Effects of sequential decay on collective flows and nuclear stopping power in heavy-ion collisions at intermediate energies

Effects of sequential decay on collective flows and nuclear stopping power in heavy-ion collisions at intermediate energies 1 2 first-author 2 3 4 2 corresp †wangyongjia@zjhu.edu.cn †wangyongjia@zjhu.edu.cn wangyongjia@zjhu.edu.cn https://orcid.org/0000-0003-2506-0010 1 https://orcid.org/0000-0002-2261-6899 2 5 corresp ‡ liqf@zjhu.edu.cn ‡ liqf@zjhu.edu.cn liqf@zjhu.edu.cn State Key Laboratory of Quantum Optics and Quantum Optics Devices & Collaborative Innovation Center of Extreme Optics, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China State Key Laboratory of Quantum Optics and Quantum Optics Devices & Collaborative Innovation Center of Extreme Optics, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China School of Science, Huzhou University, Huzhou 313000, China School of Science, Huzhou University, Huzhou 313000, China School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China Institut für Theoretische Physik, Goethe Universität Frankfurt, Frankfurt am Main 60438, Germany Institut für Theoretische Physik, Goethe Universität Frankfurt, Frankfurt am Main 60438, Germany Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, China In this study, the rapidity distribution, collective flows, and nuclear stopping power in 197 Au+ 197 Au collisions at intermediate energies were investigated using the ultrarelativistic quantum molecular dynamics (UrQMD) model with GEMINI++ code. The UrQMD model was adopted to simulate the dynamic evolution of heavy-ion collisions, whereas the GEMINI++ code was used to simulate the decay of primary fragments produced by UrQMD. The calculated results were compared with the INDRA and FOPI experimental data. It was found that the rapidity distribution, collective flows, and nuclear stopping power were affected to a certain extent by the decay of primary fragments, especially at lower beam energies. Furthermore, the experimental data of the collective flows and nuclear stopping power at the investigated beam energies were better reproduced when the sequential decay effect was included. Heavy-ion collisions Sequential decay effect Collective flow Nuclear stopping power Introduction 1 Understanding the properties of dense nuclear matter is a main topic of the basic research on heavy-ion collisions (HICs), as it is the only way of creating dense nuclear matter in terrestrial laboratories [ 1 1 - 5 5 1-5 ]. However, the typical time scale for the appearance of the created dense nuclear matter is extremely short (less than or of the order of a few fm/ c ) [ 6 6 ]; thus, direct measurement of its properties is currently impossible. Usually, the fundamental properties of dense nuclear matter are inferred by comparing experimental data and transport model simulations [ 7 7 - 11 11 7-11 ]. As powerful tools for investigating the dynamics of nonequilibrium systems at intermediate energies, two typical transport models, the Boltzmann–Uehling–Uhlenbeck (BUU)-type [ 1 1 ] and quantum molecular dynamics (QMD)-type models [ 2 2 ] and their updated versions, have been extensively used for simulating HICs to extract the structure of the initial nuclei and the properties of the created dense nuclear matter, such as the nuclear equation of state, the nuclear symmetry energy, and the in-medium nucleon-nucleon cross section [ 12 12 - 17 17 12-17 ]. Both the BUU- and QMD-type models mainly include three components: the initialization of the projectile and target nuclei, mean-field potential for each particle, and two-body scattering (collision term). Different techniques are typically used to treat the three components in different models. To establish a theoretical systematic error that quantifies the model dependence of transport predictions and disentangle the causes of different predictions, the transport model evaluation project (TMEP) has been recently performed [ 18 18 - 22 22 18-22 ]. After simulating the dynamical process of HICs within the transport models, an afterburner is usually chosen for the formation of fragments, which can then be used to construct observables, and further compared with the corresponding experimental data to extract interesting information regarding the nuclear matter. However, the formed fragments are typically excited and cannot be experimentally detected. In the experiments, the detected particles underwent long-term de-excitation. Hence, greater focus should be placed on the effect of de-excitation in the treatment of afterburners, especially for studying HICs at Fermi energies where the beam energy is comparable to the exciting energy [ 23 23 - 25 25 23-25 ]. The underestimation of the yield of light