1.2.cmml">P</mi><mrow id="S3.F5.3.m1.1.1.3" xref="S3.F5.3.m1.1.1.3.cmml"><mo id="S3.F5.3.m1.1.1.3b" xref="S3.F5.3.m1.1.1.3.cmml">−</mo><mrow id="S3.F5.3.m1.1.1.3.2" xref="S3.F5.3.m1.1.1.3.2.cmml"><mi id="S3.F5.3.m1.1.1.3.2.2" xref="S3.F5.3.m1.1.1.3.2.2.cmml">x</mi><mo id="S3.F5.3.m1.1.1.3.2.1" xref="S3.F5.3.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.F5.3.m1.1.1.3.2.3" xref="S3.F5.3.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F5.3.m1.1c"><apply id="S3.F5.3.m1.1.1.cmml" xref="S3.F5.3.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.3.m1.1.1.1.cmml" xref="S3.F5.3.m1.1.1">subscript</csymbol><ci id="S3.F5.3.m1.1.1.2.cmml" xref="S3.F5.3.m1.1.1.2">𝑃</ci><apply id="S3.F5.3.m1.1.1.3.cmml" xref="S3.F5.3.m1.1.1.3"><minus id="S3.F5.3.m1.1.1.3.1.cmml" xref="S3.F5.3.m1.1.1.3"></minus><apply id="S3.F5.3.m1.1.1.3.2.cmml" xref="S3.F5.3.m1.1.1.3.2"><times id="S3.F5.3.m1.1.1.3.2.1.cmml" xref="S3.F5.3.m1.1.1.3.2.1"></times><ci id="S3.F5.3.m1.1.1.3.2.2.cmml" xref="S3.F5.3.m1.1.1.3.2.2">𝑥</ci><ci id="S3.F5.3.m1.1.1.3.2.3.cmml" xref="S3.F5.3.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.3.m1.1d">P_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.3.m1.1e">italic_P start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) as a function of the number of neutrons removed in the initial fragmentation (abrasion stage). The maximum excitation energy parameter <math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S3.F5.4.m2.1"><semantics id="S3.F5.4.m2.1b"><msub id="S3.F5.4.m2.1.1" xref="S3.F5.4.m2.1.1.cmml"><mi id="S3.F5.4.m2.1.1.2" xref="S3.F5.4.m2.1.1.2.cmml">E</mi><mi id="S3.F5.4.m2.1.1.3" xref="S3.F5.4.m2.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F5.4.m2.1c"><apply id="S3.F5.4.m2.1.1.cmml" xref="S3.F5.4.m2.1.1"><csymbol cd="ambiguous" id="S3.F5.4.m2.1.1.1.cmml" xref="S3.F5.4.m2.1.1">subscript</csymbol><ci id="S3.F5.4.m2.1.1.2.cmml" xref="S3.F5.4.m2.1.1.2">𝐸</ci><ci id="S3.F5.4.m2.1.1.3.cmml" xref="S3.F5.4.m2.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.4.m2.1d">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.4.m2.1e">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> is selected to approximately reproduce the charge-changing cross-section data at 290 MeV/nucleon for a carbon target. </figcaption> </figure> <figure class="ltx_figure" id="S3.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="578" id="S3.F6.g1" src="x6.png" width="581"/> <figcaption class="ltx_caption ltx_centering">Figure 6: Top panel (a) represents the ratio of the estimated evaporation contributions <math alttext="(\sigma_{\mathrm{cc}}^{\text{expt.}}-\sigma_{\mathrm{cc}})" class="ltx_Math" display="inline" id="S3.F6.5.m1.1"><semantics id="S3.F6.5.m1.1b"><mrow id="S3.F6.5.m1.1.1.1" xref="S3.F6.5.m1.1.1.1.1.cmml"><mo id="S3.F6.5.m1.1.1.1.2" stretchy="false" xref="S3.F6.5.m1.1.1.1.1.cmml">(</mo><mrow id="S3.F6.5.m1.1.1.1.1" xref="S3.F6.5.m1.1.1.1.1.cmml"><msubsup id="S3.F6.5.m1.1.1.1.1.2" xref="S3.F6.5.m1.1.1.1.1.2.cmml"><mi id="S3.F6.5.m1.1.1.1.1.2.2.2" xref="S3.F6.5.m1.1.1.1.1.2.2.2.cmml">σ</mi><mi id="S3.F6.5.m1.1.1.1.1.2.2.3" xref="S3.F6.5.m1.1.1.1.1.2.2.3.cmml">cc</mi><mtext id="S3.F6.5.m1.1.1.1.1.2.3" xref="S3.F6.5.m1.1.1.1.1.2.3a.cmml">expt.</mtext></msubsup><mo id="S3.F6.5.m1.1.1.1.1.1" xref="S3.F6.5.m1.1.1.1.1.1.cmml">−</mo><msub id="S3.F6.5.m1.1.1.1.1.3" xref="S3.F6.5.m1.1.1.1.1.3.cmml"><mi id="S3.F6.5.m1.1.1.1.1.3.2" xref="S3.F6.5.m1.1.1.1.1.3.2.cmml">σ</mi><mi id="S3.F6.5.m1.1.1.1.1.3.3" xref="S3.F6.5.m1.1.1.1.1.3.3.cmml">cc</mi></msub></mrow><mo id="S3.F6.5.m1.1.1.1.3" stretchy="false" xref="S3.F6.5.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.5.m1.1c"><apply id="S3.F6.5.m1.1.1.1.1.cmml" xref="S3.F6.5.m1.1.1.1"><minus id="S3.F6.5.m1.1.1.1.1.1.cmml" xref="S3.F6.5.m1.1.1.1.1.1"></minus><apply id="S3.F6.5.m1.1.1.1.1.2.cmml" xref="S3.F6.5.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.F6.5.m1.1.1.1.1.2.1.cmml" xref="S3.F6.5.m1.1.1.1.1.2">superscript</csymbol><apply id="S3.F6.5.m1.1.1.1.1.2.2.cmml" xref="S3.F6.5.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.F6.5.m1.1.1.1.1.2.2.1.cmml" xref="S3.F6.5.m1.1.1.1.1.2">subscript</csymbol><ci id="S3.F6.5.m1.1.1.1.1.2.2.2.cmml" xref="S3.F6.5.m1.1.1.1.1.2.2.2">𝜎</ci><ci id="S3.F6.5.m1.1.1.1.1.2.2.3.cmml" xref="S3.F6.5.m1.1.1.1.1.2.2.3">cc</ci></apply><ci id="S3.F6.5.m1.1.1.1.1.2.3a.cmml" xref="S3.F6.5.m1.1.1.1.1.2.3"><mtext id="S3.F6.5.m1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S3.F6.5.m1.1.1.1.1.2.3">expt.</mtext></ci></apply><apply id="S3.F6.5.m1.1.1.1.1.3.cmml" xref="S3.F6.5.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.F6.5.m1.1.1.1.1.3.1.cmml" xref="S3.F6.5.m1.1.1.1.1.3">subscript</csymbol><ci id="S3.F6.5.m1.1.1.1.1.3.2.cmml" xref="S3.F6.5.m1.1.1.1.1.3.2">𝜎</ci><ci id="S3.F6.5.m1.1.1.1.1.3.3.cmml" xref="S3.F6.5.m1.1.1.1.1.3.3">cc</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.5.m1.1d">(\sigma_{\mathrm{cc}}^{\text{expt.}}-\sigma_{\mathrm{cc}})</annotation><annotation encoding="application/x-llamapun" id="S3.F6.5.m1.1e">( italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt. end_POSTSUPERSCRIPT - italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT )</annotation></semantics></math> to the total experimental charge-changing cross section <math alttext="(\sigma_{\rm cc}^{\text{expt.}})" class="ltx_Math" display="inline" id="S3.F6.6.m2.1"><semantics id="S3.F6.6.m2.1b"><mrow id="S3.F6.6.m2.1.1.1" xref="S3.F6.6.m2.1.1.1.1.cmml"><mo id="S3.F6.6.m2.1.1.1.2" stretchy="false" xref="S3.F6.6.m2.1.1.1.1.cmml">(</mo><msubsup id="S3.F6.6.m2.1.1.1.1" xref="S3.F6.6.m2.1.1.1.1.cmml"><mi id="S3.F6.6.m2.1.1.1.1.2.2" xref="S3.F6.6.m2.1.1.1.1.2.2.cmml">σ</mi><mi id="S3.F6.6.m2.1.1.1.1.2.3" xref="S3.F6.6.m2.1.1.1.1.2.3.cmml">cc</mi><mtext id="S3.F6.6.m2.1.1.1.1.3" xref="S3.F6.6.m2.1.1.1.1.3a.cmml">expt.</mtext></msubsup><mo id="S3.F6.6.m2.1.1.1.3" stretchy="false" xref="S3.F6.6.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.6.m2.1c"><apply id="S3.F6.6.m2.1.1.1.1.cmml" xref="S3.F6.6.m2.1.1.1"><csymbol cd="ambiguous" id="S3.F6.6.m2.1.1.1.1.1.cmml" xref="S3.F6.6.m2.1.1.1">superscript</csymbol><apply id="S3.F6.6.m2.1.1.1.1.2.cmml" xref="S3.F6.6.m2.1.1.1"><csymbol cd="ambiguous" id="S3.F6.6.m2.1.1.1.1.2.1.cmml" xref="S3.F6.6.m2.1.1.1">subscript</csymbol><ci id="S3.F6.6.m2.1.1.1.1.2.2.cmml" xref="S3.F6.6.m2.1.1.1.1.2.2">𝜎</ci><ci id="S3.F6.6.m2.1.1.1.1.2.3.cmml" xref="S3.F6.6.m2.1.1.1.1.2.3">cc</ci></apply><ci id="S3.F6.6.m2.1.1.1.1.3a.cmml" xref="S3.F6.6.m2.1.1.1.1.3"><mtext id="S3.F6.6.m2.1.1.1.1.3.cmml" mathsize="70%" xref="S3.F6.6.m2.1.1.1.1.3">expt.</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.6.m2.1d">(\sigma_{\rm cc}^{\text{expt.}})</annotation><annotation encoding="application/x-llamapun" id="S3.F6.6.m2.1e">( italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt. end_POSTSUPERSCRIPT )</annotation></semantics></math> on the left vertical axis, while the ratio <math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\rm cc}^{\text{expt.}}" class="ltx_Math" display="inline" id="S3.F6.7.m3.1"><semantics id="S3.F6.7.m3.1b"><mrow id="S3.F6.7.m3.1.1" xref="S3.F6.7.m3.1.1.cmml"><msubsup id="S3.F6.7.m3.1.1.2" xref="S3.F6.7.m3.1.1.2.cmml"><mi id="S3.F6.7.m3.1.1.2.2.2" xref="S3.F6.7.m3.1.1.2.2.2.cmml">σ</mi><mi id="S3.F6.7.m3.1.1.2.2.3" xref="S3.F6.7.m3.1.1.2.2.3.cmml">cc</mi><mtext id="S3.F6.7.m3.1.1.2.3" xref="S3.F6.7.m3.1.1.2.3a.cmml">evap</mtext></msubsup><mo id="S3.F6.7.m3.1.1.1" xref="S3.F6.7.m3.1.1.1.cmml">/</mo><msubsup id="S3.F6.7.m3.1.1.3" xref="S3.F6.7.m3.1.1.3.cmml"><mi id="S3.F6.7.m3.1.1.3.2.2" xref="S3.F6.7.m3.1.1.3.2.2.cmml">σ</mi><mi id="S3.F6.7.m3.1.1.3.2.3" xref="S3.F6.7.m3.1.1.3.2.3.cmml">cc</mi><mtext id="S3.F6.7.m3.1.1.3.3" xref="S3.F6.7.m3.1.1.3.3a.cmml">expt.</mtext></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.7.m3.1c"><apply id="S3.F6.7.m3.1.1.cmml" xref="S3.F6.7.m3.1.1"><divide id="S3.F6.7.m3.1.1.1.cmml" xref="S3.F6.7.m3.1.1.1"></divide><apply id="S3.F6.7.m3.1.1.2.cmml" xref="S3.F6.7.m3.1.1.2"><csymbol cd="ambiguous" id="S3.F6.7.m3.1.1.2.1.cmml" xref="S3.F6.7.m3.1.1.2">superscript</csymbol><apply id="S3.F6.7.m3.1.1.2.2.cmml" xref="S3.F6.7.m3.1.1.2"><csymbol cd="ambiguous" id="S3.F6.7.m3.1.1.2.2.1.cmml" xref="S3.F6.7.m3.1.1.2">subscript</csymbol><ci id="S3.F6.7.m3.1.1.2.2.2.cmml" xref="S3.F6.7.m3.1.1.2.2.2">𝜎</ci><ci id="S3.F6.7.m3.1.1.2.2.3.cmml" xref="S3.F6.7.m3.1.1.2.2.3">cc</ci></apply><ci id="S3.F6.7.m3.1.1.2.3a.cmml" xref="S3.F6.7.m3.1.1.2.3"><mtext id="S3.F6.7.m3.1.1.2.3.cmml" mathsize="70%" xref="S3.F6.7.m3.1.1.2.3">evap</mtext></ci></apply><apply id="S3.F6.7.m3.1.1.3.cmml" xref="S3.F6.7.m3.1.1.3"><csymbol cd="ambiguous" id="S3.F6.7.m3.1.1.3.1.cmml" xref="S3.F6.7.m3.1.1.3">superscript</csymbol><apply id="S3.F6.7.m3.1.1.3.2.cmml" xref="S3.F6.7.m3.1.1.3"><csymbol cd="ambiguous" id="S3.F6.7.m3.1.1.3.2.1.cmml" xref="S3.F6.7.m3.1.1.3">subscript</csymbol><ci id="S3.F6.7.m3.1.1.3.2.2.cmml" xref="S3.F6.7.m3.1.1.3.2.2">𝜎</ci><ci id="S3.F6.7.m3.1.1.3.2.3.cmml" xref="S3.F6.7.m3.1.1.3.2.3">cc</ci></apply><ci id="S3.F6.7.m3.1.1.3.3a.cmml" xref="S3.F6.7.m3.1.1.3.3"><mtext id="S3.F6.7.m3.1.1.3.3.cmml" mathsize="70%" xref="S3.F6.7.m3.1.1.3.3">expt.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.7.m3.1d">\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\rm cc}^{\text{expt.}}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.7.m3.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT / italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt. end_POSTSUPERSCRIPT</annotation></semantics></math> is shown on the right vertical axis. The bottom panel (b) displays the ratio <math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\mathrm{cc}}" class="ltx_Math" display="inline" id="S3.F6.8.m4.1"><semantics id="S3.F6.8.m4.1b"><mrow id="S3.F6.8.m4.1.1" xref="S3.F6.8.m4.1.1.cmml"><msubsup id="S3.F6.8.m4.1.1.2" xref="S3.F6.8.m4.1.1.2.cmml"><mi id="S3.F6.8.m4.1.1.2.2.2" xref="S3.F6.8.m4.1.1.2.2.2.cmml">σ</mi><mi id="S3.F6.8.m4.1.1.2.2.3" xref="S3.F6.8.m4.1.1.2.2.3.cmml">cc</mi><mtext id="S3.F6.8.m4.1.1.2.3" xref="S3.F6.8.m4.1.1.2.3a.cmml">evap</mtext></msubsup><mo id="S3.F6.8.m4.1.1.1" xref="S3.F6.8.m4.1.1.1.cmml">/</mo><msub id="S3.F6.8.m4.1.1.3" xref="S3.F6.8.m4.1.1.3.cmml"><mi id="S3.F6.8.m4.1.1.3.2" xref="S3.F6.8.m4.1.1.3.2.cmml">σ</mi><mi id="S3.F6.8.m4.1.1.3.3" xref="S3.F6.8.m4.1.1.3.3.cmml">cc</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.8.m4.1c"><apply id="S3.F6.8.m4.1.1.cmml" xref="S3.F6.8.m4.1.1"><divide id="S3.F6.8.m4.1.1.1.cmml" xref="S3.F6.8.m4.1.1.1"></divide><apply id="S3.F6.8.m4.1.1.2.cmml" xref="S3.F6.8.m4.1.1.2"><csymbol cd="ambiguous" id="S3.F6.8.m4.1.1.2.1.cmml" xref="S3.F6.8.m4.1.1.2">superscript</csymbol><apply id="S3.F6.8.m4.1.1.2.2.cmml" xref="S3.F6.8.m4.1.1.2"><csymbol cd="ambiguous" id="S3.F6.8.m4.1.1.2.2.1.cmml" xref="S3.F6.8.m4.1.1.2">subscript</csymbol><ci id="S3.F6.8.m4.1.1.2.2.2.cmml" xref="S3.F6.8.m4.1.1.2.2.2">𝜎</ci><ci id="S3.F6.8.m4.1.1.2.2.3.cmml" xref="S3.F6.8.m4.1.1.2.2.3">cc</ci></apply><ci id="S3.F6.8.m4.1.1.2.3a.cmml" xref="S3.F6.8.m4.1.1.2.3"><mtext id="S3.F6.8.m4.1.1.2.3.cmml" mathsize="70%" xref="S3.F6.8.m4.1.1.2.3">evap</mtext></ci></apply><apply id="S3.F6.8.m4.1.1.3.cmml" xref="S3.F6.8.m4.1.1.3"><csymbol cd="ambiguous" id="S3.F6.8.m4.1.1.3.1.cmml" xref="S3.F6.8.m4.1.1.3">subscript</csymbol><ci id="S3.F6.8.m4.1.1.3.2.cmml" xref="S3.F6.8.m4.1.1.3.2">𝜎</ci><ci id="S3.F6.8.m4.1.1.3.3.cmml" xref="S3.F6.8.m4.1.1.3.3">cc</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.8.m4.1d">\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\mathrm{cc}}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.8.m4.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT / italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math>. Distinct marker styles differentiate the targets, with filled markers representing carbon and unfilled markers denoting aluminum and copper. Individual data points are marked, while lines serve as visual guides to indicate trends across different projectiles and targets.</figcaption> </figure> <figure class="ltx_figure" id="S3.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="517" id="S3.F7.g1" src="x7.png" width="581"/> <figcaption class="ltx_caption ltx_centering">Figure 7: Separation energies and evaporation cross sections for various prefragment nuclei. The neutron separation energy (<math alttext="S_{1n}" class="ltx_Math" display="inline" id="S3.F7.7.m1.1"><semantics id="S3.F7.7.m1.1b"><msub id="S3.F7.7.m1.1.1" xref="S3.F7.7.m1.1.1.cmml"><mi id="S3.F7.7.m1.1.1.2" xref="S3.F7.7.m1.1.1.2.cmml">S</mi><mrow id="S3.F7.7.m1.1.1.3" xref="S3.F7.7.m1.1.1.3.cmml"><mn id="S3.F7.7.m1.1.1.3.2" xref="S3.F7.7.m1.1.1.3.2.cmml">1</mn><mo id="S3.F7.7.m1.1.1.3.1" xref="S3.F7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.F7.7.m1.1.1.3.3" xref="S3.F7.7.m1.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F7.7.m1.1c"><apply id="S3.F7.7.m1.1.1.cmml" xref="S3.F7.7.m1.1.1"><csymbol cd="ambiguous" id="S3.F7.7.m1.1.1.1.cmml" xref="S3.F7.7.m1.1.1">subscript</csymbol><ci id="S3.F7.7.m1.1.1.2.cmml" xref="S3.F7.7.m1.1.1.2">𝑆</ci><apply id="S3.F7.7.m1.1.1.3.cmml" xref="S3.F7.7.m1.1.1.3"><times id="S3.F7.7.m1.1.1.3.1.cmml" xref="S3.F7.7.m1.1.1.3.1"></times><cn id="S3.F7.7.m1.1.1.3.2.cmml" type="integer" xref="S3.F7.7.m1.1.1.3.2">1</cn><ci id="S3.F7.7.m1.1.1.3.3.cmml" xref="S3.F7.7.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.7.m1.1d">S_{1n}</annotation><annotation encoding="application/x-llamapun" id="S3.F7.7.m1.1e">italic_S start_POSTSUBSCRIPT 1 italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) is represented by bars with forward-slash patterned shading, while the proton separation energy (<math alttext="S_{1p}" class="ltx_Math" display="inline" id="S3.F7.8.m2.1"><semantics id="S3.F7.8.m2.1b"><msub id="S3.F7.8.m2.1.1" xref="S3.F7.8.m2.1.1.cmml"><mi id="S3.F7.8.m2.1.1.2" xref="S3.F7.8.m2.1.1.2.cmml">S</mi><mrow id="S3.F7.8.m2.1.1.3" xref="S3.F7.8.m2.1.1.3.cmml"><mn id="S3.F7.8.m2.1.1.3.2" xref="S3.F7.8.m2.1.1.3.2.cmml">1</mn><mo id="S3.F7.8.m2.1.1.3.1" xref="S3.F7.8.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.F7.8.m2.1.1.3.3" xref="S3.F7.8.m2.1.1.3.3.cmml">p</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F7.8.m2.1c"><apply id="S3.F7.8.m2.1.1.cmml" xref="S3.F7.8.m2.1.1"><csymbol cd="ambiguous" id="S3.F7.8.m2.1.1.1.cmml" xref="S3.F7.8.m2.1.1">subscript</csymbol><ci id="S3.F7.8.m2.1.1.2.cmml" xref="S3.F7.8.m2.1.1.2">𝑆</ci><apply id="S3.F7.8.m2.1.1.3.cmml" xref="S3.F7.8.m2.1.1.3"><times id="S3.F7.8.m2.1.1.3.1.cmml" xref="S3.F7.8.m2.1.1.3.1"></times><cn id="S3.F7.8.m2.1.1.3.2.cmml" type="integer" xref="S3.F7.8.m2.1.1.3.2">1</cn><ci id="S3.F7.8.m2.1.1.3.3.cmml" xref="S3.F7.8.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.8.m2.1d">S_{1p}</annotation><annotation encoding="application/x-llamapun" id="S3.F7.8.m2.1e">italic_S start_POSTSUBSCRIPT 1 italic_p end_POSTSUBSCRIPT</annotation></semantics></math>) is depicted with backslash-patterned shading. The difference between neutron and proton separation energies (<math alttext="\Delta S_{np}" class="ltx_Math" display="inline" id="S3.F7.9.m3.1"><semantics id="S3.F7.9.m3.1b"><mrow id="S3.F7.9.m3.1.1" xref="S3.F7.9.m3.1.1.cmml"><mi id="S3.F7.9.m3.1.1.2" mathvariant="normal" xref="S3.F7.9.m3.1.1.2.cmml">Δ</mi><mo id="S3.F7.9.m3.1.1.1" xref="S3.F7.9.m3.1.1.1.cmml">⁢</mo><msub id="S3.F7.9.m3.1.1.3" xref="S3.F7.9.m3.1.1.3.cmml"><mi id="S3.F7.9.m3.1.1.3.2" xref="S3.F7.9.m3.1.1.3.2.cmml">S</mi><mrow id="S3.F7.9.m3.1.1.3.3" xref="S3.F7.9.m3.1.1.3.3.cmml"><mi id="S3.F7.9.m3.1.1.3.3.2" xref="S3.F7.9.m3.1.1.3.3.2.cmml">n</mi><mo id="S3.F7.9.m3.1.1.3.3.1" xref="S3.F7.9.m3.1.1.3.3.1.cmml">⁢</mo><mi id="S3.F7.9.m3.1.1.3.3.3" xref="S3.F7.9.m3.1.1.3.3.3.cmml">p</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.9.m3.1c"><apply id="S3.F7.9.m3.1.1.cmml" xref="S3.F7.9.m3.1.1"><times id="S3.F7.9.m3.1.1.1.cmml" xref="S3.F7.9.m3.1.1.1"></times><ci id="S3.F7.9.m3.1.1.2.cmml" xref="S3.F7.9.m3.1.1.2">Δ</ci><apply id="S3.F7.9.m3.1.1.3.cmml" xref="S3.F7.9.m3.1.1.3"><csymbol cd="ambiguous" id="S3.F7.9.m3.1.1.3.1.cmml" xref="S3.F7.9.m3.1.1.3">subscript</csymbol><ci id="S3.F7.9.m3.1.1.3.2.cmml" xref="S3.F7.9.m3.1.1.3.2">𝑆</ci><apply id="S3.F7.9.m3.1.1.3.3.cmml" xref="S3.F7.9.m3.1.1.3.3"><times id="S3.F7.9.m3.1.1.3.3.1.cmml" xref="S3.F7.9.m3.1.1.3.3.1"></times><ci id="S3.F7.9.m3.1.1.3.3.2.cmml" xref="S3.F7.9.m3.1.1.3.3.2">𝑛</ci><ci id="S3.F7.9.m3.1.1.3.3.3.cmml" xref="S3.F7.9.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.9.m3.1d">\Delta S_{np}</annotation><annotation encoding="application/x-llamapun" id="S3.F7.9.m3.1e">roman_Δ italic_S start_POSTSUBSCRIPT italic_n italic_p end_POSTSUBSCRIPT</annotation></semantics></math>) is shown as a dashed-dotted line. Evaporation cross sections following one-neutron removal in the abrasion stage (<math alttext="\sigma_{\mathrm{cc}}^{\mathrm{evap}}(1n)" class="ltx_Math" display="inline" id="S3.F7.10.m4.1"><semantics id="S3.F7.10.m4.1b"><mrow id="S3.F7.10.m4.1.1" xref="S3.F7.10.m4.1.1.cmml"><msubsup id="S3.F7.10.m4.1.1.3" xref="S3.F7.10.m4.1.1.3.cmml"><mi id="S3.F7.10.m4.1.1.3.2.2" xref="S3.F7.10.m4.1.1.3.2.2.cmml">σ</mi><mi id="S3.F7.10.m4.1.1.3.2.3" xref="S3.F7.10.m4.1.1.3.2.3.cmml">cc</mi><mi id="S3.F7.10.m4.1.1.3.3" xref="S3.F7.10.m4.1.1.3.3.cmml">evap</mi></msubsup><mo id="S3.F7.10.m4.1.1.2" xref="S3.F7.10.m4.1.1.2.cmml">⁢</mo><mrow id="S3.F7.10.m4.1.1.1.1" xref="S3.F7.10.m4.1.1.1.1.1.cmml"><mo id="S3.F7.10.m4.1.1.1.1.2" stretchy="false" xref="S3.F7.10.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.F7.10.m4.1.1.1.1.1" xref="S3.F7.10.m4.1.1.1.1.1.cmml"><mn id="S3.F7.10.m4.1.1.1.1.1.2" xref="S3.F7.10.m4.1.1.1.1.1.2.cmml">1</mn><mo id="S3.F7.10.m4.1.1.1.1.1.1" xref="S3.F7.10.m4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.F7.10.m4.1.1.1.1.1.3" xref="S3.F7.10.m4.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="S3.F7.10.m4.1.1.1.1.3" stretchy="false" xref="S3.F7.10.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.10.m4.1c"><apply id="S3.F7.10.m4.1.1.cmml" xref="S3.F7.10.m4.1.1"><times id="S3.F7.10.m4.1.1.2.cmml" xref="S3.F7.10.m4.1.1.2"></times><apply id="S3.F7.10.m4.1.1.3.cmml" xref="S3.F7.10.m4.1.1.3"><csymbol cd="ambiguous" id="S3.F7.10.m4.1.1.3.1.cmml" xref="S3.F7.10.m4.1.1.3">superscript</csymbol><apply id="S3.F7.10.m4.1.1.3.2.cmml" xref="S3.F7.10.m4.1.1.3"><csymbol cd="ambiguous" id="S3.F7.10.m4.1.1.3.2.1.cmml" xref="S3.F7.10.m4.1.1.3">subscript</csymbol><ci id="S3.F7.10.m4.1.1.3.2.2.cmml" xref="S3.F7.10.m4.1.1.3.2.2">𝜎</ci><ci id="S3.F7.10.m4.1.1.3.2.3.cmml" xref="S3.F7.10.m4.1.1.3.2.3">cc</ci></apply><ci id="S3.F7.10.m4.1.1.3.3.cmml" xref="S3.F7.10.m4.1.1.3.3">evap</ci></apply><apply id="S3.F7.10.m4.1.1.1.1.1.cmml" xref="S3.F7.10.m4.1.1.1.1"><times id="S3.F7.10.m4.1.1.1.1.1.1.cmml" xref="S3.F7.10.m4.1.1.1.1.1.1"></times><cn id="S3.F7.10.m4.1.1.1.1.1.2.cmml" type="integer" xref="S3.F7.10.m4.1.1.1.1.1.2">1</cn><ci id="S3.F7.10.m4.1.1.1.1.1.3.cmml" xref="S3.F7.10.m4.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.10.m4.1d">\sigma_{\mathrm{cc}}^{\mathrm{evap}}(1n)</annotation><annotation encoding="application/x-llamapun" id="S3.F7.10.m4.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_evap end_POSTSUPERSCRIPT ( 1 italic_n )</annotation></semantics></math>) are represented by diamond markers with dashed lines, two-neutron removal (<math alttext="\sigma_{\mathrm{cc}}^{\mathrm{evap}}(2n)" class="ltx_Math" display="inline" id="S3.F7.11.m5.1"><semantics id="S3.F7.11.m5.1b"><mrow id="S3.F7.11.m5.1.1" xref="S3.F7.11.m5.1.1.cmml"><msubsup id="S3.F7.11.m5.1.1.3" xref="S3.F7.11.m5.1.1.3.cmml"><mi id="S3.F7.11.m5.1.1.3.2.2" xref="S3.F7.11.m5.1.1.3.2.2.cmml">σ</mi><mi id="S3.F7.11.m5.1.1.3.2.3" xref="S3.F7.11.m5.1.1.3.2.3.cmml">cc</mi><mi id="S3.F7.11.m5.1.1.3.3" xref="S3.F7.11.m5.1.1.3.3.cmml">evap</mi></msubsup><mo id="S3.F7.11.m5.1.1.2" xref="S3.F7.11.m5.1.1.2.cmml">⁢</mo><mrow id="S3.F7.11.m5.1.1.1.1" xref="S3.F7.11.m5.1.1.1.1.1.cmml"><mo id="S3.F7.11.m5.1.1.1.1.2" stretchy="false" xref="S3.F7.11.m5.1.1.1.1.1.cmml">(</mo><mrow id="S3.F7.11.m5.1.1.1.1.1" xref="S3.F7.11.m5.1.1.1.1.1.cmml"><mn id="S3.F7.11.m5.1.1.1.1.1.2" xref="S3.F7.11.m5.1.1.1.1.1.2.cmml">2</mn><mo id="S3.F7.11.m5.1.1.1.1.1.1" xref="S3.F7.11.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.F7.11.m5.1.1.1.1.1.3" xref="S3.F7.11.m5.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="S3.F7.11.m5.1.1.1.1.3" stretchy="false" xref="S3.F7.11.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.11.m5.1c"><apply id="S3.F7.11.m5.1.1.cmml" xref="S3.F7.11.m5.1.1"><times id="S3.F7.11.m5.1.1.2.cmml" xref="S3.F7.11.m5.1.1.2"></times><apply id="S3.F7.11.m5.1.1.3.cmml" xref="S3.F7.11.m5.1.1.3"><csymbol cd="ambiguous" id="S3.F7.11.m5.1.1.3.1.cmml" xref="S3.F7.11.m5.1.1.3">superscript</csymbol><apply id="S3.F7.11.m5.1.1.3.2.cmml" xref="S3.F7.11.m5.1.1.3"><csymbol cd="ambiguous" id="S3.F7.11.m5.1.1.3.2.1.cmml" xref="S3.F7.11.m5.1.1.3">subscript</csymbol><ci id="S3.F7.11.m5.1.1.3.2.2.cmml" xref="S3.F7.11.m5.1.1.3.2.2">𝜎</ci><ci id="S3.F7.11.m5.1.1.3.2.3.cmml" xref="S3.F7.11.m5.1.1.3.2.3">cc</ci></apply><ci id="S3.F7.11.m5.1.1.3.3.cmml" xref="S3.F7.11.m5.1.1.3.3">evap</ci></apply><apply id="S3.F7.11.m5.1.1.1.1.1.cmml" xref="S3.F7.11.m5.1.1.1.1"><times id="S3.F7.11.m5.1.1.1.1.1.1.cmml" xref="S3.F7.11.m5.1.1.1.1.1.1"></times><cn id="S3.F7.11.m5.1.1.1.1.1.2.cmml" type="integer" xref="S3.F7.11.m5.1.1.1.1.1.2">2</cn><ci id="S3.F7.11.m5.1.1.1.1.1.3.cmml" xref="S3.F7.11.m5.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.11.m5.1d">\sigma_{\mathrm{cc}}^{\mathrm{evap}}(2n)</annotation><annotation encoding="application/x-llamapun" id="S3.F7.11.m5.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_evap end_POSTSUPERSCRIPT ( 2 italic_n )</annotation></semantics></math>) by circular markers with dotted lines, and total evaporation cross sections (<math alttext="\sigma_{\mathrm{cc}}^{\mathrm{evap}}(\text{Total})" class="ltx_Math" display="inline" id="S3.F7.12.m6.1"><semantics id="S3.F7.12.m6.1b"><mrow id="S3.F7.12.m6.1.2" xref="S3.F7.12.m6.1.2.cmml"><msubsup id="S3.F7.12.m6.1.2.2" xref="S3.F7.12.m6.1.2.2.cmml"><mi id="S3.F7.12.m6.1.2.2.2.2" xref="S3.F7.12.m6.1.2.2.2.2.cmml">σ</mi><mi id="S3.F7.12.m6.1.2.2.2.3" xref="S3.F7.12.m6.1.2.2.2.3.cmml">cc</mi><mi id="S3.F7.12.m6.1.2.2.3" xref="S3.F7.12.m6.1.2.2.3.cmml">evap</mi></msubsup><mo id="S3.F7.12.m6.1.2.1" xref="S3.F7.12.m6.1.2.1.cmml">⁢</mo><mrow id="S3.F7.12.m6.1.2.3.2" xref="S3.F7.12.m6.1.1a.cmml"><mo id="S3.F7.12.m6.1.2.3.2.1" stretchy="false" xref="S3.F7.12.m6.1.1a.cmml">(</mo><mtext id="S3.F7.12.m6.1.1" xref="S3.F7.12.m6.1.1.cmml">Total</mtext><mo id="S3.F7.12.m6.1.2.3.2.2" stretchy="false" xref="S3.F7.12.m6.