fragments (e.g., 3 H, 3 He, and 4 He) in HICs at intermediate energies is a long-standing problem in transport model simulations, and numerous techniques have been developed and adopted to overcome it [ 26 26 ], including the statistical multifragmentation model (SMM) [ 27 27 , 28 28 ], the statistical evaporation model (HIVAP) [ 29 29 ], the statistical model GEMINI [ 30 30 ], and the Simulated Annealing Clusterisation Algorithm (SACA) [ 31 31 , 32 32 ]. From our previous studies [ 33 33 - 35 35 33-35 ], adopting the conventional phase space coalescence model ‘‘Minimum Spanning Tree” (MST) [ 36 36 - 38 38 36-38 ] and disregarding the decay of the excited primary fragments from the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model simulation, the experimental data of the collective flow of protons and deuterons at INDRA and FOPI energies can be reasonably reproduced, whereas those of α particles are different from the experimental data. To improve the corresponding collective flow distribution and explore the influence of sequential decay on the observables in the HICs at intermediate energies, the statistical code GEMINI++ was employed to describe the decay of primary fragments. In this study, the rapidity distributions and collective flows of free protons and light clusters, as well as the nuclear stopping power, were calculated based on the UrQMD model with and without considering sequential decay. The remainder of this paper is organized as follows. In Sect. 2, the relevant descriptions of the transport model, statistical model, and the observables are provided. The calculation results are presented and discussed in Sect. 3. Finally, a summary and outlook are provided in Sect. 4. MODEL DESCRIPTION AND OBSERVABLES 1 Dynamical model: UrQMD 2 The UrQMD model [ 39 39 - 42 42 39-42 ] is a typical transport model used for microscopic many-body nonequilibrium dynamics. In this model, each nucleon is represented by a Gaussian wave packet of a certain width L . Empirically, L = 2 fm 2 are chosen to simulate Au+Au collisions. The coordinates and momentum of each nucleon are propagated using Hamilton's equations of motion. The total Hamiltonian of the system <math id="M1"><mrow><mo>〈</mo><mi>H</mi><mo>〉</mo></mrow></math> consists of the kinetic energy T and the effective interaction potential energy U , which includes the Skyrme potential energy U ρ , Coulomb energy U Coul , and momentum-dependent potential energy U md : <math display="block" id="M2"><mrow><mo>〈</mo><mi>H</mi><mo>〉</mo><mo>=</mo><mi>T</mi><mo>+</mo><msub><mi>U</mi><mi>ρ</mi></msub><mo>+</mo><msub><mi>U</mi><mrow><mtext>Coul</mtext></mrow></msub><mo>+</mo><msub><mi>U</mi><mrow><mtext>md</mtext></mrow></msub><mn>.</mn></mrow></math> To study the HICs at intermediate energies, the Skyrme energy density functional was introduced in the same manner as that in the improved quantum molecular dynamics (ImQMD) model [ 43 43 , 44 44 ]. The local and momentum-dependent potential energies can be written as <math id="M3"><mrow><msub><mi>U</mi><mrow><mi>ρ</mi><mn>,</mn><mtext>md</mtext></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msub><mi>u</mi><mrow><mi>ρ</mi><mn>,</mn><mtext>md</mtext></mrow></msub><mtext>d</mtext><mi>r</mi></mrow></math> , where <math display="block" id="M4"><mtable><mtr><mtd><msub><mi>u</mi><mi>ρ</mi></msub><mo>=</mo><mfrac><mi>α</mi><mn>2</mn></mfrac><mfrac><mrow><msup><mi>ρ</mi><mn>2</mn></msup></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mi>β</mi><mrow><mi>γ</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mfrac><mrow><msup><mi>ρ</mi><mrow><mi>γ</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><msubsup><mi>ρ</mi><mn>0</mn><mi>γ</mi></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>g</mi><mrow><mtext>sur</mtext></mrow></msub></mrow><mrow><mn>2</mn><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><msup><mrow><mo>(</mo><mrow><mo>∇</mo><mi>ρ</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><msub><mi>g</mi><mrow><mtext>sur,iso</mtext></mrow></msub></mrow><mrow><mn>2</mn><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><msup><mrow><mo>[</mo><mrow><mo>∇</mo><mrow><mo>(</mo><mrow><msub><mi>ρ</mi><mi>n</mi></msub><mo>−</mo><msub><mi>ρ</mi><mi>p</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>+</mo><mrow><mo>[</mo><mrow><msub><mi>A</mi><mrow><mtext>sym</mtext></mrow></msub><mrow><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>B</mi><mrow><mtext>sym</mtext></mrow></msub><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mi>η</mi></msup><mo>+</mo><msub><mi>C</mi><mrow><mtext>sym</mtext></mrow></msub><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>]</mo></mrow><msup><mi>δ</mi><mn>2</mn></msup><mi>ρ</mi><mn>,</mn></mtd></mtr></mtable></math> and <math display="block" id="M5"><mrow><msub><mi>u</mi><mrow><mtext>md</mtext></mrow></msub><mo>=</mo><msub><mi>t</mi><mrow><mtext>md</mtext></mrow></msub><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>ln</mi></mrow></mstyle><mn>2</mn></msup><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>a</mi><mrow><mtext>md</mtext></mrow></msub><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>−</mo><msub><mi>p</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mn>.