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.12.m6.1c"><apply id="S3.F7.12.m6.1.2.cmml" xref="S3.F7.12.m6.1.2"><times id="S3.F7.12.m6.1.2.1.cmml" xref="S3.F7.12.m6.1.2.1"></times><apply id="S3.F7.12.m6.1.2.2.cmml" xref="S3.F7.12.m6.1.2.2"><csymbol cd="ambiguous" id="S3.F7.12.m6.1.2.2.1.cmml" xref="S3.F7.12.m6.1.2.2">superscript</csymbol><apply id="S3.F7.12.m6.1.2.2.2.cmml" xref="S3.F7.12.m6.1.2.2"><csymbol cd="ambiguous" id="S3.F7.12.m6.1.2.2.2.1.cmml" xref="S3.F7.12.m6.1.2.2">subscript</csymbol><ci id="S3.F7.12.m6.1.2.2.2.2.cmml" xref="S3.F7.12.m6.1.2.2.2.2">𝜎</ci><ci id="S3.F7.12.m6.1.2.2.2.3.cmml" xref="S3.F7.12.m6.1.2.2.2.3">cc</ci></apply><ci id="S3.F7.12.m6.1.2.2.3.cmml" xref="S3.F7.12.m6.1.2.2.3">evap</ci></apply><ci id="S3.F7.12.m6.1.1a.cmml" xref="S3.F7.12.m6.1.2.3.2"><mtext id="S3.F7.12.m6.1.1.cmml" xref="S3.F7.12.m6.1.1">Total</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.12.m6.1d">\sigma_{\mathrm{cc}}^{\mathrm{evap}}(\text{Total})</annotation><annotation encoding="application/x-llamapun" id="S3.F7.12.m6.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_evap end_POSTSUPERSCRIPT ( Total )</annotation></semantics></math>) by square markers with solid lines. The separation energy data are calculated using the atomic mass table from Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib64" title="">64</a>]</cite>.</figcaption> </figure> <figure class="ltx_figure" id="S3.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="407" id="S3.F8.g1" src="x8.png" width="581"/> <figcaption class="ltx_caption ltx_centering">Figure 8: Ratio of the proton to neutron decay widths as a function of the excitation energy of the prefragment after <math alttext="1n" class="ltx_Math" display="inline" id="S3.F8.2.m1.1"><semantics id="S3.F8.2.m1.1b"><mrow id="S3.F8.2.m1.1.1" xref="S3.F8.2.m1.1.1.cmml"><mn id="S3.F8.2.m1.1.1.2" xref="S3.F8.2.m1.1.1.2.cmml">1</mn><mo id="S3.F8.2.m1.1.1.1" xref="S3.F8.2.m1.1.1.1.cmml">⁢</mo><mi id="S3.F8.2.m1.1.1.3" xref="S3.F8.2.m1.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F8.2.m1.1c"><apply id="S3.F8.2.m1.1.1.cmml" xref="S3.F8.2.m1.1.1"><times id="S3.F8.2.m1.1.1.1.cmml" xref="S3.F8.2.m1.1.1.1"></times><cn id="S3.F8.2.m1.1.1.2.cmml" type="integer" xref="S3.F8.2.m1.1.1.2">1</cn><ci id="S3.F8.2.m1.1.1.3.cmml" xref="S3.F8.2.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.2.m1.1d">1n</annotation><annotation encoding="application/x-llamapun" id="S3.F8.2.m1.1e">1 italic_n</annotation></semantics></math> removal from the projectiles.</figcaption> </figure> <figure class="ltx_table" id="S3.T2"> <figcaption class="ltx_caption ltx_centering">Table 2: Proton evaporation contributions (<math alttext="\sigma_{\rm cc}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.T2.11.m1.1"><semantics id="S3.T2.11.m1.1b"><msubsup id="S3.T2.11.m1.1.1" xref="S3.T2.11.m1.1.1.cmml"><mi id="S3.T2.11.m1.1.1.2.2" xref="S3.T2.11.m1.1.1.2.2.cmml">σ</mi><mi id="S3.T2.11.m1.1.1.2.3" xref="S3.T2.11.m1.1.1.2.3.cmml">cc</mi><mtext id="S3.T2.11.m1.1.1.3" xref="S3.T2.11.m1.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.T2.11.m1.1c"><apply id="S3.T2.11.m1.1.1.cmml" xref="S3.T2.11.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.11.m1.1.1.1.cmml" xref="S3.T2.11.m1.1.1">superscript</csymbol><apply id="S3.T2.11.m1.1.1.2.cmml" xref="S3.T2.11.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.11.m1.1.1.2.1.cmml" xref="S3.T2.11.m1.1.1">subscript</csymbol><ci id="S3.T2.11.m1.1.1.2.2.cmml" xref="S3.T2.11.m1.1.1.2.2">𝜎</ci><ci id="S3.T2.11.m1.1.1.2.3.cmml" xref="S3.T2.11.m1.1.1.2.3">cc</ci></apply><ci id="S3.T2.11.m1.1.1.3a.cmml" xref="S3.T2.11.m1.1.1.3"><mtext id="S3.T2.11.m1.1.1.3.cmml" mathsize="70%" xref="S3.T2.11.m1.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.11.m1.1d">\sigma_{\rm cc}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.11.m1.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math>) [Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E11" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">11</a>)] to charge-changing cross section for various projectile nuclei on elemental targets (C, Al, Cu) at a beam energy around 290 MeV/nucleon. The partial neutron removal cross sections (<math alttext="\sigma_{-xn}" class="ltx_Math" display="inline" id="S3.T2.12.m2.1"><semantics id="S3.T2.12.m2.1b"><msub id="S3.T2.12.m2.1.1" xref="S3.T2.12.m2.1.1.cmml"><mi id="S3.T2.12.m2.1.1.2" xref="S3.T2.12.m2.1.1.2.cmml">σ</mi><mrow id="S3.T2.12.m2.1.1.3" xref="S3.T2.12.m2.1.1.3.cmml"><mo id="S3.T2.12.m2.1.1.3b" xref="S3.T2.12.m2.1.1.3.cmml">−</mo><mrow id="S3.T2.12.m2.1.1.3.2" xref="S3.T2.12.m2.1.1.3.2.cmml"><mi id="S3.T2.12.m2.1.1.3.2.2" xref="S3.T2.12.m2.1.1.3.2.2.cmml">x</mi><mo id="S3.T2.12.m2.1.1.3.2.1" xref="S3.T2.12.m2.1.1.3.2.1.cmml">⁢</mo><mi id="S3.T2.12.m2.1.1.3.2.3" xref="S3.T2.12.m2.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T2.12.m2.1c"><apply id="S3.T2.12.m2.1.1.cmml" xref="S3.T2.12.m2.1.1"><csymbol cd="ambiguous" id="S3.T2.12.m2.1.1.1.cmml" xref="S3.T2.12.m2.1.1">subscript</csymbol><ci id="S3.T2.12.m2.1.1.2.cmml" xref="S3.T2.12.m2.1.1.2">𝜎</ci><apply id="S3.T2.12.m2.1.1.3.cmml" xref="S3.T2.12.m2.1.1.3"><minus id="S3.T2.12.m2.1.1.3.1.cmml" xref="S3.T2.12.m2.1.1.3"></minus><apply id="S3.T2.12.m2.1.1.3.2.cmml" xref="S3.T2.12.m2.1.1.3.2"><times id="S3.T2.12.m2.1.1.3.2.1.cmml" xref="S3.T2.12.m2.1.1.3.2.1"></times><ci id="S3.T2.12.m2.1.1.3.2.2.cmml" xref="S3.T2.12.m2.1.1.3.2.2">𝑥</ci><ci id="S3.T2.12.m2.1.1.3.2.3.cmml" xref="S3.T2.12.m2.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.12.m2.1d">\sigma_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.12.m2.1e">italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) [Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E4" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">4</a>)] for 1<math alttext="n" class="ltx_Math" display="inline" id="S3.T2.13.m3.1"><semantics id="S3.T2.13.m3.1b"><mi id="S3.T2.13.m3.1.1" xref="S3.T2.13.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.T2.13.m3.1c"><ci id="S3.T2.13.m3.1.1.cmml" xref="S3.T2.13.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.13.m3.1d">n</annotation><annotation encoding="application/x-llamapun" id="S3.T2.13.m3.1e">italic_n</annotation></semantics></math>, 2<math alttext="n" class="ltx_Math" display="inline" id="S3.T2.14.m4.1"><semantics id="S3.T2.14.m4.1b"><mi id="S3.T2.14.m4.1.1" xref="S3.T2.14.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.T2.14.m4.1c"><ci id="S3.T2.14.m4.1.1.cmml" xref="S3.T2.14.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.14.m4.1d">n</annotation><annotation encoding="application/x-llamapun" id="S3.T2.14.m4.1e">italic_n</annotation></semantics></math>, 3<math alttext="n" class="ltx_Math" display="inline" id="S3.T2.15.m5.1"><semantics id="S3.T2.15.m5.1b"><mi id="S3.T2.15.m5.1.1" xref="S3.T2.15.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.T2.15.m5.1c"><ci id="S3.T2.15.m5.1.1.cmml" xref="S3.T2.15.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.15.m5.1d">n</annotation><annotation encoding="application/x-llamapun" id="S3.T2.15.m5.1e">italic_n</annotation></semantics></math>, and beyond 3<math alttext="n" class="ltx_Math" display="inline" id="S3.T2.16.m6.1"><semantics id="S3.T2.16.m6.1b"><mi id="S3.T2.16.m6.1.1" xref="S3.T2.16.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.T2.16.m6.1c"><ci id="S3.T2.16.m6.1.1.cmml" xref="S3.T2.16.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.16.m6.1d">n</annotation><annotation encoding="application/x-llamapun" id="S3.T2.16.m6.1e">italic_n</annotation></semantics></math> channels are also shown, along with their respective contributions to <math alttext="\sigma_{\rm cc}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.T2.17.m7.1"><semantics id="S3.T2.17.m7.1b"><msubsup id="S3.T2.17.m7.1.1" xref="S3.T2.17.m7.1.1.cmml"><mi id="S3.T2.17.m7.1.1.2.2" xref="S3.T2.17.m7.1.1.2.2.cmml">σ</mi><mi id="S3.T2.17.m7.1.1.2.3" xref="S3.T2.17.m7.1.1.2.3.cmml">cc</mi><mtext id="S3.T2.17.m7.1.1.3" xref="S3.T2.17.m7.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.T2.17.m7.1c"><apply id="S3.T2.17.m7.1.1.cmml" xref="S3.T2.17.m7.1.1"><csymbol cd="ambiguous" id="S3.T2.17.m7.1.1.1.cmml" xref="S3.T2.17.m7.1.1">superscript</csymbol><apply id="S3.T2.17.m7.1.1.2.cmml" xref="S3.T2.17.m7.1.1"><csymbol cd="ambiguous" id="S3.T2.17.m7.1.1.2.1.cmml" xref="S3.T2.17.m7.1.1">subscript</csymbol><ci id="S3.T2.17.m7.1.1.2.2.cmml" xref="S3.T2.17.m7.1.1.2.2">𝜎</ci><ci id="S3.T2.17.m7.1.1.2.3.cmml" xref="S3.T2.17.m7.1.1.2.3">cc</ci></apply><ci id="S3.T2.17.m7.1.1.3a.cmml" xref="S3.T2.17.m7.1.1.3"><mtext id="S3.T2.17.m7.1.1.3.cmml" mathsize="70%" xref="S3.T2.17.m7.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.17.m7.1d">\sigma_{\rm cc}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.17.m7.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math>. <math alttext="\sigma_{\rm cc}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.T2.18.m8.1"><semantics id="S3.T2.18.m8.1b"><msubsup id="S3.T2.18.m8.1.1" xref="S3.T2.18.m8.1.1.cmml"><mi id="S3.T2.18.m8.1.1.2.2" xref="S3.T2.18.m8.1.1.2.2.cmml">σ</mi><mi id="S3.T2.18.m8.1.1.2.3" xref="S3.T2.18.m8.1.1.2.3.cmml">cc</mi><mtext id="S3.T2.18.m8.1.1.3" xref="S3.T2.18.m8.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.T2.18.m8.1c"><apply id="S3.T2.18.m8.1.1.cmml" xref="S3.T2.18.m8.1.1"><csymbol cd="ambiguous" id="S3.T2.18.m8.1.1.1.cmml" xref="S3.T2.18.m8.1.1">superscript</csymbol><apply id="S3.T2.18.m8.1.1.2.cmml" xref="S3.T2.18.m8.1.1"><csymbol cd="ambiguous" id="S3.T2.18.m8.1.1.2.1.cmml" xref="S3.T2.18.m8.1.1">subscript</csymbol><ci id="S3.T2.18.m8.1.1.2.2.cmml" xref="S3.T2.18.m8.1.1.2.2">𝜎</ci><ci id="S3.T2.18.m8.1.1.2.3.cmml" xref="S3.T2.18.m8.1.1.2.3">cc</ci></apply><ci id="S3.T2.18.m8.1.1.3a.cmml" xref="S3.T2.18.m8.1.1.3"><mtext id="S3.T2.18.m8.1.1.3.cmml" mathsize="70%" xref="S3.T2.18.m8.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.18.m8.1d">\sigma_{\rm cc}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.18.m8.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math> is calculated by summing up all the contributions from all neutron removal channels. For comparison, the evaporation contribution is also estimated by subtracting the Glauber charge-changing cross section [<math alttext="\sigma_{\rm cc}" class="ltx_Math" display="inline" id="S3.T2.19.m9.1"><semantics id="S3.T2.19.m9.1b"><msub id="S3.T2.19.m9.1.1" xref="S3.T2.19.m9.1.1.cmml"><mi id="S3.T2.19.m9.1.1.2" xref="S3.T2.19.m9.1.1.2.cmml">σ</mi><mi id="S3.T2.19.m9.1.1.3" xref="S3.T2.19.m9.1.1.3.cmml">cc</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T2.19.m9.1c"><apply id="S3.T2.19.m9.1.1.cmml" xref="S3.T2.19.m9.1.1"><csymbol cd="ambiguous" id="S3.T2.19.m9.1.1.1.cmml" xref="S3.T2.19.m9.1.1">subscript</csymbol><ci id="S3.T2.19.m9.1.1.2.cmml" xref="S3.T2.19.m9.1.1.2">𝜎</ci><ci id="S3.T2.19.m9.1.1.3.cmml" xref="S3.T2.19.m9.1.1.3">cc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.19.m9.1d">\sigma_{\rm cc}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.19.m9.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math> (Table <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.T1" title="Table 1 ‣ III.2 Direct fragmentation or abrasion stage ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">1</a>)] from the experimental data (<math alttext="\sigma_{\rm cc}^{\text{expt}}" class="ltx_Math" display="inline" id="S3.T2.20.m10.1"><semantics id="S3.T2.20.m10.1b"><msubsup id="S3.T2.20.m10.1.1" xref="S3.T2.20.m10.1.1.cmml"><mi id="S3.T2.20.m10.1.1.2.2" xref="S3.T2.20.m10.1.1.2.2.cmml">σ</mi><mi id="S3.T2.20.m10.1.1.2.3" xref="S3.T2.20.m10.1.1.2.3.cmml">cc</mi><mtext id="S3.T2.20.m10.1.1.3" xref="S3.T2.20.m10.1.1.3a.cmml">expt</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.T2.20.m10.1c"><apply id="S3.T2.20.m10.1.1.cmml" xref="S3.T2.20.m10.1.1"><csymbol cd="ambiguous" id="S3.T2.20.m10.1.1.1.cmml" xref="S3.T2.20.m10.1.1">superscript</csymbol><apply id="S3.T2.20.m10.1.1.2.cmml" xref="S3.T2.20.m10.1.1"><csymbol cd="ambiguous" id="S3.T2.20.m10.1.1.2.1.cmml" xref="S3.T2.20.m10.1.1">subscript</csymbol><ci id="S3.T2.20.m10.1.1.2.2.cmml" xref="S3.T2.20.m10.1.1.2.2">𝜎</ci><ci id="S3.T2.20.m10.1.1.2.3.cmml" xref="S3.T2.20.m10.1.1.2.3">cc</ci></apply><ci id="S3.T2.20.m10.1.1.3a.cmml" xref="S3.T2.20.m10.1.1.3"><mtext id="S3.T2.20.m10.1.1.3.cmml" mathsize="70%" xref="S3.T2.20.m10.1.1.3">expt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.20.m10.1d">\sigma_{\rm cc}^{\text{expt}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.20.m10.1e">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt end_POSTSUPERSCRIPT</annotation></semantics></math>) taken from Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib63" title="">63</a>]</cite>. All cross sections in this table are rounded off to the nearest integer.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S3.T2.37" style="width:433.6pt;height:218.6pt;vertical-align:-0.9pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S3.T2.37.17"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T2.25.5.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_tt" id="S3.T2.21.1.1.1" rowspan="2" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.T2.21.1.1.1.1.m1.1"><semantics id="S3.T2.21.1.1.1.1.m1.1a"><msub id="S3.T2.21.1.1.1.1.m1.1.1" xref="S3.T2.21.1.1.1.1.m1.1.1.cmml"><mi id="S3.T2.21.1.1.1.1.m1.1.1.2" xref="S3.T2.21.1.1.1.1.m1.1.1.2.cmml">E</mi><mtext id="S3.T2.21.1.1.1.1.m1.1.1.3" xref="S3.T2.21.1.1.1.1.m1.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.T2.21.1.1.1.1.m1.1b"><apply id="S3.T2.21.1.1.1.1.m1.1.1.cmml" xref="S3.T2.21.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.21.1.1.1.1.m1.1.1.1.cmml" xref="S3.T2.21.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S3.T2.21.1.1.1.1.m1.1.1.2.cmml" xref="S3.T2.21.1.1.1.1.m1.1.1.2">𝐸</ci><ci id="S3.T2.21.1.1.1.1.m1.1.1.3a.cmml" xref="S3.T2.21.1.1.1.1.m1.1.1.3"><mtext id="S3.T2.21.1.1.1.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T2.21.1.1.1.1.m1.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.21.1.1.1.1.m1.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.21.1.1.1.1.m1.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> (MeV)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_tt" id="S3.T2.25.5.5.6" rowspan="2" style="padding-top:-1pt;padding-bottom:-1pt;">Nuclei</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_tt" id="S3.T2.25.5.5.7" rowspan="2" style="padding-top:-1pt;padding-bottom:-1pt;">Target</th> <td class="ltx_td ltx_align_center ltx_border_tt" colspan="4" id="S3.T2.22.2.2.2" style="padding-top:-1pt;padding-bottom:-1pt;"> <math alttext="\sigma_{{-xn}}" class="ltx_Math" display="inline" id="S3.T2.22.2.2.2.m1.1"><semantics id="S3.T2.22.2.2.2.m1.1a"><msub id="S3.T2.22.2.2.2.m1.1.1" xref="S3.T2.22.2.2.2.m1.1.1.cmml"><mi id="S3.T2.22.2.2.2.m1.1.1.2" xref="S3.T2.22.2.2.2.m1.1.1.2.cmml">σ</mi><mrow id="S3.T2.22.2.2.2.m1.1.1.3" xref="S3.T2.22.2.2.2.m1.1.1.3.cmml"><mo id="S3.T2.22.2.2.2.m1.1.1.3a" xref="S3.T2.22.2.2.2.m1.1.1.3.cmml">−</mo><mrow id="S3.T2.22.2.2.2.m1.1.1.3.2" xref="S3.T2.22.2.2.2.m1.1.1.3.2.cmml"><mi id="S3.T2.22.2.2.2.m1.1.1.3.2.2" xref="S3.T2.22.2.2.2.m1.1.1.3.2.2.cmml">x</mi><mo id="S3.T2.22.2.2.2.m1.1.1.3.2.1" xref="S3.T2.22.2.2.2.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.T2.22.2.2.2.m1.1.1.3.2.3" xref="S3.T2.22.2.2.2.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T2.22.2.2.2.m1.1b"><apply id="S3.T2.22.2.2.2.m1.1.1.cmml" xref="S3.T2.22.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.22.2.2.2.m1.1.1.1.cmml" xref="S3.T2.22.2.2.2.m1.1.1">subscript</csymbol><ci id="S3.T2.22.2.2.2.m1.1.1.2.cmml" xref="S3.T2.22.2.2.2.m1.1.1.2">𝜎</ci><apply id="S3.T2.22.2.2.2.m1.1.1.3.cmml" xref="S3.T2.22.2.2.2.m1.1.1.3"><minus id="S3.T2.22.2.2.2.m1.1.1.3.1.cmml" xref="S3.T2.22.2.2.2.m1.1.1.3"></minus><apply id="S3.T2.22.2.2.2.m1.1.1.3.2.cmml" xref="S3.T2.22.2.2.2.m1.1.1.3.2"><times id="S3.T2.22.2.2.2.m1.1.1.3.2.1.cmml" xref="S3.T2.22.2.2.2.m1.1.1.3.2.1"></times><ci id="S3.T2.22.2.2.2.m1.1.1.3.2.2.cmml" xref="S3.T2.22.2.2.2.m1.1.1.3.2.2">𝑥</ci><ci id="S3.T2.22.2.2.2.m1.1.1.3.2.3.cmml" xref="S3.T2.22.2.2.2.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.22.2.2.2.m1.1c">\sigma_{{-xn}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.22.2.2.2.m1.1d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math> (mb)</td> <td class="ltx_td ltx_align_center ltx_border_tt" colspan="4" id="S3.T2.23.3.3.3" style="padding-top:-1pt;padding-bottom:-1pt;">Contributions to <math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.T2.23.3.3.3.m1.1"><semantics id="S3.T2.23.3.3.3.m1.1a"><msubsup id="S3.T2.23.3.3.3.m1.1.1" xref="S3.T2.23.3.3.3.m1.1.1.cmml"><mi id="S3.T2.23.3.3.3.m1.1.1.2.2" xref="S3.T2.23.3.3.3.m1.1.1.2.2.cmml">σ</mi><mi id="S3.T2.23.3.3.3.m1.1.1.2.3" xref="S3.T2.23.3.3.3.m1.1.1.2.3.cmml">cc</mi><mtext id="S3.T2.23.3.3.3.m1.1.1.3" xref="S3.T2.23.3.3.3.m1.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.T2.23.3.3.3.m1.1b"><apply id="S3.T2.23.3.3.3.m1.1.1.cmml" xref="S3.T2.23.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.23.3.3.3.m1.1.1.1.cmml" xref="S3.T2.23.3.3.3.m1.1.1">superscript</csymbol><apply id="S3.T2.23.3.3.3.m1.1.1.2.cmml" xref="S3.T2.23.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.23.3.3.3.m1.1.1.2.1.cmml" xref="S3.T2.23.3.3.3.m1.1.1">subscript</csymbol><ci id="S3.T2.23.3.3.3.m1.1.1.2.2.cmml" xref="S3.T2.23.3.3.3.m1.1.1.2.2">𝜎</ci><ci id="S3.T2.23.3.3.3.m1.1.1.2.3.cmml" xref="S3.T2.23.3.3.3.m1.1.1.2.3">cc</ci></apply><ci id="S3.T2.23.3.3.3.m1.1.1.3a.cmml" xref="S3.T2.23.3.3.3.m1.1.1.3"><mtext id="S3.T2.23.3.3.3.m1.1.1.3.cmml" mathsize="70%" xref="S3.T2.23.3.3.3.m1.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.23.3.3.3.m1.1c">\sigma_{\mathrm{cc}}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.23.3.3.3.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math> (mb)</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T2.24.4.4.4" rowspan="2" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.T2.24.4.4.4.1.m1.1"><semantics id="S3.T2.24.4.4.4.1.m1.1a"><msubsup id="S3.T2.24.4.4.4.1.m1.1.1" xref="S3.T2.24.4.4.4.1.m1.1.1.cmml"><mi id="S3.T2.24.4.4.4.1.m1.1.1.2.2" xref="S3.T2.24.4.4.4.1.m1.1.1.2.2.cmml">σ</mi><mi id="S3.T2.24.4.4.4.1.m1.1.1.2.3" xref="S3.T2.24.4.4.4.1.m1.1.1.2.3.cmml">cc</mi><mtext id="S3.T2.24.4.4.4.1.m1.1.1.3" xref="S3.T2.24.4.4.4.1.m1.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.T2.24.4.4.4.1.m1.1b"><apply id="S3.T2.24.4.4.4.1.m1.1.1.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.24.4.4.4.1.m1.1.1.1.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1">superscript</csymbol><apply id="S3.T2.24.4.4.4.1.m1.1.1.2.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.24.4.4.4.1.m1.1.1.2.1.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1">subscript</csymbol><ci id="S3.T2.24.4.4.4.1.m1.1.1.2.2.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1.2.2">𝜎</ci><ci id="S3.T2.24.4.4.4.1.m1.1.1.2.3.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1.2.3">cc</ci></apply><ci id="S3.T2.24.4.4.4.1.m1.1.1.3a.cmml" xref="S3.T2.24.4.4.4.1.m1.1.1.3"><mtext id="S3.T2.24.4.4.4.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.T2.24.4.4.4.1.m1.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.24.4.4.4.1.m1.1c">\sigma_{\mathrm{cc}}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.24.4.4.4.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math> (mb)</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T2.25.5.5.5" rowspan="2" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="\sigma_{\mathrm{cc}}^{\text{expt}}-\sigma_{\mathrm{cc}}" class="ltx_Math" display="inline" id="S3.T2.25.5.5.5.1.m1.1"><semantics id="S3.T2.25.5.5.5.1.m1.1a"><mrow id="S3.T2.25.5.5.5.1.m1.1.1" xref="S3.T2.25.5.5.5.1.m1.1.1.cmml"><msubsup id="S3.T2.25.5.5.5.1.m1.1.1.2" xref="S3.T2.25.5.5.5.1.m1.1.1.2.cmml"><mi id="S3.T2.25.5.5.5.1.m1.1.1.2.2.2" xref="S3.T2.25.5.5.5.1.m1.1.1.2.2.2.cmml">σ</mi><mi id="S3.T2.25.5.5.5.1.m1.1.1.2.2.3" xref="S3.T2.25.5.5.5.1.m1.1.1.2.2.3.cmml">cc</mi><mtext id="S3.T2.25.5.5.5.1.m1.1.1.2.3" xref="S3.T2.25.5.5.5.1.m1.1.1.2.3a.cmml">expt</mtext></msubsup><mo id="S3.T2.25.5.5.5.1.m1.1.1.1" xref="S3.T2.25.5.5.5.1.m1.1.1.1.cmml">−</mo><msub id="S3.T2.25.5.5.5.1.m1.1.1.3" xref="S3.T2.25.5.5.5.1.m1.1.1.3.cmml"><mi id="S3.T2.25.5.5.5.1.m1.1.1.3.2" xref="S3.T2.25.5.5.5.1.m1.1.1.3.2.cmml">σ</mi><mi id="S3.T2.25.5.5.5.1.m1.1.1.3.3" xref="S3.T2.25.5.5.5.1.m1.1.1.3.3.cmml">cc</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.T2.25.5.5.5.1.m1.1b"><apply id="S3.T2.25.5.5.5.1.m1.1.1.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1"><minus id="S3.T2.25.5.5.5.1.m1.1.1.1.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.1"></minus><apply id="S3.T2.25.5.5.5.1.m1.1.1.2.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.T2.25.5.5.5.1.m1.1.1.2.1.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2">superscript</csymbol><apply id="S3.T2.25.5.5.5.1.m1.1.1.2.2.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.T2.25.5.5.5.1.m1.1.1.2.2.1.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S3.T2.25.5.5.5.1.m1.1.1.2.2.2.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2.2.2">𝜎</ci><ci id="S3.T2.25.5.5.5.1.m1.1.1.2.2.3.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2.2.3">cc</ci></apply><ci id="S3.T2.25.5.5.5.1.m1.1.1.2.3a.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.2.3"><mtext id="S3.T2.25.5.5.5.1.m1.1.1.2.3.cmml" mathsize="70%" xref="S3.T2.25.5.5.5.1.m1.1.1.2.3">expt</mtext></ci></apply><apply id="S3.T2.25.5.5.5.1.m1.1.1.3.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.T2.25.5.5.5.1.m1.1.1.3.1.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.3">subscript</csymbol><ci id="S3.T2.25.5.5.5.1.m1.1.1.3.2.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.3.2">𝜎</ci><ci id="S3.T2.25.5.5.5.1.m1.1.1.3.3.cmml" xref="S3.T2.25.5.5.5.1.m1.1.1.3.3">cc</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.25.5.5.5.1.m1.1c">\sigma_{\mathrm{cc}}^{\text{expt}}-\sigma_{\mathrm{cc}}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.25.5.5.5.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt end_POSTSUPERSCRIPT - italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math> (mb)</td> </tr> <tr class="ltx_tr" id="S3.T2.33.13.13"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.26.6.6.1" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="\sigma_{-1n}" class="ltx_Math" display="inline" id="S3.T2.26.6.6.1.m1.1"><semantics id="S3.T2.26.6.6.1.m1.1a"><msub id="S3.T2.26.6.6.1.m1.1.1" xref="S3.T2.26.6.6.1.m1.1.1.cmml"><mi id="S3.T2.26.6.6.1.m1.1.1.2" xref="S3.T2.26.6.6.1.m1.1.1.2.cmml">σ</mi><mrow id="S3.T2.26.6.6.1.m1.1.1.3" xref="S3.T2.26.6.6.1.m1.1.1.3.cmml"><mo id="S3.T2.26.6.6.1.m1.1.1.3a" xref="S3.T2.26.6.6.1.m1.1.1.3.cmml">−</mo><mrow id="S3.T2.26.6.6.1.m1.1.1.3.2" xref="S3.T2.26.6.6.1.m1.1.1.3.2.cmml"><mn id="S3.T2.26.6.6.1.m1.1.1.3.2.2" xref="S3.T2.26.6.6.1.m1.1.1.3.2.2.cmml">1</mn><mo id="S3.T2.26.6.6.1.m1.1.1.3.2.1" xref="S3.T2.26.6.6.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.T2.26.6.6.1.m1.1.1.3.2.3" xref="S3.T2.26.6.6.1.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T2.26.6.6.1.m1.1b"><apply id="S3.T2.26.6.6.1.m1.1.1.cmml" xref="S3.T2.26.6.6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.26.6.6.1.m1.1.1.1.cmml" xref="S3.T2.26.6.6.1.m1.1.1">subscript</csymbol><ci id="S3.T2.26.6.6.1.m1.1.1.2.cmml" xref="S3.T2.26.6.6.1.m1.1.1.2">𝜎</ci><apply id="S3.T2.26.6.6.1.m1.1.1.3.cmml" xref="S3.T2.26.6.6.1.m1.1.1.3"><minus id="S3.T2.26.6.6.1.m1.1.1.3.1.cmml" xref="S3.T2.26.6.6.1.m1.1.1.3"></minus><apply id="S3.T2.26.6.6.1.m1.1.1.3.2.cmml" xref="S3.T2.26.6.6.1.m1.1.1.3.2"><times id="S3.T2.26.6.6.1.m1.1.1.3.2.1.cmml" xref="S3.T2.26.6.6.1.m1.1.1.3.2.1"></times><cn id="S3.T2.26.6.6.1.m1.1.1.3.2.2.cmml" type="integer" xref="S3.T2.26.6.6.1.m1.1.1.3.2.2">1</cn><ci id="S3.T2.26.6.6.1.m1.1.1.3.2.3.cmml" xref="S3.T2.26.6.6.1.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.26.6.6.1.m1.1c">\sigma_{-1n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.26.6.6.1.m1.1d">italic_σ start_POSTSUBSCRIPT - 1 italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.27.7.7.2" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="\sigma_{-2n}" class="ltx_Math" display="inline" id="S3.T2.27.7.7.2.m1.1"><semantics id="S3.T2.27.7.7.2.m1.1a"><msub id="S3.T2.27.7.7.2.m1.1.1" xref="S3.T2.27.7.7.2.m1.1.1.cmml"><mi id="S3.T2.27.7.7.2.m1.1.1.2" xref="S3.T2.27.7.7.2.m1.1.1.2.cmml">σ</mi><mrow id="S3.T2.27.7.7.2.m1.1.1.3" xref="S3.T2.27.7.7.2.m1.1.1.3.cmml"><mo id="S3.T2.27.7.7.2.m1.1.1.3a" xref="S3.T2.27.7.7.2.m1.1.1.3.cmml">−</mo><mrow id="S3.T2.27.7.7.2.m1.1.1.3.2" xref="S3.T2.27.7.7.2.m1.1.1.3.2.cmml"><mn id="S3.T2.27.7.7.2.m1.1.1.3.2.2" xref="S3.T2.27.7.7.2.m1.1.1.3.2.2.cmml">2</mn><mo id="S3.T2.27.7.7.2.m1.1.1.3.2.1" xref="S3.T2.27.7.7.2.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.T2.27.7.7.2.m1.1.1.3.2.3" xref="S3.T2.27.7.7.2.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T2.27.7.7.2.m1.1b"><apply id="S3.T2.27.7.7.2.m1.1.1.cmml" xref="S3.T2.27.7.7.2.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.27.7.7.2.m1.1.1.1.cmml" xref="S3.T2.27.7.7.2.m1.1.1">subscript</csymbol><ci id="S3.T2.27.7.7.2.m1.1.1.2.cmml" xref="S3.T2.27.7.7.2.m1.1.1.2">𝜎</ci><apply id="S3.T2.27.7.7.2.m1.1.1.3.cmml" xref="S3.T2.27.7.7.2.m1.1.1.3"><minus id="S3.T2.27.7.7.2.m1.1.1.3.1.cmml" xref="S3.T2.27.7.7.2.m1.1.1.3"></minus><apply id="S3.T2.27.7.7.2.m1.1.1.3.2.cmml" xref="S3.T2.27.7.7.2.m1.1.1.3.2"><times id="S3.T2.27.7.7.2.m1.1.1.3.2.1.cmml" xref="S3.T2.27.7.7.2.m1.1.1.3.2.1"></times><cn id="S3.T2.27.7.7.2.m1.1.1.3.2.2.cmml" type="integer" xref="S3.T2.27.7.7.2.m1.1.1.3.2.2">2</cn><ci id="S3.T2.27.7.7.2.m1.1.1.3.2.3.cmml" xref="S3.T2.27.7.7.2.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.27.7.7.2.m1.1c">\sigma_{-2n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.27.7.7.2.m1.1d">italic_σ start_POSTSUBSCRIPT - 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.28.8.8.3" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="\sigma_{-3n}" class="ltx_Math" display="inline" id="S3.T2.28.8.8.3.m1.1"><semantics id="S3.T2.28.8.8.3.m1.1a"><msub id="S3.T2.28.8.8.3.m1.1.1" xref="S3.T2.28.8.8.3.m1.1.1.cmml"><mi id="S3.T2.28.8.8.3.m1.1.1.2" xref="S3.T2.28.8.8.3.m1.1.1.2.cmml">σ</mi><mrow id="S3.T2.28.8.8.3.m1.1.1.3" xref="S3.T2.28.8.8.3.m1.1.1.3.cmml"><mo id="S3.T2.28.8.8.3.m1.1.1.3a" xref="S3.T2.28.8.8.3.m1.1.1.3.cmml">−</mo><mrow id="S3.T2.28.8.8.3.m1.1.1.3.2" xref="S3.T2.28.8.8.3.m1.1.1.3.2.cmml"><mn id="S3.T2.28.8.8.3.m1.1.1.3.2.2" xref="S3.T2.28.8.8.3.m1.1.1.3.2.2.cmml">3</mn><mo id="S3.T2.28.8.8.3.m1.1.1.3.2.1" xref="S3.T2.28.8.8.3.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.T2.28.8.8.3.m1.1.1.3.2.3" xref="S3.T2.28.8.8.3.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T2.28.8.8.3.m1.1b"><apply id="S3.T2.28.8.8.3.m1.1.1.cmml" xref="S3.T2.28.8.8.3.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.28.8.8.3.m1.1.1.1.cmml" xref="S3.T2.28.8.8.3.m1.1.1">subscript</csymbol><ci id="S3.T2.28.8.8.3.m1.1.1.2.cmml" xref="S3.T2.28.8.8.3.m1.1.1.2">𝜎</ci><apply id="S3.T2.28.8.8.3.m1.1.1.3.cmml" xref="S3.T2.28.8.8.3.m1.1.1.3"><minus id="S3.T2.28.8.8.3.m1.1.1.3.1.cmml" xref="S3.T2.28.8.8.3.m1.1.1.3"></minus><apply id="S3.T2.28.8.8.3.m1.1.1.3.2.cmml" xref="S3.T2.28.8.8.3.m1.1.1.3.2"><times id="S3.T2.28.8.8.3.m1.1.1.3.2.1.cmml" xref="S3.T2.28.8.8.3.m1.1.1.3.2.1"></times><cn id="S3.T2.28.8.8.3.m1.1.1.3.2.2.cmml" type="integer" xref="S3.T2.28.8.8.3.m1.1.1.3.2.2">3</cn><ci id="S3.T2.28.8.8.3.m1.1.1.3.2.3.cmml" xref="S3.T2.28.8.8.3.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.28.8.8.3.m1.1c">\sigma_{-3n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.28.8.8.3.m1.1d">italic_σ start_POSTSUBSCRIPT - 3 italic_n end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.29.9.9.4" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="\sigma_{-xn,~{}(x>3)}" class="ltx_Math" display="inline" id="S3.T2.29.9.9.4.m1.2"><semantics id="S3.T2.29.9.9.4.m1.2a"><msub id="S3.T2.29.9.9.4.m1.2.3" xref="S3.T2.29.9.9.4.m1.2.3.cmml"><mi id="S3.T2.29.9.9.4.m1.2.3.2" xref="S3.T2.29.9.9.4.m1.2.3.2.cmml">σ</mi><mrow id="S3.T2.29.9.9.4.m1.2.2.2.2" xref="S3.T2.29.9.9.4.m1.2.2.2.3.cmml"><mrow id="S3.T2.29.9.9.4.m1.1.1.1.1.1" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.cmml"><mo id="S3.T2.29.9.9.4.m1.1.1.1.1.1a" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.cmml">−</mo><mrow id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.cmml"><mi id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.2" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.2.cmml">x</mi><mo id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.1" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.3" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.3.cmml">n</mi></mrow></mrow><mo id="S3.T2.29.9.9.4.m1.2.2.2.2.3" rspace="0.397em" xref="S3.T2.29.9.9.4.m1.2.2.2.3.cmml">,</mo><mrow id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.cmml"><mo id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.2" stretchy="false" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.cmml">(</mo><mrow id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.cmml"><mi id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.2" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.2.cmml">x</mi><mo id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.1" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.1.cmml">></mo><mn id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.3" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.3.cmml">3</mn></mrow><mo id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.3" stretchy="false" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T2.29.9.9.4.m1.2b"><apply id="S3.T2.29.9.9.4.m1.2.3.cmml" xref="S3.T2.29.9.9.4.m1.2.3"><csymbol cd="ambiguous" id="S3.T2.29.9.9.4.m1.2.3.1.cmml" xref="S3.T2.29.9.9.4.m1.2.3">subscript</csymbol><ci id="S3.T2.29.9.9.4.m1.2.3.2.cmml" xref="S3.T2.29.9.9.4.m1.2.3.2">𝜎</ci><list id="S3.T2.29.9.9.4.m1.2.2.2.3.cmml" xref="S3.T2.29.9.9.4.m1.2.2.2.2"><apply id="S3.T2.29.9.9.4.m1.1.1.1.1.1.cmml" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1"><minus id="S3.T2.29.9.9.4.m1.1.1.1.1.1.1.cmml" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1"></minus><apply id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.cmml" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2"><times id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.1.cmml" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.1"></times><ci id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.2.cmml" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.2">𝑥</ci><ci id="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.3.cmml" xref="S3.T2.29.9.9.4.m1.1.1.1.1.1.2.3">𝑛</ci></apply></apply><apply id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.cmml" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1"><gt id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.1.cmml" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.1"></gt><ci id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.2.cmml" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.2">𝑥</ci><cn id="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.3.cmml" type="integer" xref="S3.T2.29.9.9.4.m1.2.2.2.2.2.1.1.3">3</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.29.9.9.4.m1.2c">\sigma_{-xn,~{}(x>3)}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.29.9.9.4.m1.2d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n , ( italic_x > 3 ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.30.10.10.5" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="{-1n}" class="ltx_Math" display="inline" id="S3.T2.30.10.10.5.m1.1"><semantics id="S3.T2.30.10.10.5.m1.1a"><mrow id="S3.T2.30.10.10.5.m1.1.1" xref="S3.T2.30.10.10.5.m1.1.1.cmml"><mo id="S3.T2.30.10.10.5.m1.1.1a" xref="S3.T2.30.10.10.5.m1.1.1.cmml">−</mo><mrow id="S3.T2.30.10.10.5.m1.1.1.2" xref="S3.T2.30.10.10.5.m1.1.1.2.cmml"><mn id="S3.T2.30.10.10.5.m1.1.1.2.2" xref="S3.T2.30.10.10.5.m1.1.1.2.2.cmml">1</mn><mo id="S3.T2.30.10.10.5.m1.1.1.2.1" xref="S3.T2.30.10.10.5.m1.1.1.2.1.cmml">⁢</mo><mi id="S3.T2.30.10.10.5.m1.1.1.2.3" xref="S3.T2.30.10.10.5.m1.1.1.2.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.T2.30.10.10.5.m1.1b"><apply id="S3.T2.30.10.10.5.m1.1.1.cmml" xref="S3.T2.30.10.10.5.m1.1.1"><minus id="S3.T2.30.10.10.5.m1.1.1.1.cmml" xref="S3.T2.30.10.10.5.m1.1.1"></minus><apply id="S3.T2.30.10.10.5.m1.1.1.2.cmml" xref="S3.T2.30.10.10.5.m1.1.1.2"><times id="S3.T2.30.10.10.5.m1.1.1.2.1.cmml" xref="S3.T2.30.10.10.5.m1.1.1.2.1"></times><cn id="S3.T2.30.10.10.5.m1.1.1.2.2.cmml" type="integer" xref="S3.T2.30.10.10.5.m1.1.1.2.2">1</cn><ci id="S3.T2.30.10.10.5.m1.1.1.2.3.cmml" xref="S3.T2.30.10.10.5.m1.1.1.2.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.30.10.10.5.m1.1c">{-1n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.30.10.10.5.m1.1d">- 1 italic_n</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.31.11.11.6" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="{-2n}" class="ltx_Math" display="inline" id="S3.T2.31.11.11.6.m1.1"><semantics id="S3.T2.31.11.11.6.m1.1a"><mrow id="S3.T2.31.11.11.6.m1.1.1" xref="S3.T2.31.11.11.6.m1.1.1.cmml"><mo id="S3.T2.31.11.11.6.m1.1.1a" xref="S3.T2.31.11.11.6.m1.1.1.cmml">−</mo><mrow id="S3.T2.31.11.11.6.m1.1.1.2" xref="S3.T2.31.11.11.6.m1.1.1.2.cmml"><mn id="S3.T2.31.11.11.6.m1.1.1.2.2" xref="S3.T2.31.11.11.6.m1.1.1.2.2.cmml">2</mn><mo id="S3.T2.31.11.11.6.m1.1.1.2.1" xref="S3.T2.31.11.11.6.m1.1.1.2.1.cmml">⁢</mo><mi id="S3.T2.31.11.11.6.m1.1.1.2.3" xref="S3.T2.31.11.11.6.m1.1.1.2.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.T2.31.11.11.6.m1.1b"><apply id="S3.T2.31.11.11.6.m1.1.1.cmml" xref="S3.T2.31.11.11.6.m1.1.1"><minus id="S3.T2.31.11.11.6.m1.1.1.1.cmml" xref="S3.T2.31.11.11.6.m1.1.1"></minus><apply id="S3.T2.31.11.11.6.m1.1.1.2.cmml" xref="S3.T2.31.11.11.6.m1.1.1.2"><times id="S3.T2.31.11.11.6.m1.1.1.2.1.cmml" xref="S3.T2.31.11.11.6.m1.1.1.2.1"></times><cn id="S3.T2.31.11.11.6.m1.1.1.2.2.cmml" type="integer" xref="S3.T2.31.11.11.6.m1.1.1.2.2">2</cn><ci id="S3.T2.31.11.11.6.m1.1.1.2.3.cmml" xref="S3.T2.31.11.11.6.m1.1.1.2.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.31.11.11.6.m1.1c">{-2n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.31.11.11.6.m1.1d">- 2 italic_n</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.32.12.12.7" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="{-3n}" class="ltx_Math" display="inline" id="S3.T2.32.12.12.7.m1.1"><semantics id="S3.T2.32.12.12.7.m1.1a"><mrow id="S3.T2.32.12.12.7.m1.1.1" xref="S3.T2.32.12.12.7.m1.1.1.cmml"><mo id="S3.T2.32.12.12.7.m1.1.1a" xref="S3.T2.32.12.12.7.m1.1.1.cmml">−</mo><mrow id="S3.T2.32.12.12.7.m1.1.1.2" xref="S3.T2.32.12.12.7.m1.1.1.2.cmml"><mn id="S3.T2.32.12.12.7.m1.1.1.2.2" xref="S3.T2.32.12.12.7.m1.1.1.2.2.cmml">3</mn><mo id="S3.T2.32.12.12.7.m1.1.1.2.1" xref="S3.T2.32.12.12.7.m1.1.1.2.1.cmml">⁢</mo><mi id="S3.T2.32.12.12.7.m1.1.1.2.3" xref="S3.T2.32.12.12.7.m1.1.1.2.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.T2.32.12.12.7.m1.1b"><apply id="S3.T2.32.12.12.7.m1.1.1.cmml" xref="S3.T2.32.12.12.7.m1.1.1"><minus id="S3.T2.32.12.12.7.m1.1.1.1.cmml" xref="S3.T2.32.12.12.7.m1.1.1"></minus><apply id="S3.T2.32.12.12.7.m1.1.1.2.cmml" xref="S3.T2.32.12.12.7.m1.1.1.2"><times id="S3.T2.32.12.12.7.m1.1.1.2.1.cmml" xref="S3.T2.32.12.12.7.m1.1.1.2.1"></times><cn id="S3.T2.32.12.12.7.m1.1.1.2.2.cmml" type="integer" xref="S3.T2.32.12.12.7.m1.1.1.2.2">3</cn><ci id="S3.T2.32.12.12.7.m1.1.1.2.3.cmml" xref="S3.T2.32.12.12.7.m1.1.1.2.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.32.12.12.7.m1.1c">{-3n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.32.12.12.7.m1.1d">- 3 italic_n</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.33.13.13.8" style="padding-top:-1pt;padding-bottom:-1pt;"><math alttext="{>-3n}" class="ltx_Math" display="inline" id="S3.T2.33.13.13.8.m1.1"><semantics id="S3.T2.33.13.13.8.m1.1a"><mrow id="S3.T2.33.13.13.8.m1.1.1" xref="S3.T2.33.13.13.8.m1.1.1.cmml"><mi id="S3.T2.33.13.13.8.m1.1.1.2" xref="S3.T2.33.13.13.8.m1.1.1.2.cmml"></mi><mo id="S3.T2.33.13.13.8.m1.1.1.1" xref="S3.T2.33.13.13.8.m1.1.1.1.cmml">></mo><mrow id="S3.T2.33.13.13.8.m1.1.1.3" xref="S3.T2.33.13.13.8.m1.1.1.3.cmml"><mo id="S3.T2.33.13.13.8.m1.1.1.3a" xref="S3.T2.33.13.13.8.m1.1.1.3.cmml">−</mo><mrow id="S3.T2.33.13.13.8.m1.1.1.3.2" xref="S3.T2.33.13.13.8.m1.1.1.3.2.cmml"><mn id="S3.T2.33.13.13.8.m1.1.1.3.2.2" xref="S3.T2.33.13.13.8.m1.1.1.3.2.2.cmml">3</mn><mo id="S3.T2.33.13.13.8.m1.1.1.3.2.1" xref="S3.T2.33.13.13.8.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.T2.33.13.13.8.m1.1.1.3.2.3" xref="S3.T2.33.13.13.8.m1.1.1.3.2.3.cmml">n</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.T2.33.13.13.8.m1.1b"><apply id="S3.T2.33.13.13.8.m1.1.1.cmml" xref="S3.T2.33.13.13.8.m1.1.1"><gt id="S3.T2.33.13.13.8.m1.1.1.1.cmml" xref="S3.T2.33.13.13.8.m1.1.1.1"></gt><csymbol cd="latexml" id="S3.T2.33.13.13.8.m1.1.1.2.cmml" xref="S3.T2.33.13.13.8.m1.1.1.2">absent</csymbol><apply id="S3.T2.33.13.13.8.m1.1.1.3.cmml" xref="S3.T2.33.13.13.8.m1.1.1.3"><minus id="S3.T2.33.13.13.8.m1.1.1.3.1.cmml" xref="S3.T2.33.13.13.8.m1.1.1.3"></minus><apply id="S3.T2.33.13.13.8.m1.1.1.3.2.cmml" xref="S3.T2.33.13.13.8.m1.1.1.3.2"><times id="S3.T2.33.13.13.8.m1.1.1.3.2.1.cmml" xref="S3.T2.33.13.13.8.m1.1.1.3.2.1"></times><cn id="S3.T2.33.13.13.8.m1.1.1.3.2.2.cmml" type="integer" xref="S3.T2.33.13.13.8.m1.1.1.3.2.2">3</cn><ci id="S3.T2.33.13.13.8.m1.1.1.3.2.3.cmml" xref="S3.T2.33.13.13.8.m1.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.33.13.13.8.m1.1c">{>-3n}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.33.13.13.8.m1.1d">> - 3 italic_n</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T2.34.14.14"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.34.14.14.2" style="padding-top:-1pt;padding-bottom:-1pt;">25</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.34.14.14.1" style="padding-top:-1pt;padding-bottom:-1pt;"> 12C</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.34.14.14.3" style="padding-top:-1pt;padding-bottom:-1pt;">C</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.4" style="padding-top:-1pt;padding-bottom:-1pt;">114</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.5" style="padding-top:-1pt;padding-bottom:-1pt;">28</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.6" style="padding-top:-1pt;padding-bottom:-1pt;">8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.7" style="padding-top:-1pt;padding-bottom:-1pt;">1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.8" style="padding-top:-1pt;padding-bottom:-1pt;">50</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.9" style="padding-top:-1pt;padding-bottom:-1pt;">26</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.10" style="padding-top:-1pt;padding-bottom:-1pt;">8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.11" style="padding-top:-1pt;padding-bottom:-1pt;">1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.12" style="padding-top:-1pt;padding-bottom:-1pt;">85</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.34.14.14.13" style="padding-top:-1pt;padding-bottom:-1pt;">81</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.18.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.18.1.1" style="padding-top:-1pt;padding-bottom:-1pt;">15</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.18.1.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.18.1.3" style="padding-top:-1pt;padding-bottom:-1pt;">Al</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.4" style="padding-top:-1pt;padding-bottom:-1pt;">136</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.5" style="padding-top:-1pt;padding-bottom:-1pt;">30</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.6" style="padding-top:-1pt;padding-bottom:-1pt;">9</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.7" style="padding-top:-1pt;padding-bottom:-1pt;">1</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.8" style="padding-top:-1pt;padding-bottom:-1pt;">26</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.9" style="padding-top:-1pt;padding-bottom:-1pt;">25</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.10" style="padding-top:-1pt;padding-bottom:-1pt;">8</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.11" style="padding-top:-1pt;padding-bottom:-1pt;">1</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.12" style="padding-top:-1pt;padding-bottom:-1pt;">60</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.18.1.13" style="padding-top:-1pt;padding-bottom:-1pt;">51</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.19.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.19.2.1" style="padding-top:-1pt;padding-bottom:-1pt;">10</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.19.2.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.19.2.3" style="padding-top:-1pt;padding-bottom:-1pt;">Cu</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.4" style="padding-top:-1pt;padding-bottom:-1pt;">177</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.5" style="padding-top:-1pt;padding-bottom:-1pt;">40</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.6" style="padding-top:-1pt;padding-bottom:-1pt;">11</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.7" style="padding-top:-1pt;padding-bottom:-1pt;">2</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.8" style="padding-top:-1pt;padding-bottom:-1pt;">16</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.9" style="padding-top:-1pt;padding-bottom:-1pt;">31</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.10" style="padding-top:-1pt;padding-bottom:-1pt;">10</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.11" style="padding-top:-1pt;padding-bottom:-1pt;">2</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.12" style="padding-top:-1pt;padding-bottom:-1pt;">59</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.19.2.13" style="padding-top:-1pt;padding-bottom:-1pt;">57</td> </tr> <tr class="ltx_tr" id="S3.T2.35.15.15"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.35.15.15.2" style="padding-top:-1pt;padding-bottom:-1pt;">60</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.35.15.15.1" style="padding-top:-1pt;padding-bottom:-1pt;"> 14N</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.35.15.15.3" style="padding-top:-1pt;padding-bottom:-1pt;">C</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.4" style="padding-top:-1pt;padding-bottom:-1pt;">115</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.5" style="padding-top:-1pt;padding-bottom:-1pt;">29</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.6" style="padding-top:-1pt;padding-bottom:-1pt;">8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.7" style="padding-top:-1pt;padding-bottom:-1pt;">3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.8" style="padding-top:-1pt;padding-bottom:-1pt;">100</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.9" style="padding-top:-1pt;padding-bottom:-1pt;">29</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.10" style="padding-top:-1pt;padding-bottom:-1pt;">8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.11" style="padding-top:-1pt;padding-bottom:-1pt;">3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.12" style="padding-top:-1pt;padding-bottom:-1pt;">140</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.35.15.15.13" style="padding-top:-1pt;padding-bottom:-1pt;">156</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.20.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.20.3.1" style="padding-top:-1pt;padding-bottom:-1pt;">25</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.20.3.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.20.3.3" style="padding-top:-1pt;padding-bottom:-1pt;">Al</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.4" style="padding-top:-1pt;padding-bottom:-1pt;">137</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.5" style="padding-top:-1pt;padding-bottom:-1pt;">34</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.6" style="padding-top:-1pt;padding-bottom:-1pt;">10</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.7" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.8" style="padding-top:-1pt;padding-bottom:-1pt;">88</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.9" style="padding-top:-1pt;padding-bottom:-1pt;">33</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.10" style="padding-top:-1pt;padding-bottom:-1pt;">9</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.11" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.12" style="padding-top:-1pt;padding-bottom:-1pt;">134</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.20.3.13" style="padding-top:-1pt;padding-bottom:-1pt;">120</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.21.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.21.4.1" style="padding-top:-1pt;padding-bottom:-1pt;">15</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.21.4.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.21.4.3" style="padding-top:-1pt;padding-bottom:-1pt;">Cu</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.4" style="padding-top:-1pt;padding-bottom:-1pt;">165</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.5" style="padding-top:-1pt;padding-bottom:-1pt;">42</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.6" style="padding-top:-1pt;padding-bottom:-1pt;">12</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.7" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.8" style="padding-top:-1pt;padding-bottom:-1pt;">92</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.9" style="padding-top:-1pt;padding-bottom:-1pt;">41</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.10" style="padding-top:-1pt;padding-bottom:-1pt;">11</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.11" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.12" style="padding-top:-1pt;padding-bottom:-1pt;">148</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.21.4.13" style="padding-top:-1pt;padding-bottom:-1pt;">146</td> </tr> <tr class="ltx_tr" id="S3.T2.36.16.16"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.36.16.16.2" style="padding-top:-1pt;padding-bottom:-1pt;">35</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.36.16.16.1" style="padding-top:-1pt;padding-bottom:-1pt;"> 16O</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.36.16.16.3" style="padding-top:-1pt;padding-bottom:-1pt;">C</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.4" style="padding-top:-1pt;padding-bottom:-1pt;">115</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.5" style="padding-top:-1pt;padding-bottom:-1pt;">29</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.6" style="padding-top:-1pt;padding-bottom:-1pt;">9</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.7" style="padding-top:-1pt;padding-bottom:-1pt;">3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.8" style="padding-top:-1pt;padding-bottom:-1pt;">69</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.9" style="padding-top:-1pt;padding-bottom:-1pt;">27</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.10" style="padding-top:-1pt;padding-bottom:-1pt;">9</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.11" style="padding-top:-1pt;padding-bottom:-1pt;">3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.12" style="padding-top:-1pt;padding-bottom:-1pt;">108</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.36.16.16.13" style="padding-top:-1pt;padding-bottom:-1pt;">106</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.22.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.22.5.1" style="padding-top:-1pt;padding-bottom:-1pt;">20</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.22.5.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.22.5.3" style="padding-top:-1pt;padding-bottom:-1pt;">Al</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.4" style="padding-top:-1pt;padding-bottom:-1pt;">135</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.5" style="padding-top:-1pt;padding-bottom:-1pt;">34</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.6" style="padding-top:-1pt;padding-bottom:-1pt;">10</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.7" style="padding-top:-1pt;padding-bottom:-1pt;">3</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.8" style="padding-top:-1pt;padding-bottom:-1pt;">49</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.9" style="padding-top:-1pt;padding-bottom:-1pt;">32</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.10" style="padding-top:-1pt;padding-bottom:-1pt;">9</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.11" style="padding-top:-1pt;padding-bottom:-1pt;">3</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.12" style="padding-top:-1pt;padding-bottom:-1pt;">93</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.22.5.13" style="padding-top:-1pt;padding-bottom:-1pt;">89</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.23.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.23.6.1" style="padding-top:-1pt;padding-bottom:-1pt;">15</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.23.6.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.23.6.3" style="padding-top:-1pt;padding-bottom:-1pt;">Cu</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.4" style="padding-top:-1pt;padding-bottom:-1pt;">163</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.5" style="padding-top:-1pt;padding-bottom:-1pt;">41</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.6" style="padding-top:-1pt;padding-bottom:-1pt;">12</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.7" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.8" style="padding-top:-1pt;padding-bottom:-1pt;">35</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.9" style="padding-top:-1pt;padding-bottom:-1pt;">31</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.10" style="padding-top:-1pt;padding-bottom:-1pt;">12</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.11" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.12" style="padding-top:-1pt;padding-bottom:-1pt;">82</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.23.6.13" style="padding-top:-1pt;padding-bottom:-1pt;">76</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.17"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.37.17.17.2" style="padding-top:-1pt;padding-bottom:-1pt;">35</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.37.17.17.1" style="padding-top:-1pt;padding-bottom:-1pt;"> 20Ne</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S3.T2.37.17.17.3" style="padding-top:-1pt;padding-bottom:-1pt;">C</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.4" style="padding-top:-1pt;padding-bottom:-1pt;">137</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.5" style="padding-top:-1pt;padding-bottom:-1pt;">35</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.6" style="padding-top:-1pt;padding-bottom:-1pt;">11</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.7" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.8" style="padding-top:-1pt;padding-bottom:-1pt;">106</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.9" style="padding-top:-1pt;padding-bottom:-1pt;">34</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.10" style="padding-top:-1pt;padding-bottom:-1pt;">11</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.11" style="padding-top:-1pt;padding-bottom:-1pt;">4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.12" style="padding-top:-1pt;padding-bottom:-1pt;">155</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.37.17.17.13" style="padding-top:-1pt;padding-bottom:-1pt;">154</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.24.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.24.7.1" style="padding-top:-1pt;padding-bottom:-1pt;">20</th> <th class="ltx_td ltx_th ltx_th_row" id="S3.T2.37.17.24.7.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S3.T2.37.17.24.7.3" style="padding-top:-1pt;padding-bottom:-1pt;">Al</th> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.4" style="padding-top:-1pt;padding-bottom:-1pt;">159</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.5" style="padding-top:-1pt;padding-bottom:-1pt;">40</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.6" style="padding-top:-1pt;padding-bottom:-1pt;">12</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.7" style="padding-top:-1pt;padding-bottom:-1pt;">6</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.8" style="padding-top:-1pt;padding-bottom:-1pt;">92</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.9" style="padding-top:-1pt;padding-bottom:-1pt;">36</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.10" style="padding-top:-1pt;padding-bottom:-1pt;">12</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.11" style="padding-top:-1pt;padding-bottom:-1pt;">6</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.12" style="padding-top:-1pt;padding-bottom:-1pt;">146</td> <td class="ltx_td ltx_align_center" id="S3.T2.37.17.24.7.13" style="padding-top:-1pt;padding-bottom:-1pt;">137</td> </tr> <tr class="ltx_tr" id="S3.T2.37.17.25.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b" id="S3.T2.37.17.25.8.1" style="padding-top:-1pt;padding-bottom:-1pt;">10</th> <th class="ltx_td ltx_th ltx_th_row ltx_border_b" id="S3.T2.37.17.25.8.2" style="padding-top:-1pt;padding-bottom:-1pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b" id="S3.T2.37.17.25.8.3" style="padding-top:-1pt;padding-bottom:-1pt;">Cu</th> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.4" style="padding-top:-1pt;padding-bottom:-1pt;">186</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.5" style="padding-top:-1pt;padding-bottom:-1pt;">47</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.6" style="padding-top:-1pt;padding-bottom:-1pt;">14</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.7" style="padding-top:-1pt;padding-bottom:-1pt;">7</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.8" style="padding-top:-1pt;padding-bottom:-1pt;">48</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.9" style="padding-top:-1pt;padding-bottom:-1pt;">36</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.10" style="padding-top:-1pt;padding-bottom:-1pt;">14</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.11" style="padding-top:-1pt;padding-bottom:-1pt;">7</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.12" style="padding-top:-1pt;padding-bottom:-1pt;">105</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S3.T2.37.17.25.8.13" style="padding-top:-1pt;padding-bottom:-1pt;">94</td> </tr> </tbody> </table> </div> </figure> <div class="ltx_para" id="S3.SS3.SSS2.p3"> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.T2" title="Table 2 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">2</a> presents the contributions of proton evaporation (<math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.1.m1.1"><semantics id="S3.SS3.SSS2.p3.1.m1.1a"><msubsup id="S3.SS3.SSS2.p3.1.m1.1.1" xref="S3.SS3.SSS2.p3.1.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p3.1.m1.1.1.2.2" xref="S3.SS3.SSS2.p3.1.m1.1.1.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p3.1.m1.1.1.2.3" xref="S3.SS3.SSS2.p3.1.m1.1.1.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p3.1.m1.1.1.3" xref="S3.SS3.SSS2.p3.1.m1.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.1.m1.1b"><apply id="S3.SS3.SSS2.p3.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.1.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1">superscript</csymbol><apply id="S3.SS3.SSS2.p3.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.1.m1.1.1.2.1.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p3.1.m1.1.1.2.2.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p3.1.m1.1.1.2.3.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p3.1.m1.1.1.3a.cmml" xref="S3.SS3.SSS2.p3.1.m1.1.1.3"><mtext id="S3.SS3.SSS2.p3.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p3.1.m1.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.1.m1.1c">\sigma_{\mathrm{cc}}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math>) [Eq.( <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E11" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">11</a>)] to the charge-changing cross section for various projectile nuclei (12C, 14N, 16O, 20Ne) on different elemental targets (C, Al, Cu) at a beam energy of 290 MeV/nucleon. The table is structured to show the neutron-removal cross sections (<math alttext="\sigma_{-xn}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.6.m6.1"><semantics id="S3.SS3.SSS2.p3.6.m6.1a"><msub id="S3.SS3.SSS2.p3.6.m6.1.1" xref="S3.SS3.SSS2.p3.6.m6.1.1.cmml"><mi id="S3.SS3.SSS2.p3.6.m6.1.1.2" xref="S3.SS3.SSS2.p3.6.m6.1.1.2.cmml">σ</mi><mrow id="S3.SS3.SSS2.p3.6.m6.1.1.3" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.cmml"><mo id="S3.SS3.SSS2.p3.6.m6.1.1.3a" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.cmml">−</mo><mrow id="S3.SS3.SSS2.p3.6.m6.1.1.3.2" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.cmml"><mi id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.2" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.2.cmml">x</mi><mo id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.1" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.3" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.3.cmml">n</mi></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.6.m6.1b"><apply id="S3.SS3.SSS2.p3.6.m6.1.1.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.6.m6.1.1.1.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p3.6.m6.1.1.2.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.2">𝜎</ci><apply id="S3.SS3.SSS2.p3.6.m6.1.1.3.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.3"><minus id="S3.SS3.SSS2.p3.6.m6.1.1.3.1.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.3"></minus><apply id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2"><times id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.1"></times><ci id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.2.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.2">𝑥</ci><ci id="S3.SS3.SSS2.p3.6.m6.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p3.6.m6.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.6.m6.1c">\sigma_{-xn}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.6.m6.1d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n end_POSTSUBSCRIPT</annotation></semantics></math>), which are calculated in the abrasion step using Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E4" title="In II.1 Primary fragmentation or abrasion stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">4</a>), for 1<math alttext="n" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.7.m7.1"><semantics id="S3.SS3.SSS2.p3.7.m7.1a"><mi id="S3.SS3.SSS2.p3.7.m7.1.1" xref="S3.SS3.SSS2.p3.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.7.m7.1b"><ci id="S3.SS3.SSS2.p3.7.m7.1.1.cmml" xref="S3.SS3.SSS2.p3.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.7.m7.1d">italic_n</annotation></semantics></math>, 2<math alttext="n" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.8.m8.1"><semantics id="S3.SS3.SSS2.p3.8.m8.1a"><mi id="S3.SS3.SSS2.p3.8.m8.1.1" xref="S3.SS3.SSS2.p3.8.m8.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.8.m8.1b"><ci id="S3.SS3.SSS2.p3.8.m8.1.1.cmml" xref="S3.SS3.SSS2.p3.8.m8.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.8.m8.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.8.m8.1d">italic_n</annotation></semantics></math>, and 3<math alttext="n" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.9.m9.1"><semantics id="S3.SS3.SSS2.p3.9.m9.1a"><mi id="S3.SS3.SSS2.p3.9.m9.1.1" xref="S3.SS3.SSS2.p3.9.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.9.m9.1b"><ci id="S3.SS3.SSS2.p3.9.m9.1.1.cmml" xref="S3.SS3.SSS2.p3.9.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.9.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.9.m9.1d">italic_n</annotation></semantics></math> removal channels, along with their respective contributions to proton evaporation. The individual contributions from each neutron-removal channel to <math alttext="\sigma_{\rm cc}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.10.m10.1"><semantics id="S3.SS3.SSS2.p3.10.m10.1a"><msubsup id="S3.SS3.SSS2.p3.10.m10.1.1" xref="S3.SS3.SSS2.p3.10.m10.1.1.cmml"><mi id="S3.SS3.SSS2.p3.10.m10.1.1.2.2" xref="S3.SS3.SSS2.p3.10.m10.1.1.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p3.10.m10.1.1.2.3" xref="S3.SS3.SSS2.p3.10.m10.1.1.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p3.10.m10.1.1.3" xref="S3.SS3.SSS2.p3.10.m10.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.10.m10.1b"><apply id="S3.SS3.SSS2.p3.10.m10.1.1.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.10.m10.1.1.1.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1">superscript</csymbol><apply id="S3.SS3.SSS2.p3.10.m10.1.1.2.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.10.m10.1.1.2.1.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p3.10.m10.1.1.2.2.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p3.10.m10.1.1.2.3.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p3.10.m10.1.1.3a.cmml" xref="S3.SS3.SSS2.p3.10.m10.1.1.3"><mtext id="S3.SS3.SSS2.p3.10.m10.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p3.10.m10.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.10.m10.1c">\sigma_{\rm cc}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.10.m10.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math> [Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S2.E11" title="In II.2 Secondary fragmentation or ablation stage ‣ II Theoretical Formalism ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">11</a>)] are explicitly provided. The contributions, <math alttext="\sigma_{-xn,~{}(x>3)}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.11.m11.2"><semantics id="S3.SS3.SSS2.p3.11.m11.2a"><msub id="S3.SS3.SSS2.p3.11.m11.2.3" xref="S3.SS3.SSS2.p3.11.m11.2.3.cmml"><mi id="S3.SS3.SSS2.p3.11.m11.2.3.2" xref="S3.SS3.SSS2.p3.11.m11.2.3.2.cmml">σ</mi><mrow id="S3.SS3.SSS2.p3.11.m11.2.2.2.2" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.3.cmml"><mrow id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.cmml"><mo id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1a" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.cmml">−</mo><mrow id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.cmml"><mi id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.2" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.2.cmml">x</mi><mo id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.1" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.3" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.3.cmml">n</mi></mrow></mrow><mo id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.3" rspace="0.397em" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.3.cmml">,</mo><mrow id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.cmml"><mo id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.2" stretchy="false" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.cmml">(</mo><mrow id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.cmml"><mi id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.2" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.2.cmml">x</mi><mo id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.1" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.1.cmml">></mo><mn id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.3" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.3.cmml">3</mn></mrow><mo id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.3" stretchy="false" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.11.m11.2b"><apply id="S3.SS3.SSS2.p3.11.m11.2.3.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.11.m11.2.3.1.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.3">subscript</csymbol><ci id="S3.SS3.SSS2.p3.11.m11.2.3.2.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.3.2">𝜎</ci><list id="S3.SS3.SSS2.p3.11.m11.2.2.2.3.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2"><apply id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1"><minus id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1"></minus><apply id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.cmml" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2"><times id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.1.cmml" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.1"></times><ci id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.2.cmml" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.2">𝑥</ci><ci id="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.3.cmml" xref="S3.SS3.SSS2.p3.11.m11.1.1.1.1.1.2.3">𝑛</ci></apply></apply><apply id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1"><gt id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.1.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.1"></gt><ci id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.2.cmml" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.2">𝑥</ci><cn id="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.SSS2.p3.11.m11.2.2.2.2.2.1.1.3">3</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.11.m11.2c">\sigma_{-xn,~{}(x>3)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.11.m11.2d">italic_σ start_POSTSUBSCRIPT - italic_x italic_n , ( italic_x > 3 ) end_POSTSUBSCRIPT</annotation></semantics></math> from neutron removal channels beyond <math alttext="-3n" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.12.m12.1"><semantics id="S3.SS3.SSS2.p3.12.m12.1a"><mrow id="S3.SS3.SSS2.p3.12.m12.1.1" xref="S3.SS3.SSS2.p3.12.m12.1.1.cmml"><mo id="S3.SS3.SSS2.p3.12.m12.1.1a" xref="S3.SS3.SSS2.p3.12.m12.1.1.cmml">−</mo><mrow id="S3.SS3.SSS2.p3.12.m12.1.1.2" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.cmml"><mn id="S3.SS3.SSS2.p3.12.m12.1.1.2.2" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.2.cmml">3</mn><mo id="S3.SS3.SSS2.p3.12.m12.1.1.2.1" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p3.12.m12.1.1.2.3" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.12.m12.1b"><apply id="S3.SS3.SSS2.p3.12.m12.1.1.cmml" xref="S3.SS3.SSS2.p3.12.m12.1.1"><minus id="S3.SS3.SSS2.p3.12.m12.1.1.1.cmml" xref="S3.SS3.SSS2.p3.12.m12.1.1"></minus><apply id="S3.SS3.SSS2.p3.12.m12.1.1.2.cmml" xref="S3.SS3.SSS2.p3.12.m12.1.1.2"><times id="S3.SS3.SSS2.p3.12.m12.1.1.2.1.cmml" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.1"></times><cn id="S3.SS3.SSS2.p3.12.m12.1.1.2.2.cmml" type="integer" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.2">3</cn><ci id="S3.SS3.SSS2.p3.12.m12.1.1.2.3.cmml" xref="S3.SS3.SSS2.p3.12.m12.1.1.2.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.12.m12.1c">-3n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.12.m12.1d">- 3 italic_n</annotation></semantics></math>, such as the removal of four or more neutrons in the abrasion stage, are calculated and summed to estimate the contributions to the ‘<math alttext=">-3n" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.13.m13.1"><semantics id="S3.SS3.SSS2.p3.13.m13.1a"><mrow id="S3.SS3.SSS2.p3.13.m13.1.1" xref="S3.SS3.SSS2.p3.13.m13.1.1.cmml"><mi id="S3.SS3.SSS2.p3.13.m13.1.1.2" xref="S3.SS3.SSS2.p3.13.m13.1.1.2.cmml"></mi><mo id="S3.SS3.SSS2.p3.13.m13.1.1.1" xref="S3.SS3.SSS2.p3.13.m13.1.1.1.cmml">></mo><mrow id="S3.SS3.SSS2.p3.13.m13.1.1.3" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.cmml"><mo id="S3.SS3.SSS2.p3.13.m13.1.1.3a" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.cmml">−</mo><mrow id="S3.SS3.SSS2.p3.13.m13.1.1.3.2" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.cmml"><mn id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.2" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.2.cmml">3</mn><mo id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.1" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.3" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.3.cmml">n</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.13.m13.1b"><apply id="S3.SS3.SSS2.p3.13.m13.1.1.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1"><gt id="S3.SS3.SSS2.p3.13.m13.1.1.1.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.1"></gt><csymbol cd="latexml" id="S3.SS3.SSS2.p3.13.m13.1.1.2.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.2">absent</csymbol><apply id="S3.SS3.SSS2.p3.13.m13.1.1.3.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.3"><minus id="S3.SS3.SSS2.p3.13.m13.1.1.3.1.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.3"></minus><apply id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2"><times id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.1"></times><cn id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.2.cmml" type="integer" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.2">3</cn><ci id="S3.SS3.SSS2.p3.13.m13.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p3.13.m13.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.13.m13.1c">>-3n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.13.m13.1d">> - 3 italic_n</annotation></semantics></math>’ channel. The total evaporation contribution, <math alttext="\sigma_{\rm cc}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p3.14.m14.1"><semantics id="S3.SS3.SSS2.p3.14.m14.1a"><msubsup id="S3.SS3.SSS2.p3.14.m14.1.1" xref="S3.SS3.SSS2.p3.14.m14.1.1.cmml"><mi id="S3.SS3.SSS2.p3.14.m14.1.1.2.2" xref="S3.SS3.SSS2.p3.14.m14.1.1.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p3.14.m14.1.1.2.3" xref="S3.SS3.SSS2.p3.14.m14.1.1.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p3.14.m14.1.1.3" xref="S3.SS3.SSS2.p3.14.m14.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p3.14.m14.1b"><apply id="S3.SS3.SSS2.p3.14.m14.1.1.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.14.m14.1.1.1.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1">superscript</csymbol><apply id="S3.SS3.SSS2.p3.14.m14.1.1.2.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p3.14.m14.1.1.2.1.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p3.14.m14.1.1.2.2.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p3.14.m14.1.1.2.3.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p3.14.m14.1.1.3a.cmml" xref="S3.SS3.SSS2.p3.14.m14.1.1.3"><mtext id="S3.SS3.SSS2.p3.14.m14.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p3.14.m14.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p3.14.m14.1c">\sigma_{\rm cc}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p3.14.m14.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math>, is then determined by summing the contributions from all neutron removal channels. </div> <div class="ltx_para" id="S3.SS3.SSS2.p4"> The last column of Table <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.T2" title="Table 2 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">2</a> is the evaporation estimate from experimental data <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib63" title="">63</a>]</cite>, and is calculated by subtracting the charge-changing cross section (<math alttext="\sigma_{\rm cc}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p4.1.m1.1"><semantics id="S3.SS3.SSS2.p4.1.m1.1a"><msub id="S3.SS3.SSS2.p4.1.m1.1.1" xref="S3.SS3.SSS2.p4.1.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p4.1.m1.1.1.2" xref="S3.SS3.SSS2.p4.1.m1.1.1.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p4.1.m1.1.1.3" xref="S3.SS3.SSS2.p4.1.m1.1.1.3.cmml">cc</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p4.1.m1.1b"><apply id="S3.SS3.SSS2.p4.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p4.1.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p4.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p4.1.m1.1.1.2">𝜎</ci><ci id="S3.SS3.SSS2.p4.1.m1.1.1.3.cmml" xref="S3.SS3.SSS2.p4.1.m1.1.1.3">cc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p4.1.m1.1c">\sigma_{\rm cc}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p4.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math>) (Table <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.T1" title="Table 1 ‣ III.2 Direct fragmentation or abrasion stage ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">1</a>) using the Glauber model. The first column of the table represents the parameter <math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p4.2.m2.1"><semantics id="S3.SS3.SSS2.p4.2.m2.1a"><msub id="S3.SS3.SSS2.p4.2.m2.1.1" xref="S3.SS3.SSS2.p4.2.m2.1.1.cmml"><mi id="S3.SS3.SSS2.p4.2.m2.1.1.2" xref="S3.SS3.SSS2.p4.2.m2.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p4.2.m2.1.1.3" xref="S3.SS3.SSS2.p4.2.m2.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p4.2.m2.1b"><apply id="S3.SS3.SSS2.p4.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p4.2.m2.1.1.1.cmml" xref="S3.SS3.SSS2.p4.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p4.2.m2.1.1.2.cmml" xref="S3.SS3.SSS2.p4.2.m2.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p4.2.m2.1.1.3a.cmml" xref="S3.SS3.SSS2.p4.2.m2.1.1.3"><mtext id="S3.SS3.SSS2.p4.2.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p4.2.m2.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p4.2.m2.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p4.2.m2.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>, which is varied to achieve an approximate match of <math alttext="\sigma_{\rm cc}^{\text{evap}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p4.3.m3.1"><semantics id="S3.SS3.SSS2.p4.3.m3.1a"><msubsup id="S3.SS3.SSS2.p4.3.m3.1.1" xref="S3.SS3.SSS2.p4.3.m3.1.1.cmml"><mi id="S3.SS3.SSS2.p4.3.m3.1.1.2.2" xref="S3.SS3.SSS2.p4.3.m3.1.1.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p4.3.m3.1.1.2.3" xref="S3.SS3.SSS2.p4.3.m3.1.1.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p4.3.m3.1.1.3" xref="S3.SS3.SSS2.p4.3.m3.1.1.3a.cmml">evap</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p4.3.m3.1b"><apply id="S3.SS3.SSS2.p4.3.m3.1.1.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p4.3.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1">superscript</csymbol><apply id="S3.SS3.SSS2.p4.3.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p4.3.m3.1.1.2.1.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p4.3.m3.1.1.2.2.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p4.3.m3.1.1.2.3.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p4.3.m3.1.1.3a.cmml" xref="S3.SS3.SSS2.p4.3.m3.1.1.3"><mtext id="S3.SS3.SSS2.p4.3.m3.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p4.3.m3.1.1.3">evap</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p4.3.m3.1c">\sigma_{\rm cc}^{\text{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p4.3.m3.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT</annotation></semantics></math> with the estimated value extracted from the experimental data. </div> <div class="ltx_para" id="S3.SS3.SSS2.p5"> Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F6" title="Figure 6 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">6</a> compares the ratio of evaporation contributions to charge-changing cross sections. The top panel (a) shows the ratio of the estimated evaporation contributions to the experimental charge-changing cross section, given by <math alttext="(\sigma_{\mathrm{cc}}^{\text{expt}}-\sigma_{\mathrm{cc}})/\sigma_{\mathrm{cc}}% ^{\text{expt}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p5.1.m1.1"><semantics id="S3.SS3.SSS2.p5.1.m1.1a"><mrow id="S3.SS3.SSS2.p5.1.m1.1.1" xref="S3.SS3.SSS2.p5.1.m1.1.1.cmml"><mrow id="S3.SS3.SSS2.p5.1.m1.1.1.1.1" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.cmml"><msubsup id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.cmml"><mi id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.2" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.3a.cmml">expt</mtext></msubsup><mo id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.1" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.1.cmml">−</mo><msub id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.cmml"><mi id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.2" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.3.cmml">cc</mi></msub></mrow><mo id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS3.SSS2.p5.1.m1.1.1.2" xref="S3.SS3.SSS2.p5.1.m1.1.1.2.cmml">/</mo><msubsup id="S3.SS3.SSS2.p5.1.m1.1.1.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.cmml"><mi id="S3.SS3.SSS2.p5.1.m1.1.1.3.2.2" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.1.m1.1.1.3.2.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p5.1.m1.1.1.3.3" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.3a.cmml">expt</mtext></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p5.1.m1.1b"><apply id="S3.SS3.SSS2.p5.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1"><divide id="S3.SS3.SSS2.p5.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.2"></divide><apply id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1"><minus id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.1"></minus><apply id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2">superscript</csymbol><apply id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.3.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.3a.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.3"><mtext id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.2.3">expt</mtext></ci></apply><apply id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.3.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.1.1.1.3.3">cc</ci></apply></apply><apply id="S3.SS3.SSS2.p5.1.m1.1.1.3.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.1.m1.1.1.3.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3">superscript</csymbol><apply id="S3.SS3.SSS2.p5.1.m1.1.1.3.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.1.m1.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p5.1.m1.1.1.3.2.2.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.1.m1.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p5.1.m1.1.1.3.3a.cmml" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.3"><mtext id="S3.SS3.SSS2.p5.1.m1.1.1.3.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p5.1.m1.1.1.3.3">expt</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p5.1.m1.1c">(\sigma_{\mathrm{cc}}^{\text{expt}}-\sigma_{\mathrm{cc}})/\sigma_{\mathrm{cc}}% ^{\text{expt}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p5.1.m1.1d">( italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt end_POSTSUPERSCRIPT - italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT ) / italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt end_POSTSUPERSCRIPT</annotation></semantics></math> on the left vertical axis, while the ratio <math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\mathrm{cc}}^{\text{expt}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p5.2.m2.1"><semantics id="S3.SS3.SSS2.p5.2.m2.1a"><mrow id="S3.SS3.SSS2.p5.2.m2.1.1" xref="S3.SS3.SSS2.p5.2.m2.1.1.cmml"><msubsup id="S3.SS3.SSS2.p5.2.m2.1.1.2" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.cmml"><mi id="S3.SS3.SSS2.p5.2.m2.1.1.2.2.2" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.2.m2.1.1.2.2.3" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p5.2.m2.1.1.2.3" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.3a.cmml">evap</mtext></msubsup><mo id="S3.SS3.SSS2.p5.2.m2.1.1.1" xref="S3.SS3.SSS2.p5.2.m2.1.1.1.cmml">/</mo><msubsup id="S3.SS3.SSS2.p5.2.m2.1.1.3" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.cmml"><mi id="S3.SS3.SSS2.p5.2.m2.1.1.3.2.2" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.2.m2.1.1.3.2.3" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p5.2.m2.1.1.3.3" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.3a.cmml">expt</mtext></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p5.2.m2.1b"><apply id="S3.SS3.SSS2.p5.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1"><divide id="S3.SS3.SSS2.p5.2.m2.1.1.1.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.1"></divide><apply id="S3.SS3.SSS2.p5.2.m2.1.1.2.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.2.m2.1.1.2.1.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2">superscript</csymbol><apply id="S3.SS3.SSS2.p5.2.m2.1.1.2.2.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.2.m2.1.1.2.2.1.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p5.2.m2.1.1.2.2.2.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.2.m2.1.1.2.2.3.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p5.2.m2.1.1.2.3a.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.3"><mtext id="S3.SS3.SSS2.p5.2.m2.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p5.2.m2.1.1.2.3">evap</mtext></ci></apply><apply id="S3.SS3.SSS2.p5.2.m2.1.1.3.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.2.m2.1.1.3.1.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3">superscript</csymbol><apply id="S3.SS3.SSS2.p5.2.m2.1.1.3.2.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.2.m2.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p5.2.m2.1.1.3.2.2.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.2.m2.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p5.2.m2.1.1.3.3a.cmml" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.3"><mtext id="S3.SS3.SSS2.p5.2.m2.1.1.3.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p5.2.m2.1.1.3.3">expt</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p5.2.m2.1c">\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\mathrm{cc}}^{\text{expt}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p5.2.m2.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT / italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt end_POSTSUPERSCRIPT</annotation></semantics></math> is displayed on the right vertical axis. The bottom panel (b) represents the ratio of evaporation contributions in the abrasion stage, given by <math alttext="\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\mathrm{cc}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p5.3.m3.1"><semantics id="S3.SS3.SSS2.p5.3.m3.1a"><mrow id="S3.SS3.SSS2.p5.3.m3.1.1" xref="S3.SS3.SSS2.p5.3.m3.1.1.cmml"><msubsup id="S3.SS3.SSS2.p5.3.m3.1.1.2" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.cmml"><mi id="S3.SS3.SSS2.p5.3.m3.1.1.2.2.2" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.3.m3.1.1.2.2.3" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p5.3.m3.1.1.2.3" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.3a.cmml">evap</mtext></msubsup><mo id="S3.SS3.SSS2.p5.3.m3.1.1.1" xref="S3.SS3.SSS2.p5.3.m3.1.1.1.cmml">/</mo><msub id="S3.SS3.SSS2.p5.3.m3.1.1.3" xref="S3.SS3.SSS2.p5.3.m3.1.1.3.cmml"><mi id="S3.SS3.SSS2.p5.3.m3.1.1.3.2" xref="S3.SS3.SSS2.p5.3.m3.1.1.3.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p5.3.m3.1.1.3.3" xref="S3.SS3.SSS2.p5.3.m3.1.1.3.3.cmml">cc</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p5.3.m3.1b"><apply id="S3.SS3.SSS2.p5.3.m3.1.1.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1"><divide id="S3.SS3.SSS2.p5.3.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.1"></divide><apply id="S3.SS3.SSS2.p5.3.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.3.m3.1.1.2.1.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2">superscript</csymbol><apply id="S3.SS3.SSS2.p5.3.m3.1.1.2.2.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.3.m3.1.1.2.2.1.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p5.3.m3.1.1.2.2.2.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.3.m3.1.1.2.2.3.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p5.3.m3.1.1.2.3a.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.3"><mtext id="S3.SS3.SSS2.p5.3.m3.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p5.3.m3.1.1.2.3">evap</mtext></ci></apply><apply id="S3.SS3.SSS2.p5.3.m3.1.1.3.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p5.3.m3.1.1.3.1.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p5.3.m3.1.1.3.2.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.3.2">𝜎</ci><ci id="S3.SS3.SSS2.p5.3.m3.1.1.3.3.cmml" xref="S3.SS3.SSS2.p5.3.m3.1.1.3.3">cc</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p5.3.m3.1c">\sigma_{\mathrm{cc}}^{\text{evap}}/\sigma_{\mathrm{cc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p5.3.m3.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT evap end_POSTSUPERSCRIPT / italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math>, which illustrates the influence of abrasion-stage calculations on the evaporation contributions. The plot compares these ratios for the projectiles 12C, 14N, 16O, and 20Ne interacting with three different elemental targets at a beam energy of approximately 290 MeV/nucleon. </div> <div class="ltx_para" id="S3.SS3.SSS2.p6"> The data in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.T2" title="Table 2 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">2</a> demonstrate that reconciling theoretical proton evaporation contributions (<math alttext="\sigma_{\mathrm{cc}}^{\mathrm{evap}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p6.1.m1.1"><semantics id="S3.SS3.SSS2.p6.1.m1.1a"><msubsup id="S3.SS3.SSS2.p6.1.m1.1.1" xref="S3.SS3.SSS2.p6.1.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p6.1.m1.1.1.2.2" xref="S3.SS3.SSS2.p6.1.m1.1.1.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p6.1.m1.1.1.2.3" xref="S3.SS3.SSS2.p6.1.m1.1.1.2.3.cmml">cc</mi><mi id="S3.SS3.SSS2.p6.1.m1.1.1.3" xref="S3.SS3.SSS2.p6.1.m1.1.1.3.cmml">evap</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p6.1.m1.1b"><apply id="S3.SS3.SSS2.p6.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p6.1.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1">superscript</csymbol><apply id="S3.SS3.SSS2.p6.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p6.1.m1.1.1.2.1.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p6.1.m1.1.1.2.2.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p6.1.m1.1.1.2.3.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p6.1.m1.1.1.3.cmml" xref="S3.SS3.SSS2.p6.1.m1.1.1.3">evap</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p6.1.m1.1c">\sigma_{\mathrm{cc}}^{\mathrm{evap}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p6.1.m1.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_evap end_POSTSUPERSCRIPT</annotation></semantics></math>) with the estimation from experimental data (<math alttext="\sigma_{\mathrm{cc}}^{\text{expt}}-\sigma_{\mathrm{cc}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p6.2.m2.1"><semantics id="S3.SS3.SSS2.p6.2.m2.1a"><mrow id="S3.SS3.SSS2.p6.2.m2.1.1" xref="S3.SS3.SSS2.p6.2.m2.1.1.cmml"><msubsup id="S3.SS3.SSS2.p6.2.m2.1.1.2" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.cmml"><mi id="S3.SS3.SSS2.p6.2.m2.1.1.2.2.2" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p6.2.m2.1.1.2.2.3" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.2.3.cmml">cc</mi><mtext id="S3.SS3.SSS2.p6.2.m2.1.1.2.3" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.3a.cmml">expt</mtext></msubsup><mo id="S3.SS3.SSS2.p6.2.m2.1.1.1" xref="S3.SS3.SSS2.p6.2.m2.1.1.1.cmml">−</mo><msub id="S3.SS3.SSS2.p6.2.m2.1.1.3" xref="S3.SS3.SSS2.p6.2.m2.1.1.3.cmml"><mi id="S3.SS3.SSS2.p6.2.m2.1.1.3.2" xref="S3.SS3.SSS2.p6.2.m2.1.1.3.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p6.2.m2.1.1.3.3" xref="S3.SS3.SSS2.p6.2.m2.1.1.3.3.cmml">cc</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p6.2.m2.1b"><apply id="S3.SS3.SSS2.p6.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1"><minus id="S3.SS3.SSS2.p6.2.m2.1.1.1.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.1"></minus><apply id="S3.SS3.SSS2.p6.2.m2.1.1.2.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p6.2.m2.1.1.2.1.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2">superscript</csymbol><apply id="S3.SS3.SSS2.p6.2.m2.1.1.2.2.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p6.2.m2.1.1.2.2.1.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p6.2.m2.1.1.2.2.2.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p6.2.m2.1.1.2.2.3.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p6.2.m2.1.1.2.3a.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.3"><mtext id="S3.SS3.SSS2.p6.2.m2.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p6.2.m2.1.1.2.3">expt</mtext></ci></apply><apply id="S3.SS3.SSS2.p6.2.m2.1.1.3.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p6.2.m2.1.1.3.1.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p6.2.m2.1.1.3.2.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.3.2">𝜎</ci><ci id="S3.SS3.SSS2.p6.2.m2.1.1.3.3.cmml" xref="S3.SS3.SSS2.p6.2.m2.1.1.3.3">cc</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p6.2.m2.1c">\sigma_{\mathrm{cc}}^{\text{expt}}-\sigma_{\mathrm{cc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p6.2.m2.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT expt end_POSTSUPERSCRIPT - italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT</annotation></semantics></math>) requires lowering the parameter <math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p6.3.m3.1"><semantics id="S3.SS3.SSS2.p6.3.m3.1a"><msub id="S3.SS3.SSS2.p6.3.m3.1.1" xref="S3.SS3.SSS2.p6.3.m3.1.1.cmml"><mi id="S3.SS3.SSS2.p6.3.m3.1.1.2" xref="S3.SS3.SSS2.p6.3.m3.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p6.3.m3.1.1.3" xref="S3.SS3.SSS2.p6.3.m3.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p6.3.m3.1b"><apply id="S3.SS3.SSS2.p6.3.m3.1.1.cmml" xref="S3.SS3.SSS2.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p6.3.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p6.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p6.3.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p6.3.m3.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p6.3.m3.1.1.3a.cmml" xref="S3.SS3.SSS2.p6.3.m3.1.1.3"><mtext id="S3.SS3.SSS2.p6.3.m3.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p6.3.m3.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p6.3.m3.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p6.3.m3.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> (to <math alttext="\sim 15~{}\text{MeV}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p6.4.m4.1"><semantics id="S3.SS3.SSS2.p6.4.m4.1a"><mrow id="S3.SS3.SSS2.p6.4.m4.1.1" xref="S3.SS3.SSS2.p6.4.m4.1.1.cmml"><mi id="S3.SS3.SSS2.p6.4.m4.1.1.2" xref="S3.SS3.SSS2.p6.4.m4.1.1.2.cmml"></mi><mo id="S3.SS3.SSS2.p6.4.m4.1.1.1" xref="S3.SS3.SSS2.p6.4.m4.1.1.1.cmml">∼</mo><mrow id="S3.SS3.SSS2.p6.4.m4.1.1.3" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.cmml"><mn id="S3.SS3.SSS2.p6.4.m4.1.1.3.2" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.2.cmml">15</mn><mo id="S3.SS3.SSS2.p6.4.m4.1.1.3.1" lspace="0.330em" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.1.cmml">⁢</mo><mtext id="S3.SS3.SSS2.p6.4.m4.1.1.3.3" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.3a.cmml">MeV</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p6.4.m4.1b"><apply id="S3.SS3.SSS2.p6.4.m4.1.1.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1"><csymbol cd="latexml" id="S3.SS3.SSS2.p6.4.m4.1.1.1.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS3.SSS2.p6.4.m4.1.1.2.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1.2">absent</csymbol><apply id="S3.SS3.SSS2.p6.4.m4.1.1.3.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1.3"><times id="S3.SS3.SSS2.p6.4.m4.1.1.3.1.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.1"></times><cn id="S3.SS3.SSS2.p6.4.m4.1.1.3.2.cmml" type="integer" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.2">15</cn><ci id="S3.SS3.SSS2.p6.4.m4.1.1.3.3a.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.3"><mtext id="S3.SS3.SSS2.p6.4.m4.1.1.3.3.cmml" xref="S3.SS3.SSS2.p6.4.m4.1.1.3.3">MeV</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p6.4.m4.1c">\sim 15~{}\text{MeV}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p6.4.m4.1d">∼ 15 MeV</annotation></semantics></math>) for targets with a higher mass number and atomic number, e.g., Al, Cu, compared to lighter ones like C. This adjustment arises from target-dependent differences in initial neutron removal. Heavier targets induce 2–3 neutron removals during the projectile-target interaction in the abrasion stage, creating a neutron-deficient prefragment with an elevated proton-to-neutron ratio. Simultaneously, the neutron-deficient prefragment exhibits reduced binding energy, favoring competing decay pathways—particularly neutron re-emission—which deplete excitation energy before protons can evaporate. Since neutron evaporation is not hindered by the Coulomb barrier, it becomes the dominant decay mode, effectively shifting the available excitation spectrum away from high-energy proton emission. </div> <div class="ltx_para" id="S3.SS3.SSS2.p7"> The parameter <math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p7.1.m1.1"><semantics id="S3.SS3.SSS2.p7.1.m1.1a"><msub id="S3.SS3.SSS2.p7.1.m1.1.1" xref="S3.SS3.SSS2.p7.1.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p7.1.m1.1.1.2" xref="S3.SS3.SSS2.p7.1.m1.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p7.1.m1.1.1.3" xref="S3.SS3.SSS2.p7.1.m1.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p7.1.m1.1b"><apply id="S3.SS3.SSS2.p7.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p7.1.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p7.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p7.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p7.1.m1.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p7.1.m1.1.1.3a.cmml" xref="S3.SS3.SSS2.p7.1.m1.1.1.3"><mtext id="S3.SS3.SSS2.p7.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p7.1.m1.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p7.1.m1.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p7.1.m1.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>, which defines the upper energy limit for proton emission in statistical decay models, must therefore be reduced for Al and Cu targets to reflect the redistribution of excitation energy and the increased dominance of neutron evaporation. Importantly, the target’s role is limited to the initial neutron removal; subsequent evaporation occurs solely from the prefragment, independent of the target. </div> <div class="ltx_para" id="S3.SS3.SSS2.p8"> This suggests that proton evaporation becomes less significant as the mass of the target nucleus increases. In such cases, proton or charge particle removal from the projectile primarily occurs during the abrasion stage or direct fragmentation processes, where the nucleus breaks apart with less contribution from proton evaporation. The trend is more clearly illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F6" title="Figure 6 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">6</a>(b), which indicates that the ratio of evaporation contributions is greater with the carbon target in comparison to the other targets. </div> <div class="ltx_para" id="S3.SS3.SSS2.p9"> Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F6" title="Figure 6 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">6</a> also reveals distinct trends among the projectiles, with 14N consistently exhibiting the highest ratio for all target elements. The evaporation of neutrons and protons from the prefragment is governed by different underlying factors. Neutron emission is primarily determined by the neutron separation energy (<math alttext="S_{n}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.2.m2.1"><semantics id="S3.SS3.SSS2.p9.2.m2.1a"><msub id="S3.SS3.SSS2.p9.2.m2.1.1" xref="S3.SS3.SSS2.p9.2.m2.1.1.cmml"><mi id="S3.SS3.SSS2.p9.2.m2.1.1.2" xref="S3.SS3.SSS2.p9.2.m2.1.1.2.cmml">S</mi><mi id="S3.SS3.SSS2.p9.2.m2.1.1.3" xref="S3.SS3.SSS2.p9.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.2.m2.1b"><apply id="S3.SS3.SSS2.p9.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p9.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.2.m2.1.1.1.cmml" xref="S3.SS3.SSS2.p9.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p9.2.m2.1.1.2.cmml" xref="S3.SS3.SSS2.p9.2.m2.1.1.2">𝑆</ci><ci id="S3.SS3.SSS2.p9.2.m2.1.1.3.cmml" xref="S3.SS3.SSS2.p9.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.2.m2.1c">S_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.2.m2.1d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) and is favored in neutron-rich nuclei at low excitation energies (<math alttext="E_{\text{ex}}\sim S_{n}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.3.m3.1"><semantics id="S3.SS3.SSS2.p9.3.m3.1a"><mrow id="S3.SS3.SSS2.p9.3.m3.1.1" xref="S3.SS3.SSS2.p9.3.m3.1.1.cmml"><msub id="S3.SS3.SSS2.p9.3.m3.1.1.2" xref="S3.SS3.SSS2.p9.3.m3.1.1.2.cmml"><mi id="S3.SS3.SSS2.p9.3.m3.1.1.2.2" xref="S3.SS3.SSS2.p9.3.m3.1.1.2.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p9.3.m3.1.1.2.3" xref="S3.SS3.SSS2.p9.3.m3.1.1.2.3a.cmml">ex</mtext></msub><mo id="S3.SS3.SSS2.p9.3.m3.1.1.1" xref="S3.SS3.SSS2.p9.3.m3.1.1.1.cmml">∼</mo><msub id="S3.SS3.SSS2.p9.3.m3.1.1.3" xref="S3.SS3.SSS2.p9.3.m3.1.1.3.cmml"><mi id="S3.SS3.SSS2.p9.3.m3.1.1.3.2" xref="S3.SS3.SSS2.p9.3.m3.1.1.3.2.cmml">S</mi><mi id="S3.SS3.SSS2.p9.3.m3.1.1.3.3" xref="S3.SS3.SSS2.p9.3.m3.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.3.m3.1b"><apply id="S3.SS3.SSS2.p9.3.m3.1.1.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1"><csymbol cd="latexml" id="S3.SS3.SSS2.p9.3.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.1">similar-to</csymbol><apply id="S3.SS3.SSS2.p9.3.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.3.m3.1.1.2.1.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p9.3.m3.1.1.2.2.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.2.2">𝐸</ci><ci id="S3.SS3.SSS2.p9.3.m3.1.1.2.3a.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.2.3"><mtext id="S3.SS3.SSS2.p9.3.m3.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p9.3.m3.1.1.2.3">ex</mtext></ci></apply><apply id="S3.SS3.SSS2.p9.3.m3.1.1.3.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.3.m3.1.1.3.1.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p9.3.m3.1.1.3.2.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.3.2">𝑆</ci><ci id="S3.SS3.SSS2.p9.3.m3.1.1.3.3.cmml" xref="S3.SS3.SSS2.p9.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.3.m3.1c">E_{\text{ex}}\sim S_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.3.m3.1d">italic_E start_POSTSUBSCRIPT ex end_POSTSUBSCRIPT ∼ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>), where the absence of a Coulomb barrier allows evaporation even with limited energy. In contrast, proton evaporation requires overcoming both the proton separation energy (<math alttext="S_{p}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.4.m4.1"><semantics id="S3.SS3.SSS2.p9.4.m4.1a"><msub id="S3.SS3.SSS2.p9.4.m4.1.1" xref="S3.SS3.SSS2.p9.4.m4.1.1.cmml"><mi id="S3.SS3.SSS2.p9.4.m4.1.1.2" xref="S3.SS3.SSS2.p9.4.m4.1.1.2.cmml">S</mi><mi id="S3.SS3.SSS2.p9.4.m4.1.1.3" xref="S3.SS3.SSS2.p9.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.4.m4.1b"><apply id="S3.SS3.SSS2.p9.4.m4.1.1.cmml" xref="S3.SS3.SSS2.p9.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.4.m4.1.1.1.cmml" xref="S3.SS3.SSS2.p9.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p9.4.m4.1.1.2.cmml" xref="S3.SS3.SSS2.p9.4.m4.1.1.2">𝑆</ci><ci id="S3.SS3.SSS2.p9.4.m4.1.1.3.cmml" xref="S3.SS3.SSS2.p9.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.4.m4.1c">S_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.4.m4.1d">italic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>) and the Coulomb barrier (<math alttext="V_{c}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.5.m5.1"><semantics id="S3.SS3.SSS2.p9.5.m5.1a"><msub id="S3.SS3.SSS2.p9.5.m5.1.1" xref="S3.SS3.SSS2.p9.5.m5.1.1.cmml"><mi id="S3.SS3.SSS2.p9.5.m5.1.1.2" xref="S3.SS3.SSS2.p9.5.m5.1.1.2.cmml">V</mi><mi id="S3.SS3.SSS2.p9.5.m5.1.1.3" xref="S3.SS3.SSS2.p9.5.m5.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.5.m5.1b"><apply id="S3.SS3.SSS2.p9.5.m5.1.1.cmml" xref="S3.SS3.SSS2.p9.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.5.m5.1.1.1.cmml" xref="S3.SS3.SSS2.p9.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p9.5.m5.1.1.2.cmml" xref="S3.SS3.SSS2.p9.5.m5.1.1.2">𝑉</ci><ci id="S3.SS3.SSS2.p9.5.m5.1.1.3.cmml" xref="S3.SS3.SSS2.p9.5.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation 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When the initial abrasion stage preferentially removes neutrons from the projectile, the resulting prefragment becomes proton rich, further increasing the probability of proton evaporation. In such cases, the higher proton-to-neutron ratio makes proton emission more favorable compared to neutron emission, even at moderate excitation energies. In neutron rich nuclei, neutron evaporation remains dominant due to lower energy thresholds and higher nuclear level densities at low excitation energy. In neutron-deficient prefragments proton evaporation can become the preferred mode of deexcitation, often reaching near-unity probability as the Coulomb barrier effect diminishes. The analysis as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F7" title="Figure 7 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">7</a> highlights the critical role of the separation energy difference, <math alttext="\Delta{S_{np}}=S_{1n}-S_{1p}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.7.m7.1"><semantics id="S3.SS3.SSS2.p9.7.m7.1a"><mrow id="S3.SS3.SSS2.p9.7.m7.1.1" xref="S3.SS3.SSS2.p9.7.m7.1.1.cmml"><mrow id="S3.SS3.SSS2.p9.7.m7.1.1.2" xref="S3.SS3.SSS2.p9.7.m7.1.1.2.cmml"><mi id="S3.SS3.SSS2.p9.7.m7.1.1.2.2" mathvariant="normal" xref="S3.SS3.SSS2.p9.7.m7.1.1.2.2.cmml">Δ</mi><mo id="S3.SS3.SSS2.p9.7.m7.1.1.2.1" xref="S3.SS3.SSS2.p9.7.m7.1.1.2.1.cmml">⁢</mo><msub id="S3.SS3.SSS2.p9.7.m7.1.1.2.3" xref="S3.SS3.SSS2.p9.7.m7.1.1.2.3.cmml"><mi 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xref="S3.SS3.SSS2.p9.7.m7.1.1.3"><minus id="S3.SS3.SSS2.p9.7.m7.1.1.3.1.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.1"></minus><apply id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.1.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2">subscript</csymbol><ci id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.2.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2.2">𝑆</ci><apply id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3"><times id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.1.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.1"></times><cn id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.2.cmml" type="integer" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.2">1</cn><ci id="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.3.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.2.3.3">𝑛</ci></apply></apply><apply id="S3.SS3.SSS2.p9.7.m7.1.1.3.3.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.7.m7.1.1.3.3.1.cmml" xref="S3.SS3.SSS2.p9.7.m7.1.1.3.3">subscript</csymbol><ci 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abrasion for carbon target. Higher <math alttext="\Delta{S_{np}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.8.m8.1"><semantics id="S3.SS3.SSS2.p9.8.m8.1a"><mrow id="S3.SS3.SSS2.p9.8.m8.1.1" xref="S3.SS3.SSS2.p9.8.m8.1.1.cmml"><mi id="S3.SS3.SSS2.p9.8.m8.1.1.2" mathvariant="normal" xref="S3.SS3.SSS2.p9.8.m8.1.1.2.cmml">Δ</mi><mo id="S3.SS3.SSS2.p9.8.m8.1.1.1" xref="S3.SS3.SSS2.p9.8.m8.1.1.1.cmml">⁢</mo><msub id="S3.SS3.SSS2.p9.8.m8.1.1.3" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.cmml"><mi id="S3.SS3.SSS2.p9.8.m8.1.1.3.2" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.2.cmml">S</mi><mrow id="S3.SS3.SSS2.p9.8.m8.1.1.3.3" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.cmml"><mi id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.2" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.2.cmml">n</mi><mo id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.1" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.3" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.3.cmml">p</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.8.m8.1b"><apply id="S3.SS3.SSS2.p9.8.m8.1.1.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1"><times id="S3.SS3.SSS2.p9.8.m8.1.1.1.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.1"></times><ci id="S3.SS3.SSS2.p9.8.m8.1.1.2.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.2">Δ</ci><apply id="S3.SS3.SSS2.p9.8.m8.1.1.3.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.8.m8.1.1.3.1.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p9.8.m8.1.1.3.2.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.2">𝑆</ci><apply id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3"><times id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.1.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.1"></times><ci id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.2.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.2">𝑛</ci><ci id="S3.SS3.SSS2.p9.8.m8.1.1.3.3.3.cmml" xref="S3.SS3.SSS2.p9.8.m8.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.8.m8.1c">\Delta{S_{np}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.8.m8.1d">roman_Δ italic_S start_POSTSUBSCRIPT italic_n italic_p end_POSTSUBSCRIPT</annotation></semantics></math> values, such as for <math alttext="{}^{13}\text{N}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.9.m9.1"><semantics id="S3.SS3.SSS2.p9.9.m9.1a"><mmultiscripts id="S3.SS3.SSS2.p9.9.m9.1.1" xref="S3.SS3.SSS2.p9.9.m9.1.1.cmml"><mtext id="S3.SS3.SSS2.p9.9.m9.1.1.2" xref="S3.SS3.SSS2.p9.9.m9.1.1.2a.cmml">N</mtext><mprescripts id="S3.SS3.SSS2.p9.9.m9.1.1a" xref="S3.SS3.SSS2.p9.9.m9.1.1.cmml"></mprescripts><mrow id="S3.SS3.SSS2.p9.9.m9.1.1b" xref="S3.SS3.SSS2.p9.9.m9.1.1.cmml"></mrow><mn id="S3.SS3.SSS2.p9.9.m9.1.1.3" xref="S3.SS3.SSS2.p9.9.m9.1.1.3.cmml">13</mn></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.9.m9.1b"><apply id="S3.SS3.SSS2.p9.9.m9.1.1.cmml" xref="S3.SS3.SSS2.p9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.9.m9.1.1.1.cmml" xref="S3.SS3.SSS2.p9.9.m9.1.1">superscript</csymbol><ci id="S3.SS3.SSS2.p9.9.m9.1.1.2a.cmml" xref="S3.SS3.SSS2.p9.9.m9.1.1.2"><mtext id="S3.SS3.SSS2.p9.9.m9.1.1.2.cmml" xref="S3.SS3.SSS2.p9.9.m9.1.1.2">N</mtext></ci><cn id="S3.SS3.SSS2.p9.9.m9.1.1.3.cmml" type="integer" xref="S3.SS3.SSS2.p9.9.m9.1.1.3">13</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.9.m9.1c">{}^{13}\text{N}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.9.m9.1d">start_FLOATSUPERSCRIPT 13 end_FLOATSUPERSCRIPT N</annotation></semantics></math> (<math alttext="\Delta S_{np}=18.12\,\text{MeV}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.10.m10.1"><semantics id="S3.SS3.SSS2.p9.10.m10.1a"><mrow id="S3.SS3.SSS2.p9.10.m10.1.1" xref="S3.SS3.SSS2.p9.10.m10.1.1.cmml"><mrow id="S3.SS3.SSS2.p9.10.m10.1.1.2" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.cmml"><mi id="S3.SS3.SSS2.p9.10.m10.1.1.2.2" mathvariant="normal" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.2.cmml">Δ</mi><mo id="S3.SS3.SSS2.p9.10.m10.1.1.2.1" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.1.cmml">⁢</mo><msub id="S3.SS3.SSS2.p9.10.m10.1.1.2.3" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.cmml"><mi id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.2" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.2.cmml">S</mi><mrow id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.cmml"><mi id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.2" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.2.cmml">n</mi><mo id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.1" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.3" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.3.3.cmml">p</mi></mrow></msub></mrow><mo id="S3.SS3.SSS2.p9.10.m10.1.1.1" xref="S3.SS3.SSS2.p9.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS3.SSS2.p9.10.m10.1.1.3" xref="S3.SS3.SSS2.p9.10.m10.1.1.3.cmml"><mn id="S3.SS3.SSS2.p9.10.m10.1.1.3.2" xref="S3.SS3.SSS2.p9.10.m10.1.1.3.2.cmml">18.12</mn><mo id="S3.SS3.SSS2.p9.10.m10.1.1.3.1" lspace="0.170em" xref="S3.SS3.SSS2.p9.10.m10.1.1.3.1.cmml">⁢</mo><mtext id="S3.SS3.SSS2.p9.10.m10.1.1.3.3" xref="S3.SS3.SSS2.p9.10.m10.1.1.3.3a.cmml">MeV</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.10.m10.1b"><apply id="S3.SS3.SSS2.p9.10.m10.1.1.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1"><eq id="S3.SS3.SSS2.p9.10.m10.1.1.1.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.1"></eq><apply id="S3.SS3.SSS2.p9.10.m10.1.1.2.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.2"><times id="S3.SS3.SSS2.p9.10.m10.1.1.2.1.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.1"></times><ci id="S3.SS3.SSS2.p9.10.m10.1.1.2.2.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.2">Δ</ci><apply id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.1.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3">subscript</csymbol><ci id="S3.SS3.SSS2.p9.10.m10.1.1.2.3.2.cmml" xref="S3.SS3.SSS2.p9.10.m10.1.1.2.3.2">𝑆</ci><apply 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id="S3.SS3.SSS2.p9.13.m13.1.1.2.1.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.1"></times><ci id="S3.SS3.SSS2.p9.13.m13.1.1.2.2.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.2">Δ</ci><apply id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.1.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3">subscript</csymbol><ci id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.2.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3.2">𝑆</ci><apply id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3"><times id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.1.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.1"></times><ci id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.2.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.2">𝑛</ci><ci id="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.3.cmml" xref="S3.SS3.SSS2.p9.13.m13.1.1.2.3.3.3">𝑝</ci></apply></apply></apply><cn id="S3.SS3.SSS2.p9.13.m13.1.1.3.cmml" type="float" xref="S3.SS3.SSS2.p9.13.m13.1.1.3">18.56</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.13.m13.1c">\Delta S_{np}=18.56</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.13.m13.1d">roman_Δ italic_S start_POSTSUBSCRIPT italic_n italic_p end_POSTSUBSCRIPT = 18.56</annotation></semantics></math> MeV), driven by symmetry energy and Coulomb effects <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib65" title="">65</a>]</cite>. These trends underscore <math alttext="\Delta{S_{np}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p9.14.m14.1"><semantics id="S3.SS3.SSS2.p9.14.m14.1a"><mrow id="S3.SS3.SSS2.p9.14.m14.1.1" xref="S3.SS3.SSS2.p9.14.m14.1.1.cmml"><mi id="S3.SS3.SSS2.p9.14.m14.1.1.2" mathvariant="normal" xref="S3.SS3.SSS2.p9.14.m14.1.1.2.cmml">Δ</mi><mo id="S3.SS3.SSS2.p9.14.m14.1.1.1" xref="S3.SS3.SSS2.p9.14.m14.1.1.1.cmml">⁢</mo><msub id="S3.SS3.SSS2.p9.14.m14.1.1.3" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.cmml"><mi id="S3.SS3.SSS2.p9.14.m14.1.1.3.2" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.2.cmml">S</mi><mrow id="S3.SS3.SSS2.p9.14.m14.1.1.3.3" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.cmml"><mi id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.2" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.2.cmml">n</mi><mo id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.1" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.3" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.3.cmml">p</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p9.14.m14.1b"><apply id="S3.SS3.SSS2.p9.14.m14.1.1.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1"><times id="S3.SS3.SSS2.p9.14.m14.1.1.1.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.1"></times><ci id="S3.SS3.SSS2.p9.14.m14.1.1.2.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.2">Δ</ci><apply id="S3.SS3.SSS2.p9.14.m14.1.1.3.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p9.14.m14.1.1.3.1.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p9.14.m14.1.1.3.2.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.2">𝑆</ci><apply id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3"><times id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.1.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.1"></times><ci id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.2.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.2">𝑛</ci><ci id="S3.SS3.SSS2.p9.14.m14.1.1.3.3.3.cmml" xref="S3.SS3.SSS2.p9.14.m14.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p9.14.m14.1c">\Delta{S_{np}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p9.14.m14.1d">roman_Δ italic_S start_POSTSUBSCRIPT italic_n italic_p end_POSTSUBSCRIPT</annotation></semantics></math> as a key predictor of evaporation pathways in systems with fewer neutrons relative to protons. </div> <div class="ltx_para" id="S3.SS3.SSS2.p10"> The overall dominance of proton evaporation from the prefragment 13N can be better understood by using a simplified discussion on the ratio of proton to neutron decay width <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib66" title="">66</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib67" title="">67</a>]</cite>, <table class="ltx_equation ltx_eqn_table" id="S3.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{\Gamma_{p}}{\Gamma_{n}}\approx\frac{\exp\left[2\sqrt{a(E_{\mathrm{ex}}-S% _{p}-V_{c}-\delta_{\rm eff}})\right]}{\exp\left[2\sqrt{a(E_{\mathrm{ex}}-S_{n}% -\delta_{\rm eff}})\right]}," class="ltx_math_unparsed" display="block" id="S3.E13.m1.1"><semantics id="S3.E13.m1.1a"><mrow id="S3.E13.m1.1.1.1"><mrow id="S3.E13.m1.1.1.1.1"><mfrac id="S3.E13.m1.1.1.1.1.2"><msub id="S3.E13.m1.1.1.1.1.2.2"><mi id="S3.E13.m1.1.1.1.1.2.2.2" mathvariant="normal">Γ</mi><mi id="S3.E13.m1.1.1.1.1.2.2.3">p</mi></msub><msub id="S3.E13.m1.1.1.1.1.2.3"><mi id="S3.E13.m1.1.1.1.1.2.3.2" mathvariant="normal">Γ</mi><mi id="S3.E13.m1.1.1.1.1.2.3.3">n</mi></msub></mfrac><mo id="S3.E13.m1.1.1.1.1.1">≈</mo><mfrac id="S3.E13.m1.1.1.1.1.3"><mrow id="S3.E13.m1.1.1.1.1.3.2"><mi id="S3.E13.m1.1.1.1.1.3.2.1">exp</mi><mrow id="S3.E13.m1.1.1.1.1.3.2.2"><mo id="S3.E13.m1.1.1.1.1.3.2.2.1">[</mo><mn id="S3.E13.m1.1.1.1.1.3.2.2.2">2</mn><msqrt id="S3.E13.m1.1.1.1.1.3.2.2.3"><mrow id="S3.E13.m1.1.1.1.1.3.2.2.3.2"><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.1">a</mi><mrow id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2"><mo id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.1" stretchy="false">(</mo><msub id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.2"><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.2.2">E</mi><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.2.3">ex</mi></msub><mo id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.3">−</mo><msub id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.4"><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.4.2">S</mi><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.4.3">p</mi></msub><mo id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.5">−</mo><msub id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.6"><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.6.2">V</mi><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.6.3">c</mi></msub><mo id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.7">−</mo><msub id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.8"><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.8.2">δ</mi><mi id="S3.E13.m1.1.1.1.1.3.2.2.3.2.2.8.3">eff</mi></msub></mrow></mrow></msqrt><mo id="S3.E13.m1.1.1.1.1.3.2.2.4" stretchy="false">)</mo></mrow><mo id="S3.E13.m1.1.1.1.1.3.2.3">]</mo></mrow><mrow id="S3.E13.m1.1.1.1.1.3.3"><mi id="S3.E13.m1.1.1.1.1.3.3.1">exp</mi><mrow id="S3.E13.m1.1.1.1.1.3.3.2"><mo id="S3.E13.m1.1.1.1.1.3.3.2.1">[</mo><mn id="S3.E13.m1.1.1.1.1.3.3.2.2">2</mn><msqrt id="S3.E13.m1.1.1.1.1.3.3.2.3"><mrow id="S3.E13.m1.1.1.1.1.3.3.2.3.2"><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.1">a</mi><mrow id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2"><mo id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.1" stretchy="false">(</mo><msub id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.2"><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.2.2">E</mi><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.2.3">ex</mi></msub><mo id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.3">−</mo><msub id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.4"><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.4.2">S</mi><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.4.3">n</mi></msub><mo id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.5">−</mo><msub id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.6"><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.6.2">δ</mi><mi id="S3.E13.m1.1.1.1.1.3.3.2.3.2.2.6.3">eff</mi></msub></mrow></mrow></msqrt><mo id="S3.E13.m1.1.1.1.1.3.3.2.4" stretchy="false">)</mo></mrow><mo id="S3.E13.m1.1.1.1.1.3.3.3">]</mo></mrow></mfrac></mrow><mo id="S3.E13.m1.1.1.1.2">,</mo></mrow><annotation encoding="application/x-tex" id="S3.E13.m1.1b">\frac{\Gamma_{p}}{\Gamma_{n}}\approx\frac{\exp\left[2\sqrt{a(E_{\mathrm{ex}}-S% _{p}-V_{c}-\delta_{\rm eff}})\right]}{\exp\left[2\sqrt{a(E_{\mathrm{ex}}-S_{n}% -\delta_{\rm eff}})\right]},</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.1c">divide start_ARG roman_Γ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ≈ divide start_ARG roman_exp [ 2 square-root start_ARG italic_a ( italic_E start_POSTSUBSCRIPT roman_ex end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG ) ] end_ARG start_ARG roman_exp [ 2 square-root start_ARG italic_a ( italic_E start_POSTSUBSCRIPT roman_ex end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG ) ] end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(13)</td> </tr></tbody> </table> with <math alttext="\delta_{\rm eff}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.2.m1.1"><semantics id="S3.SS3.SSS2.p10.2.m1.1a"><msub id="S3.SS3.SSS2.p10.2.m1.1.1" xref="S3.SS3.SSS2.p10.2.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p10.2.m1.1.1.2" xref="S3.SS3.SSS2.p10.2.m1.1.1.2.cmml">δ</mi><mi id="S3.SS3.SSS2.p10.2.m1.1.1.3" xref="S3.SS3.SSS2.p10.2.m1.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.2.m1.1b"><apply id="S3.SS3.SSS2.p10.2.m1.1.1.cmml" xref="S3.SS3.SSS2.p10.2.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.2.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p10.2.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p10.2.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p10.2.m1.1.1.2">𝛿</ci><ci id="S3.SS3.SSS2.p10.2.m1.1.1.3.cmml" xref="S3.SS3.SSS2.p10.2.m1.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.2.m1.1c">\delta_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.2.m1.1d">italic_δ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> as the pairing energy correction. The level density parameter is given by <table class="ltx_equation ltx_eqn_table" id="S3.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="a=\tilde{a}\left[1+\frac{\Delta S}{U}\left(1+\exp(-\gamma U)\right)\right]," class="ltx_Math" display="block" id="S3.E14.m1.2"><semantics id="S3.E14.m1.2a"><mrow id="S3.E14.m1.2.2.1" xref="S3.E14.m1.2.2.1.1.cmml"><mrow id="S3.E14.m1.2.2.1.1" xref="S3.E14.m1.2.2.1.1.cmml"><mi id="S3.E14.m1.2.2.1.1.3" xref="S3.E14.m1.2.2.1.1.3.cmml">a</mi><mo id="S3.E14.m1.2.2.1.1.2" xref="S3.E14.m1.2.2.1.1.2.cmml">=</mo><mrow id="S3.E14.m1.2.2.1.1.1" xref="S3.E14.m1.2.2.1.1.1.cmml"><mover accent="true" id="S3.E14.m1.2.2.1.1.1.3" xref="S3.E14.m1.2.2.1.1.1.3.cmml"><mi id="S3.E14.m1.2.2.1.1.1.3.2" xref="S3.E14.m1.2.2.1.1.1.3.2.cmml">a</mi><mo id="S3.E14.m1.2.2.1.1.1.3.1" xref="S3.E14.m1.2.2.1.1.1.3.1.cmml">~</mo></mover><mo 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ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(14)</td> </tr></tbody> </table> as suggested by Ignatyuk <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#bib.bib68" title="">68</a>]</cite>, where <math alttext="\tilde{a}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.3.m1.1"><semantics id="S3.SS3.SSS2.p10.3.m1.1a"><mover accent="true" id="S3.SS3.SSS2.p10.3.m1.1.1" xref="S3.SS3.SSS2.p10.3.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p10.3.m1.1.1.2" xref="S3.SS3.SSS2.p10.3.m1.1.1.2.cmml">a</mi><mo id="S3.SS3.SSS2.p10.3.m1.1.1.1" xref="S3.SS3.SSS2.p10.3.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.3.m1.1b"><apply id="S3.SS3.SSS2.p10.3.m1.1.1.cmml" xref="S3.SS3.SSS2.p10.3.m1.1.1"><ci id="S3.SS3.SSS2.p10.3.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p10.3.m1.1.1.1">~</ci><ci id="S3.SS3.SSS2.p10.3.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p10.3.m1.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.3.m1.1c">\tilde{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.3.m1.1d">over~ start_ARG italic_a end_ARG</annotation></semantics></math> is the asymptotic level density parameter, this parameter is defined <math alttext="\tilde{a}=0.073A+0.095A^{2/3}." class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.4.m2.1"><semantics id="S3.SS3.SSS2.p10.4.m2.1a"><mrow id="S3.SS3.SSS2.p10.4.m2.1.1.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.cmml"><mrow id="S3.SS3.SSS2.p10.4.m2.1.1.1.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.cmml"><mover accent="true" id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.2" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.2.cmml"><mi id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.2.2" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.2.2.cmml">a</mi><mo id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.2.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.2.1.cmml">~</mo></mover><mo id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.1.cmml">=</mo><mrow id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.cmml"><mrow id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.cmml"><mn id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.2" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.2.cmml">0.073</mn><mo id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.3" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.2.3.cmml">A</mi></mrow><mo id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.1.cmml">+</mo><mrow id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.cmml"><mn id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.2" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.2.cmml">0.095</mn><mo id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.1" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.1.cmml">⁢</mo><msup id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3" 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xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.1"></times><cn id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.2.cmml" type="float" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.2">0.095</cn><apply id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.cmml" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.1.cmml" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3">superscript</csymbol><ci id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.2.cmml" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.2">𝐴</ci><apply id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.cmml" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3"><divide id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.1.cmml" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.1"></divide><cn id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.2.cmml" type="integer" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.2">2</cn><cn id="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.3.cmml" type="integer" xref="S3.SS3.SSS2.p10.4.m2.1.1.1.1.3.3.3.3.3">3</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.4.m2.1c">\tilde{a}=0.073A+0.095A^{2/3}.</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.4.m2.1d">over~ start_ARG italic_a end_ARG = 0.073 italic_A + 0.095 italic_A start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT .</annotation></semantics></math> <math alttext="\Delta S" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.5.m3.1"><semantics id="S3.SS3.SSS2.p10.5.m3.1a"><mrow id="S3.SS3.SSS2.p10.5.m3.1.1" xref="S3.SS3.SSS2.p10.5.m3.1.1.cmml"><mi id="S3.SS3.SSS2.p10.5.m3.1.1.2" mathvariant="normal" xref="S3.SS3.SSS2.p10.5.m3.1.1.2.cmml">Δ</mi><mo id="S3.SS3.SSS2.p10.5.m3.1.1.1" xref="S3.SS3.SSS2.p10.5.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p10.5.m3.1.1.3" xref="S3.SS3.SSS2.p10.5.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.5.m3.1b"><apply id="S3.SS3.SSS2.p10.5.m3.1.1.cmml" xref="S3.SS3.SSS2.p10.5.m3.1.1"><times id="S3.SS3.SSS2.p10.5.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p10.5.m3.1.1.1"></times><ci id="S3.SS3.SSS2.p10.5.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p10.5.m3.1.1.2">Δ</ci><ci id="S3.SS3.SSS2.p10.5.m3.1.1.3.cmml" xref="S3.SS3.SSS2.p10.5.m3.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.5.m3.1c">\Delta S</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.5.m3.1d">roman_Δ italic_S</annotation></semantics></math> as the shell correction obtained from the difference between the experimental and the liquid drop model masses, <math alttext="U" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.6.m4.1"><semantics id="S3.SS3.SSS2.p10.6.m4.1a"><mi id="S3.SS3.SSS2.p10.6.m4.1.1" xref="S3.SS3.SSS2.p10.6.m4.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.6.m4.1b"><ci id="S3.SS3.SSS2.p10.6.m4.1.1.cmml" xref="S3.SS3.SSS2.p10.6.m4.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.6.m4.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.6.m4.1d">italic_U</annotation></semantics></math> is the effective excitation energy and <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.7.m5.1"><semantics id="S3.SS3.SSS2.p10.7.m5.1a"><mi id="S3.SS3.SSS2.p10.7.m5.1.1" xref="S3.SS3.SSS2.p10.7.m5.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.7.m5.1b"><ci id="S3.SS3.SSS2.p10.7.m5.1.1.cmml" xref="S3.SS3.SSS2.p10.7.m5.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.7.m5.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.7.m5.1d">italic_γ</annotation></semantics></math> as the shell damping factor. The 13N prefragment, formed from 14N after one-neutron removal in the abrasion stage, is characterized by a high level density, a low proton separation energy (<math alttext="S_{p}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.8.m6.1"><semantics id="S3.SS3.SSS2.p10.8.m6.1a"><msub id="S3.SS3.SSS2.p10.8.m6.1.1" xref="S3.SS3.SSS2.p10.8.m6.1.1.cmml"><mi id="S3.SS3.SSS2.p10.8.m6.1.1.2" xref="S3.SS3.SSS2.p10.8.m6.1.1.2.cmml">S</mi><mi id="S3.SS3.SSS2.p10.8.m6.1.1.3" xref="S3.SS3.SSS2.p10.8.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.8.m6.1b"><apply id="S3.SS3.SSS2.p10.8.m6.1.1.cmml" xref="S3.SS3.SSS2.p10.8.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.8.m6.1.1.1.cmml" xref="S3.SS3.SSS2.p10.8.m6.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p10.8.m6.1.1.2.cmml" xref="S3.SS3.SSS2.p10.8.m6.1.1.2">𝑆</ci><ci id="S3.SS3.SSS2.p10.8.m6.1.1.3.cmml" xref="S3.SS3.SSS2.p10.8.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.8.m6.1c">S_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.8.m6.1d">italic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>), a high neutron separation energy (<math alttext="S_{n}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.9.m7.1"><semantics id="S3.SS3.SSS2.p10.9.m7.1a"><msub id="S3.SS3.SSS2.p10.9.m7.1.1" xref="S3.SS3.SSS2.p10.9.m7.1.1.cmml"><mi id="S3.SS3.SSS2.p10.9.m7.1.1.2" xref="S3.SS3.SSS2.p10.9.m7.1.1.2.cmml">S</mi><mi id="S3.SS3.SSS2.p10.9.m7.1.1.3" xref="S3.SS3.SSS2.p10.9.m7.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.9.m7.1b"><apply id="S3.SS3.SSS2.p10.9.m7.1.1.cmml" xref="S3.SS3.SSS2.p10.9.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.9.m7.1.1.1.cmml" xref="S3.SS3.SSS2.p10.9.m7.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p10.9.m7.1.1.2.cmml" xref="S3.SS3.SSS2.p10.9.m7.1.1.2">𝑆</ci><ci id="S3.SS3.SSS2.p10.9.m7.1.1.3.cmml" xref="S3.SS3.SSS2.p10.9.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.9.m7.1c">S_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.9.m7.1d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>), and a low Coulomb barrier. These factors contribute to a significantly high decay width ratio (<math alttext="{\Gamma_{p}/\Gamma_{n}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.10.m8.1"><semantics id="S3.SS3.SSS2.p10.10.m8.1a"><mrow id="S3.SS3.SSS2.p10.10.m8.1.1" xref="S3.SS3.SSS2.p10.10.m8.1.1.cmml"><msub id="S3.SS3.SSS2.p10.10.m8.1.1.2" xref="S3.SS3.SSS2.p10.10.m8.1.1.2.cmml"><mi id="S3.SS3.SSS2.p10.10.m8.1.1.2.2" mathvariant="normal" xref="S3.SS3.SSS2.p10.10.m8.1.1.2.2.cmml">Γ</mi><mi id="S3.SS3.SSS2.p10.10.m8.1.1.2.3" xref="S3.SS3.SSS2.p10.10.m8.1.1.2.3.cmml">p</mi></msub><mo id="S3.SS3.SSS2.p10.10.m8.1.1.1" xref="S3.SS3.SSS2.p10.10.m8.1.1.1.cmml">/</mo><msub id="S3.SS3.SSS2.p10.10.m8.1.1.3" xref="S3.SS3.SSS2.p10.10.m8.1.1.3.cmml"><mi id="S3.SS3.SSS2.p10.10.m8.1.1.3.2" mathvariant="normal" xref="S3.SS3.SSS2.p10.10.m8.1.1.3.2.cmml">Γ</mi><mi id="S3.SS3.SSS2.p10.10.m8.1.1.3.3" xref="S3.SS3.SSS2.p10.10.m8.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.10.m8.1b"><apply id="S3.SS3.SSS2.p10.10.m8.1.1.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1"><divide id="S3.SS3.SSS2.p10.10.m8.1.1.1.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.1"></divide><apply id="S3.SS3.SSS2.p10.10.m8.1.1.2.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.10.m8.1.1.2.1.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p10.10.m8.1.1.2.2.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.2.2">Γ</ci><ci id="S3.SS3.SSS2.p10.10.m8.1.1.2.3.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.2.3">𝑝</ci></apply><apply id="S3.SS3.SSS2.p10.10.m8.1.1.3.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.10.m8.1.1.3.1.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p10.10.m8.1.1.3.2.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.3.2">Γ</ci><ci id="S3.SS3.SSS2.p10.10.m8.1.1.3.3.cmml" xref="S3.SS3.SSS2.p10.10.m8.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.10.m8.1c">{\Gamma_{p}/\Gamma_{n}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.10.m8.1d">roman_Γ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>), as illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F8" title="Figure 8 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">8</a>. This strongly favours proton evaporation, making it a dominant contributor to the charge-changing cross sections through proton emission. 11C, the prefragment of 12C after <math alttext="1n" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.11.m9.1"><semantics id="S3.SS3.SSS2.p10.11.m9.1a"><mrow id="S3.SS3.SSS2.p10.11.m9.1.1" xref="S3.SS3.SSS2.p10.11.m9.1.1.cmml"><mn id="S3.SS3.SSS2.p10.11.m9.1.1.2" xref="S3.SS3.SSS2.p10.11.m9.1.1.2.cmml">1</mn><mo id="S3.SS3.SSS2.p10.11.m9.1.1.1" xref="S3.SS3.SSS2.p10.11.m9.1.1.1.cmml">⁢</mo><mi id="S3.SS3.SSS2.p10.11.m9.1.1.3" xref="S3.SS3.SSS2.p10.11.m9.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.11.m9.1b"><apply id="S3.SS3.SSS2.p10.11.m9.1.1.cmml" xref="S3.SS3.SSS2.p10.11.m9.1.1"><times id="S3.SS3.SSS2.p10.11.m9.1.1.1.cmml" xref="S3.SS3.SSS2.p10.11.m9.1.1.1"></times><cn id="S3.SS3.SSS2.p10.11.m9.1.1.2.cmml" type="integer" xref="S3.SS3.SSS2.p10.11.m9.1.1.2">1</cn><ci id="S3.SS3.SSS2.p10.11.m9.1.1.3.cmml" xref="S3.SS3.SSS2.p10.11.m9.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.11.m9.1c">1n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.11.m9.1d">1 italic_n</annotation></semantics></math> removal in the abrasion stage, has the lowest decay width ratio (<math alttext="{\Gamma_{p}/\Gamma_{n}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.12.m10.1"><semantics id="S3.SS3.SSS2.p10.12.m10.1a"><mrow id="S3.SS3.SSS2.p10.12.m10.1.1" xref="S3.SS3.SSS2.p10.12.m10.1.1.cmml"><msub id="S3.SS3.SSS2.p10.12.m10.1.1.2" xref="S3.SS3.SSS2.p10.12.m10.1.1.2.cmml"><mi id="S3.SS3.SSS2.p10.12.m10.1.1.2.2" mathvariant="normal" xref="S3.SS3.SSS2.p10.12.m10.1.1.2.2.cmml">Γ</mi><mi id="S3.SS3.SSS2.p10.12.m10.1.1.2.3" xref="S3.SS3.SSS2.p10.12.m10.1.1.2.3.cmml">p</mi></msub><mo id="S3.SS3.SSS2.p10.12.m10.1.1.1" xref="S3.SS3.SSS2.p10.12.m10.1.1.1.cmml">/</mo><msub id="S3.SS3.SSS2.p10.12.m10.1.1.3" xref="S3.SS3.SSS2.p10.12.m10.1.1.3.cmml"><mi id="S3.SS3.SSS2.p10.12.m10.1.1.3.2" mathvariant="normal" xref="S3.SS3.SSS2.p10.12.m10.1.1.3.2.cmml">Γ</mi><mi id="S3.SS3.SSS2.p10.12.m10.1.1.3.3" xref="S3.SS3.SSS2.p10.12.m10.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.12.m10.1b"><apply id="S3.SS3.SSS2.p10.12.m10.1.1.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1"><divide id="S3.SS3.SSS2.p10.12.m10.1.1.1.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.1"></divide><apply id="S3.SS3.SSS2.p10.12.m10.1.1.2.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.12.m10.1.1.2.1.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.2">subscript</csymbol><ci id="S3.SS3.SSS2.p10.12.m10.1.1.2.2.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.2.2">Γ</ci><ci id="S3.SS3.SSS2.p10.12.m10.1.1.2.3.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.2.3">𝑝</ci></apply><apply id="S3.SS3.SSS2.p10.12.m10.1.1.3.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.12.m10.1.1.3.1.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.3">subscript</csymbol><ci id="S3.SS3.SSS2.p10.12.m10.1.1.3.2.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.3.2">Γ</ci><ci id="S3.SS3.SSS2.p10.12.m10.1.1.3.3.cmml" xref="S3.SS3.SSS2.p10.12.m10.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.12.m10.1c">{\Gamma_{p}/\Gamma_{n}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.12.m10.1d">roman_Γ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>) and consequently exhibits the smallest contribution to (<math alttext="\sigma_{\rm cc}^{\rm evap}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p10.13.m11.1"><semantics id="S3.SS3.SSS2.p10.13.m11.1a"><msubsup id="S3.SS3.SSS2.p10.13.m11.1.1" xref="S3.SS3.SSS2.p10.13.m11.1.1.cmml"><mi id="S3.SS3.SSS2.p10.13.m11.1.1.2.2" xref="S3.SS3.SSS2.p10.13.m11.1.1.2.2.cmml">σ</mi><mi id="S3.SS3.SSS2.p10.13.m11.1.1.2.3" xref="S3.SS3.SSS2.p10.13.m11.1.1.2.3.cmml">cc</mi><mi id="S3.SS3.SSS2.p10.13.m11.1.1.3" xref="S3.SS3.SSS2.p10.13.m11.1.1.3.cmml">evap</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p10.13.m11.1b"><apply id="S3.SS3.SSS2.p10.13.m11.1.1.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.13.m11.1.1.1.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1">superscript</csymbol><apply id="S3.SS3.SSS2.p10.13.m11.1.1.2.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p10.13.m11.1.1.2.1.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p10.13.m11.1.1.2.2.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1.2.2">𝜎</ci><ci id="S3.SS3.SSS2.p10.13.m11.1.1.2.3.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1.2.3">cc</ci></apply><ci id="S3.SS3.SSS2.p10.13.m11.1.1.3.cmml" xref="S3.SS3.SSS2.p10.13.m11.1.1.3">evap</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p10.13.m11.1c">\sigma_{\rm cc}^{\rm evap}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p10.13.m11.1d">italic_σ start_POSTSUBSCRIPT roman_cc end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_evap end_POSTSUPERSCRIPT</annotation></semantics></math>) in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F6" title="Figure 6 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">6</a>, making it a limited contributor to charge-changing cross sections through proton emission. </div> <div class="ltx_para" id="S3.SS3.SSS2.p11"> The one neutron separation energy (<math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.1.m1.1"><semantics 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influences the residual excitation energy available for evaporation. High <math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.2.m2.1"><semantics id="S3.SS3.SSS2.p11.2.m2.1a"><msub id="S3.SS3.SSS2.p11.2.m2.1.1" xref="S3.SS3.SSS2.p11.2.m2.1.1.cmml"><mi id="S3.SS3.SSS2.p11.2.m2.1.1.2" xref="S3.SS3.SSS2.p11.2.m2.1.1.2.cmml">S</mi><mn id="S3.SS3.SSS2.p11.2.m2.1.1.3" xref="S3.SS3.SSS2.p11.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p11.2.m2.1b"><apply id="S3.SS3.SSS2.p11.2.m2.1.1.cmml" xref="S3.SS3.SSS2.p11.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p11.2.m2.1.1.1.cmml" xref="S3.SS3.SSS2.p11.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p11.2.m2.1.1.2.cmml" xref="S3.SS3.SSS2.p11.2.m2.1.1.2">𝑆</ci><cn id="S3.SS3.SSS2.p11.2.m2.1.1.3.cmml" type="integer" xref="S3.SS3.SSS2.p11.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p11.2.m2.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p11.2.m2.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> in neutron-deficient projectiles depletes excitation energy during neutron removal, reducing ablation, whereas low <math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.3.m3.1"><semantics id="S3.SS3.SSS2.p11.3.m3.1a"><msub id="S3.SS3.SSS2.p11.3.m3.1.1" xref="S3.SS3.SSS2.p11.3.m3.1.1.cmml"><mi id="S3.SS3.SSS2.p11.3.m3.1.1.2" xref="S3.SS3.SSS2.p11.3.m3.1.1.2.cmml">S</mi><mn id="S3.SS3.SSS2.p11.3.m3.1.1.3" xref="S3.SS3.SSS2.p11.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p11.3.m3.1b"><apply id="S3.SS3.SSS2.p11.3.m3.1.1.cmml" xref="S3.SS3.SSS2.p11.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p11.3.m3.1.1.1.cmml" xref="S3.SS3.SSS2.p11.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p11.3.m3.1.1.2.cmml" xref="S3.SS3.SSS2.p11.3.m3.1.1.2">𝑆</ci><cn id="S3.SS3.SSS2.p11.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.SSS2.p11.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p11.3.m3.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p11.3.m3.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> in neutron-rich systems preserves energy, enhancing ablation. Thus, projectiles with lower <math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.4.m4.1"><semantics id="S3.SS3.SSS2.p11.4.m4.1a"><msub id="S3.SS3.SSS2.p11.4.m4.1.1" xref="S3.SS3.SSS2.p11.4.m4.1.1.cmml"><mi id="S3.SS3.SSS2.p11.4.m4.1.1.2" xref="S3.SS3.SSS2.p11.4.m4.1.1.2.cmml">S</mi><mn id="S3.SS3.SSS2.p11.4.m4.1.1.3" xref="S3.SS3.SSS2.p11.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p11.4.m4.1b"><apply id="S3.SS3.SSS2.p11.4.m4.1.1.cmml" xref="S3.SS3.SSS2.p11.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p11.4.m4.1.1.1.cmml" xref="S3.SS3.SSS2.p11.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p11.4.m4.1.1.2.cmml" xref="S3.SS3.SSS2.p11.4.m4.1.1.2">𝑆</ci><cn id="S3.SS3.SSS2.p11.4.m4.1.1.3.cmml" type="integer" xref="S3.SS3.SSS2.p11.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p11.4.m4.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p11.4.m4.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> set a higher <math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.5.m5.1"><semantics id="S3.SS3.SSS2.p11.5.m5.1a"><msub id="S3.SS3.SSS2.p11.5.m5.1.1" xref="S3.SS3.SSS2.p11.5.m5.1.1.cmml"><mi id="S3.SS3.SSS2.p11.5.m5.1.1.2" xref="S3.SS3.SSS2.p11.5.m5.1.1.2.cmml">E</mi><mi id="S3.SS3.SSS2.p11.5.m5.1.1.3" xref="S3.SS3.SSS2.p11.5.m5.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p11.5.m5.1b"><apply id="S3.SS3.SSS2.p11.5.m5.1.1.cmml" xref="S3.SS3.SSS2.p11.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p11.5.m5.1.1.1.cmml" xref="S3.SS3.SSS2.p11.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p11.5.m5.1.1.2.cmml" xref="S3.SS3.SSS2.p11.5.m5.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p11.5.m5.1.1.3.cmml" xref="S3.SS3.SSS2.p11.5.m5.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p11.5.m5.1c">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p11.5.m5.1d">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math>, as evidenced by the linear <math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.6.m6.1"><semantics id="S3.SS3.SSS2.p11.6.m6.1a"><msub id="S3.SS3.SSS2.p11.6.m6.1.1" xref="S3.SS3.SSS2.p11.6.m6.1.1.cmml"><mi id="S3.SS3.SSS2.p11.6.m6.1.1.2" xref="S3.SS3.SSS2.p11.6.m6.1.1.2.cmml">E</mi><mi id="S3.SS3.SSS2.p11.6.m6.1.1.3" xref="S3.SS3.SSS2.p11.6.m6.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p11.6.m6.1b"><apply id="S3.SS3.SSS2.p11.6.m6.1.1.cmml" xref="S3.SS3.SSS2.p11.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p11.6.m6.1.1.1.cmml" xref="S3.SS3.SSS2.p11.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p11.6.m6.1.1.2.cmml" xref="S3.SS3.SSS2.p11.6.m6.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p11.6.m6.1.1.3.cmml" xref="S3.SS3.SSS2.p11.6.m6.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p11.6.m6.1c">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p11.6.m6.1d">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math>-<math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p11.7.m7.1"><semantics id="S3.SS3.SSS2.p11.7.m7.1a"><msub id="S3.SS3.SSS2.p11.7.m7.1.1" xref="S3.SS3.SSS2.p11.7.m7.1.1.cmml"><mi id="S3.SS3.SSS2.p11.7.m7.1.1.2" xref="S3.SS3.SSS2.p11.7.m7.1.1.2.cmml">S</mi><mn id="S3.SS3.SSS2.p11.7.m7.1.1.3" xref="S3.SS3.SSS2.p11.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p11.7.m7.1b"><apply id="S3.SS3.SSS2.p11.7.m7.1.1.cmml" xref="S3.SS3.SSS2.p11.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p11.7.m7.1.1.1.cmml" xref="S3.SS3.SSS2.p11.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p11.7.m7.1.1.2.cmml" xref="S3.SS3.SSS2.p11.7.m7.1.1.2">𝑆</ci><cn id="S3.SS3.SSS2.p11.7.m7.1.1.3.cmml" type="integer" xref="S3.SS3.SSS2.p11.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p11.7.m7.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p11.7.m7.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> correlation in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.12017v1#S3.F9" title="Figure 9 ‣ III.3.2 Proton evaporation probability ‣ III.3 Proton evaporation contributions ‣ III Results and discussions ‣ Quantification of the evaporation process during fragmentation of space-relevant nuclei on elemental targets">9</a>, which holds across all targets. </div> <div class="ltx_para" id="S3.SS3.SSS2.p12"> These effects highlight the necessity of dynamically adjusting the maximum excitation energy parameter (<math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.SS3.SSS2.p12.1.m1.1"><semantics id="S3.SS3.SSS2.p12.1.m1.1a"><msub id="S3.SS3.SSS2.p12.1.m1.1.1" xref="S3.SS3.SSS2.p12.1.m1.1.1.cmml"><mi id="S3.SS3.SSS2.p12.1.m1.1.1.2" xref="S3.SS3.SSS2.p12.1.m1.1.1.2.cmml">E</mi><mtext id="S3.SS3.SSS2.p12.1.m1.1.1.3" xref="S3.SS3.SSS2.p12.1.m1.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS2.p12.1.m1.1b"><apply id="S3.SS3.SSS2.p12.1.m1.1.1.cmml" xref="S3.SS3.SSS2.p12.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS2.p12.1.m1.1.1.1.cmml" xref="S3.SS3.SSS2.p12.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS2.p12.1.m1.1.1.2.cmml" xref="S3.SS3.SSS2.p12.1.m1.1.1.2">𝐸</ci><ci id="S3.SS3.SSS2.p12.1.m1.1.1.3a.cmml" xref="S3.SS3.SSS2.p12.1.m1.1.1.3"><mtext id="S3.SS3.SSS2.p12.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.SSS2.p12.1.m1.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS2.p12.1.m1.1c">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS2.p12.1.m1.1d">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>) in the evaporation model. Incorporating projectile specific properties—including neutron separation energy, neutron orbital occupancy, and beam energy—is crucial for accurately reproducing the observed dominance of abrasion (direct nucleon removal) in heavier target systems and ablation (evaporation) in lighter or neutron rich projectiles. </div> <figure class="ltx_figure" id="S3.F9"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="422" id="S3.F9.g1" src="x9.png" width="581"/> <figcaption class="ltx_caption ltx_centering">Figure 9: Correlation between one neutron separation energy (<math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.F9.7.m1.1"><semantics id="S3.F9.7.m1.1b"><msub id="S3.F9.7.m1.1.1" xref="S3.F9.7.m1.1.1.cmml"><mi id="S3.F9.7.m1.1.1.2" xref="S3.F9.7.m1.1.1.2.cmml">S</mi><mn id="S3.F9.7.m1.1.1.3" xref="S3.F9.7.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.F9.7.m1.1c"><apply id="S3.F9.7.m1.1.1.cmml" xref="S3.F9.7.m1.1.1"><csymbol cd="ambiguous" id="S3.F9.7.m1.1.1.1.cmml" xref="S3.F9.7.m1.1.1">subscript</csymbol><ci id="S3.F9.7.m1.1.1.2.cmml" xref="S3.F9.7.m1.1.1.2">𝑆</ci><cn id="S3.F9.7.m1.1.1.3.cmml" type="integer" xref="S3.F9.7.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.7.m1.1d">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.7.m1.1e">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>) from the projectile and the maximum excitation energy parameter (<math alttext="E_{\text{max}}" class="ltx_Math" display="inline" id="S3.F9.8.m2.1"><semantics id="S3.F9.8.m2.1b"><msub id="S3.F9.8.m2.1.1" xref="S3.F9.8.m2.1.1.cmml"><mi id="S3.F9.8.m2.1.1.2" xref="S3.F9.8.m2.1.1.2.cmml">E</mi><mtext id="S3.F9.8.m2.1.1.3" xref="S3.F9.8.m2.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.F9.8.m2.1c"><apply id="S3.F9.8.m2.1.1.cmml" xref="S3.F9.8.m2.1.1"><csymbol cd="ambiguous" id="S3.F9.8.m2.1.1.1.cmml" xref="S3.F9.8.m2.1.1">subscript</csymbol><ci id="S3.F9.8.m2.1.1.2.cmml" xref="S3.F9.8.m2.1.1.2">𝐸</ci><ci id="S3.F9.8.m2.1.1.3a.cmml" xref="S3.F9.8.m2.1.1.3"><mtext id="S3.F9.8.m2.1.1.3.cmml" mathsize="70%" xref="S3.F9.8.m2.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.8.m2.1d">E_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.8.m2.1e">italic_E start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>) for different target materials (C, Al, Cu). The scatter points represent individual data for the projectiles 12C, 14N, 16O, and 20Ne, while the fitted lines illustrate the underlying trends for each target.</figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> IV Summary and conclusions</h2> <div class="ltx_para" id="S4.p1"> In this study, we calculated nuclear charge-changing cross sections for projectile nuclei, including 12C, 14N, 16O, and 20Ne, interacting with elemental targets such as carbon, aluminum, and copper at a beam energy of approximately 290 MeV/nucleon. We employed the abrasion-ablation model, which describes the fragmentation process in two stages – an initial abrasion phase, where nucleons are removed due to high-energy collisions, followed by an ablation phase, where the excited residual nucleus undergoes particle evaporation. Our calculations utilized the finite-range optical limit approximation within the Glauber model to describe the abrasion step of nuclear fragmentation, incorporating realistic nuclear density distributions. Additionally, we analyzed the ablation process by evaluating the evaporation contribution to charge-changing cross sections through a statistical decay model and by subtracting the abrasion component from the experimental data. We further examined how factors such as excitation energy distribution, particle decay widths, and separation energies influence charged particle evaporation and its impact on charge-changing cross sections. Our findings indicate that for heavier targets, the maximum excitation energy parameter (<math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><msub id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><mi id="S4.p1.5.m5.1.1.2" xref="S4.p1.5.m5.1.1.2.cmml">E</mi><mi id="S4.p1.5.m5.1.1.3" xref="S4.p1.5.m5.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1">subscript</csymbol><ci id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2">𝐸</ci><ci id="S4.p1.5.m5.1.1.3.cmml" xref="S4.p1.5.m5.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math>) tends to be lower when fitted to experimental data. Specifically, for a carbon target, we determined a higher <math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><msub id="S4.p1.6.m6.1.1" xref="S4.p1.6.m6.1.1.cmml"><mi id="S4.p1.6.m6.1.1.2" xref="S4.p1.6.m6.1.1.2.cmml">E</mi><mi id="S4.p1.6.m6.1.1.3" xref="S4.p1.6.m6.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.6.m6.1b"><apply id="S4.p1.6.m6.1.1.cmml" xref="S4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.p1.6.m6.1.1.1.cmml" xref="S4.p1.6.m6.1.1">subscript</csymbol><ci id="S4.p1.6.m6.1.1.2.cmml" xref="S4.p1.6.m6.1.1.2">𝐸</ci><ci id="S4.p1.6.m6.1.1.3.cmml" xref="S4.p1.6.m6.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> value around <math alttext="35\pm 10" class="ltx_Math" display="inline" id="S4.p1.7.m7.1"><semantics id="S4.p1.7.m7.1a"><mrow id="S4.p1.7.m7.1.1" xref="S4.p1.7.m7.1.1.cmml"><mn id="S4.p1.7.m7.1.1.2" xref="S4.p1.7.m7.1.1.2.cmml">35</mn><mo id="S4.p1.7.m7.1.1.1" xref="S4.p1.7.m7.1.1.1.cmml">±</mo><mn id="S4.p1.7.m7.1.1.3" xref="S4.p1.7.m7.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.7.m7.1b"><apply id="S4.p1.7.m7.1.1.cmml" xref="S4.p1.7.m7.1.1"><csymbol cd="latexml" id="S4.p1.7.m7.1.1.1.cmml" xref="S4.p1.7.m7.1.1.1">plus-or-minus</csymbol><cn id="S4.p1.7.m7.1.1.2.cmml" type="integer" xref="S4.p1.7.m7.1.1.2">35</cn><cn id="S4.p1.7.m7.1.1.3.cmml" type="integer" xref="S4.p1.7.m7.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m7.1c">35\pm 10</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m7.1d">35 ± 10</annotation></semantics></math> MeV, with the exception of 14N, which exhibited an anomalous fitting around 60 MeV. In contrast, for aluminum and copper targets, the excitation energy parameter was consistently lower, around <math alttext="15\pm 10" class="ltx_Math" display="inline" id="S4.p1.9.m9.1"><semantics id="S4.p1.9.m9.1a"><mrow id="S4.p1.9.m9.1.1" xref="S4.p1.9.m9.1.1.cmml"><mn id="S4.p1.9.m9.1.1.2" xref="S4.p1.9.m9.1.1.2.cmml">15</mn><mo id="S4.p1.9.m9.1.1.1" xref="S4.p1.9.m9.1.1.1.cmml">±</mo><mn id="S4.p1.9.m9.1.1.3" xref="S4.p1.9.m9.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.9.m9.1b"><apply id="S4.p1.9.m9.1.1.cmml" xref="S4.p1.9.m9.1.1"><csymbol cd="latexml" id="S4.p1.9.m9.1.1.1.cmml" xref="S4.p1.9.m9.1.1.1">plus-or-minus</csymbol><cn id="S4.p1.9.m9.1.1.2.cmml" type="integer" xref="S4.p1.9.m9.1.1.2">15</cn><cn id="S4.p1.9.m9.1.1.3.cmml" type="integer" xref="S4.p1.9.m9.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.9.m9.1c">15\pm 10</annotation><annotation encoding="application/x-llamapun" id="S4.p1.9.m9.1d">15 ± 10</annotation></semantics></math> MeV. Furthermore, our analysis revealed an inverse linear correlation between <math alttext="E_{\mathrm{max}}" class="ltx_Math" display="inline" id="S4.p1.10.m10.1"><semantics id="S4.p1.10.m10.1a"><msub id="S4.p1.10.m10.1.1" xref="S4.p1.10.m10.1.1.cmml"><mi id="S4.p1.10.m10.1.1.2" xref="S4.p1.10.m10.1.1.2.cmml">E</mi><mi id="S4.p1.10.m10.1.1.3" xref="S4.p1.10.m10.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.10.m10.1b"><apply id="S4.p1.10.m10.1.1.cmml" xref="S4.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.p1.10.m10.1.1.1.cmml" xref="S4.p1.10.m10.1.1">subscript</csymbol><ci id="S4.p1.10.m10.1.1.2.cmml" xref="S4.p1.10.m10.1.1.2">𝐸</ci><ci id="S4.p1.10.m10.1.1.3.cmml" xref="S4.p1.10.m10.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.10.m10.1c">E_{\mathrm{max}}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.10.m10.1d">italic_E start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> and the neutron separation energy of the projectile, highlighting the importance of projectile structure in determining the fragmentation process. </div> <div class="ltx_para" id="S4.p2"> The ablation stage plays a crucial role in shaping the charge-changing cross-section measurements by influencing the final distribution of nuclear fragments. Our study underscores the necessity of precise modeling of the ablation process to improve the accuracy of theoretical predictions and experimental interpretations. In space, charge-changing cross sections are used to predict cosmic ray interactions with shielding and human tissues, aiding astronaut safety. In cancer therapy, they influence ion beam fragmentation, impacting dose delivery and treatment precision. Understanding these processes thus enhance space radiation protection, hadron therapy accuracy, and nuclear reactor safety. Moreover, our findings contribute to broader applications in astrophysics and cosmic ray propagation studies, where accurate modeling of nuclear fragmentation is essential. </div> <div class="ltx_para" id="S4.p3"> Future work could explore refinements to the excitation energy parameterization by incorporating microscopic nuclear structure effects and extending calculations to a wider range of projectile-target combinations. Additionally, further experimental validation of the ablation model could help refine predictions, ultimately enhancing our understanding of nuclear fragmentation mechanisms. </div> <div class="ltx_acknowledgements"> <h6 class="ltx_title ltx_title_acknowledgements">Acknowledgements.</h6> The Indo-Japan Cooperative Science Programme (IJCSP)-2023 under the DST-JSPS bilateral programs JPJSBP120247715 and DST/INT/JSPS/P-393/2024(G). [S.D] also acknowledges financial support from the Ministry of Education (MoE), India, through a doctoral fellowship. </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> Wilson et al. [1984] J. W. Wilson, L. Townsend, H. Bidasaria, W. Schimmerling, M. Wong, and J. 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Murakami, Total charge-changing and partial cross-section measurements in the reactions of <math alttext="\sim 110-250~{}\mathrm{M}\mathrm{e}\mathrm{V}/\mathrm{n}\mathrm{u}\mathrm{c}% \mathrm{l}\mathrm{e}\mathrm{o}\mathrm{n}{}^{12}\mathrm{C}" class="ltx_Math" display="inline" id="bib.bib55.1.m1.1"><semantics id="bib.bib55.1.m1.1a"><mrow id="bib.bib55.1.m1.1.1" xref="bib.bib55.1.m1.1.1.cmml"><mi id="bib.bib55.1.m1.1.1.2" xref="bib.bib55.1.m1.1.1.2.cmml"></mi><mo id="bib.bib55.1.m1.1.1.1" xref="bib.bib55.1.m1.1.1.1.cmml">∼</mo><mrow id="bib.bib55.1.m1.1.1.3" xref="bib.bib55.1.m1.1.1.3.cmml"><mn id="bib.bib55.1.m1.1.1.3.2" xref="bib.bib55.1.m1.1.1.3.2.cmml">110</mn><mo id="bib.bib55.1.m1.1.1.3.1" xref="bib.bib55.1.m1.1.1.3.1.cmml">−</mo><mrow id="bib.bib55.1.m1.1.1.3.3" xref="bib.bib55.1.m1.1.1.3.3.cmml"><mrow id="bib.bib55.1.m1.1.1.3.3.2" xref="bib.bib55.1.m1.1.1.3.3.2.cmml"><mrow id="bib.bib55.1.m1.1.1.3.3.2.2" xref="bib.bib55.1.m1.1.1.3.3.2.2.cmml"><mn id="bib.bib55.1.m1.1.1.3.3.2.2.2" xref="bib.bib55.1.m1.1.1.3.3.2.2.2.cmml">250</mn><mo id="bib.bib55.1.m1.1.1.3.3.2.2.1" lspace="0.330em" xref="bib.bib55.1.m1.1.1.3.3.2.2.1.cmml">⁢</mo><mi id="bib.bib55.1.m1.1.1.3.3.2.2.3" xref="bib.bib55.1.m1.1.1.3.3.2.2.3.cmml">MeV</mi></mrow><mo id="bib.bib55.1.m1.1.1.3.3.2.1" xref="bib.bib55.1.m1.1.1.3.3.2.1.cmml">/</mo><mi id="bib.bib55.1.m1.1.1.3.3.2.3" xref="bib.bib55.1.m1.1.1.3.3.2.3.cmml">nucleon</mi></mrow><mo id="bib.bib55.1.m1.1.1.3.3.1" xref="bib.bib55.1.m1.1.1.3.3.1.cmml">⁢</mo><mmultiscripts id="bib.bib55.1.m1.1.1.3.3.3" xref="bib.bib55.1.m1.1.1.3.3.3.cmml"><mi id="bib.bib55.1.m1.1.1.3.3.3.2" mathvariant="normal" xref="bib.bib55.1.m1.1.1.3.3.3.2.cmml">C</mi><mprescripts id="bib.bib55.1.m1.1.1.3.3.3a" xref="bib.bib55.1.m1.1.1.3.3.3.cmml"></mprescripts><mrow id="bib.bib55.1.m1.1.1.3.3.3b" xref="bib.bib55.1.m1.1.1.3.3.3.cmml"></mrow><mn id="bib.bib55.1.m1.1.1.3.3.3.3" xref="bib.bib55.1.m1.1.1.3.3.3.3.cmml">12</mn></mmultiscripts></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="bib.bib55.1.m1.1b"><apply id="bib.bib55.1.m1.1.1.cmml" xref="bib.bib55.1.m1.1.1"><csymbol cd="latexml" id="bib.bib55.1.m1.1.1.1.cmml" xref="bib.bib55.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="bib.bib55.1.m1.1.1.2.cmml" xref="bib.bib55.1.m1.1.1.2">absent</csymbol><apply id="bib.bib55.1.m1.1.1.3.cmml" xref="bib.bib55.1.m1.1.1.3"><minus id="bib.bib55.1.m1.1.1.3.1.cmml" xref="bib.bib55.1.m1.1.1.3.1"></minus><cn id="bib.bib55.1.m1.1.1.3.2.cmml" type="integer" xref="bib.bib55.1.m1.1.1.3.2">110</cn><apply id="bib.bib55.1.m1.1.1.3.3.cmml" xref="bib.bib55.1.m1.1.1.3.3"><times id="bib.bib55.1.m1.1.1.3.3.1.cmml" xref="bib.bib55.1.m1.1.1.3.3.1"></times><apply id="bib.bib55.1.m1.1.1.3.3.2.cmml" xref="bib.bib55.1.m1.1.1.3.3.2"><divide id="bib.bib55.1.m1.1.1.3.3.2.1.cmml" xref="bib.bib55.1.m1.1.1.3.3.2.1"></divide><apply id="bib.bib55.1.m1.1.1.3.3.2.2.cmml" xref="bib.bib55.1.m1.1.1.3.3.2.2"><times id="bib.bib55.1.m1.1.1.3.3.2.2.1.cmml" xref="bib.bib55.1.m1.1.1.3.3.2.2.1"></times><cn id="bib.bib55.1.m1.1.1.3.3.2.2.2.cmml" type="integer" xref="bib.bib55.1.m1.1.1.3.3.2.2.2">250</cn><ci id="bib.bib55.1.m1.1.1.3.3.2.2.3.cmml" xref="bib.bib55.1.m1.1.1.3.3.2.2.3">MeV</ci></apply><ci id="bib.bib55.1.m1.1.1.3.3.2.3.cmml" xref="bib.bib55.1.m1.1.1.3.3.2.3">nucleon</ci></apply><apply id="bib.bib55.1.m1.1.1.3.3.3.cmml" xref="bib.bib55.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="bib.bib55.1.m1.1.1.3.3.3.1.cmml" xref="bib.bib55.1.m1.1.1.3.3.3">superscript</csymbol><ci id="bib.bib55.1.m1.1.1.3.3.3.2.cmml" xref="bib.bib55.1.m1.1.1.3.3.3.2">C</ci><cn id="bib.bib55.1.m1.1.1.3.3.3.3.cmml" type="integer" xref="bib.bib55.1.m1.1.1.3.3.3.3">12</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib55.1.m1.1c">\sim 110-250~{}\mathrm{M}\mathrm{e}\mathrm{V}/\mathrm{n}\mathrm{u}\mathrm{c}% \mathrm{l}\mathrm{e}\mathrm{o}\mathrm{n}{}^{12}\mathrm{C}</annotation><annotation encoding="application/x-llamapun" id="bib.bib55.1.m1.1d">∼ 110 - 250 roman_MeV / roman_nucleon start_FLOATSUPERSCRIPT 12 end_FLOATSUPERSCRIPT roman_C</annotation></semantics></math> in carbon, paraffin, and water, <a class="ltx_ref ltx_href" href="https://doi.org/10.1103/PhysRevC.66.014609" title="">Phys. Rev. C 66, 014609 (2002)</a>. </li> <li class="ltx_bibitem" id="bib.bib56"> Cecchini et al. [2008] S. Cecchini, T. Chiarusi, G. Giacomelli, M. Giorgini, A. Kumar, G. Mandrioli, S. Manzoor, A. Margiotta, E. Medinaceli, L. Patrizii, V. Popa, I. Qureshi, G. Sirri, M. Spurio, and V. Togo, Fragmentation cross sections of <math alttext="\text{Fe}^{26+}" class="ltx_Math" display="inline" id="bib.bib56.1.m1.1"><semantics id="bib.bib56.1.m1.1a"><msup id="bib.bib56.1.m1.1.1" xref="bib.bib56.1.m1.1.1.cmml"><mtext id="bib.bib56.1.m1.1.1.2" xref="bib.bib56.1.m1.1.1.2a.cmml">Fe</mtext><mrow id="bib.bib56.1.m1.1.1.3" xref="bib.bib56.1.m1.1.1.3.cmml"><mn id="bib.bib56.1.m1.1.1.3.2" xref="bib.bib56.1.m1.1.1.3.2.cmml">26</mn><mo id="bib.bib56.1.m1.1.1.3.3" xref="bib.bib56.1.m1.1.1.3.3.cmml">+</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="bib.bib56.1.m1.1b"><apply id="bib.bib56.1.m1.1.1.cmml" xref="bib.bib56.1.m1.1.1"><csymbol cd="ambiguous" id="bib.bib56.1.m1.1.1.1.cmml" xref="bib.bib56.1.m1.1.1">superscript</csymbol><ci id="bib.bib56.1.m1.1.1.2a.cmml" xref="bib.bib56.1.m1.1.1.2"><mtext id="bib.bib56.1.m1.1.1.2.cmml" xref="bib.bib56.1.m1.1.1.2">Fe</mtext></ci><apply id="bib.bib56.1.m1.1.1.3.cmml" xref="bib.bib56.1.m1.1.1.3"><csymbol cd="latexml" id="bib.bib56.1.m1.1.1.3.1.cmml" xref="bib.bib56.1.m1.1.1.3">limit-from</csymbol><cn id="bib.bib56.1.m1.1.1.3.2.cmml" type="integer" xref="bib.bib56.1.m1.1.1.3.2">26</cn><plus id="bib.bib56.1.m1.1.1.3.3.cmml" xref="bib.bib56.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib56.1.m1.1c">\text{Fe}^{26+}</annotation><annotation encoding="application/x-llamapun" id="bib.bib56.1.m1.1d">Fe start_POSTSUPERSCRIPT 26 + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\text{Si}^{14+}" class="ltx_Math" display="inline" id="bib.bib56.2.m2.1"><semantics id="bib.bib56.2.m2.1a"><msup id="bib.bib56.2.m2.1.1" xref="bib.bib56.2.m2.1.1.cmml"><mtext id="bib.bib56.2.m2.1.1.2" xref="bib.bib56.2.m2.1.1.2a.cmml">Si</mtext><mrow id="bib.bib56.2.m2.1.1.3" xref="bib.bib56.2.m2.1.1.3.cmml"><mn id="bib.bib56.2.m2.1.1.3.2" xref="bib.bib56.2.m2.1.1.3.2.cmml">14</mn><mo id="bib.bib56.2.m2.1.1.3.3" xref="bib.bib56.2.m2.1.1.3.3.cmml">+</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="bib.bib56.2.m2.1b"><apply id="bib.bib56.2.m2.1.1.cmml" xref="bib.bib56.2.m2.1.1"><csymbol cd="ambiguous" id="bib.bib56.2.m2.1.1.1.cmml" xref="bib.bib56.2.m2.1.1">superscript</csymbol><ci id="bib.bib56.2.m2.1.1.2a.cmml" xref="bib.bib56.2.m2.1.1.2"><mtext id="bib.bib56.2.m2.1.1.2.cmml" xref="bib.bib56.2.m2.1.1.2">Si</mtext></ci><apply id="bib.bib56.2.m2.1.1.3.cmml" xref="bib.bib56.2.m2.1.1.3"><csymbol cd="latexml" id="bib.bib56.2.m2.1.1.3.1.cmml" xref="bib.bib56.2.m2.1.1.3">limit-from</csymbol><cn id="bib.bib56.2.m2.1.1.3.2.cmml" type="integer" xref="bib.bib56.2.m2.1.1.3.2">14</cn><plus id="bib.bib56.2.m2.1.1.3.3.cmml" xref="bib.bib56.2.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib56.2.m2.1c">\text{Si}^{14+}</annotation><annotation encoding="application/x-llamapun" id="bib.bib56.2.m2.1d">Si start_POSTSUPERSCRIPT 14 + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\text{C}^{6+}" class="ltx_Math" display="inline" id="bib.bib56.3.m3.1"><semantics id="bib.bib56.3.m3.1a"><msup id="bib.bib56.3.m3.1.1" xref="bib.bib56.3.m3.1.1.cmml"><mtext id="bib.bib56.3.m3.1.1.2" xref="bib.bib56.3.m3.1.1.2a.cmml">C</mtext><mrow id="bib.bib56.3.m3.1.1.3" xref="bib.bib56.3.m3.1.1.3.cmml"><mn id="bib.bib56.3.m3.1.1.3.2" xref="bib.bib56.3.m3.1.1.3.2.cmml">6</mn><mo id="bib.bib56.3.m3.1.1.3.3" xref="bib.bib56.3.m3.1.1.3.3.cmml">+</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="bib.bib56.3.m3.1b"><apply id="bib.bib56.3.m3.1.1.cmml" xref="bib.bib56.3.m3.1.1"><csymbol cd="ambiguous" id="bib.bib56.3.m3.1.1.1.cmml" xref="bib.bib56.3.m3.1.1">superscript</csymbol><ci id="bib.bib56.3.m3.1.1.2a.cmml" xref="bib.bib56.3.m3.1.1.2"><mtext id="bib.bib56.3.m3.1.1.2.cmml" xref="bib.bib56.3.m3.1.1.2">C</mtext></ci><apply id="bib.bib56.3.m3.1.1.3.cmml" xref="bib.bib56.3.m3.1.1.3"><csymbol cd="latexml" id="bib.bib56.3.m3.1.1.3.1.cmml" xref="bib.bib56.3.m3.1.1.3">limit-from</csymbol><cn id="bib.bib56.3.m3.1.1.3.2.cmml" type="integer" xref="bib.bib56.3.m3.1.1.3.2">6</cn><plus id="bib.bib56.3.m3.1.1.3.3.cmml" xref="bib.bib56.3.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib56.3.m3.1c">\text{C}^{6+}</annotation><annotation encoding="application/x-llamapun" id="bib.bib56.3.m3.1d">C start_POSTSUPERSCRIPT 6 + end_POSTSUPERSCRIPT</annotation></semantics></math> ions of 0.3–10 A GeV on polyethylene, CR39 and aluminum targets, <a class="ltx_ref ltx_href" href="https://doi.org/10.1016/j.nuclphysa.2008.03.017" title="">Nucl. 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