</mn></mrow></math> Here, δ = <math id="M6"><mrow><mrow><mo>(</mo><mrow><msub><mi>ρ</mi><mtext>n</mtext></msub><mo>−</mo><msub><mi>ρ</mi><mtext>p</mtext></msub></mrow><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><msub><mi>ρ</mi><mtext>n</mtext></msub><mo>+</mo><msub><mi>ρ</mi><mtext>p</mtext></msub></mrow><mo>)</mo></mrow></mrow></math> denotes the isospin asymmetry defined by the neutron ( ρ n ) and proton ( ρ p ) densities. In the present work, the normal nuclear matter density ρ 0 = 0.16 fm -3 , α = -393 MeV, β = 320 MeV, γ = 1.14, g sur = 19.5 MeV fm 2 , g sur,iso = -11.3 MeV fm 2 , A sym = 20.4 MeV, B sym = 10.8 MeV, C sym = -9.3 MeV, η = 1.3, t md = 1.57 MeV, and a md = 500 (GeV/ c ) -2 are chosen, corresponding to a soft and momentum-dependent (SM) equation of state with the incompressibility K 0 = 200 MeV and the symmetry-energy slope parameter L = 80.95 MeV, which are in their commonly accepted regions [ 42 42 , 45 45 - 49 49 45-49 ]. In this work, the in-medium nucleon-nucleon (NN) elastic cross-section is considered as a density- and momentum-dependent medium correction factor <math id="M7"><mrow><mi mathvariant="script">F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo></mrow></math> times the free nucleon-nucleon cross-section <math id="M8"><mrow><msubsup><mi>σ</mi><mrow><mtext>NN</mtext></mrow><mrow><mtext>free</mtext></mrow></msubsup></mrow></math> , which is available from the experimental data. <math display="block" id="M9"><mrow><msubsup><mi>σ</mi><mrow><mtext>NN</mtext></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>=</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mrow><mi>ρ</mi><mn>,</mn><mi>p</mi></mrow><mo>)</mo></mrow><mo>*</mo><msubsup><mi>σ</mi><mrow><mtext>NN</mtext></mrow><mrow><mtext>free</mtext></mrow></msubsup><mn>,</mn></mrow></math> with <math display="block" id="M10"><mrow><mi mathvariant="script">F</mi><mrow><mo>(</mo><mrow><mi>ρ</mi><mn>,</mn><mi>p</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>f</mi><mn>0</mn></msub><mn>,</mn><mtext>     </mtext><msub><mi>p</mi><mrow><mtext>NN</mtext></mrow></msub><mo>></mo><mn>1</mn><mtext>G</mtext><mi>e</mi><mi>V</mi><mo>/</mo><mi>c</mi><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mi>λ</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mi>ρ</mi><mrow><mi>ζ</mi><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow></msup><mo>−</mo><msub><mi>f</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>NN</mtext></mrow></msub><mo>/</mo><msub><mi>p</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mi>κ</mi></msup></mrow></mfrac><mo>+</mo><msub><mi>f</mi><mn>0</mn></msub><mn>,</mn><mtext>     </mtext><msub><mi>p</mi><mrow><mtext>NN</mtext></mrow></msub><mo>≤</mo><mn>1</mn><mtext>GeV</mtext><mo>/</mo><mi>c</mi><mn>.</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> The parameters f 0 = 1, λ = 1/6, ζ = 1/3, p 0 = 0.3 GeV/ c , and κ = 8 were adopted, with p NN being the momentum in the two-nucleon center-of-mass frame. Further details regarding the correction factor <math id="M11"><mrow><mi mathvariant="script">F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo></mrow></math> can be found in Refs. [ 41 41 , 42 42 , 50 50 ]. In addition to the in-medium NN cross-section, Pauli blocking plays an important role in determining the rate of collisions. The Pauli blocking treatment is the same as that described in Refs. [ 41 41 , 19 19 ]; it involves two steps. First, the phase space densities f i and f j of the two outgoing particles for each NN collision are calculated. <math display="block" id="M12"><mtable><mtr><mtd><msub><mi>f</mi><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder><mrow><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mi>π</mi><mi>ℏ</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac></mrow></mstyle><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo>−</mo><msub><mi>r</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><msubsup><mi>σ</mi><mi>r</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>−</mo><msub><mi>p</mi><mi>k</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>⋅</mo><mn>2</mn><msubsup><mi>σ</mi><mi>r</mi><mn>2</mn></msubsup></mrow><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> Here, <math id="M13"><mrow><msubsup><mi>σ</mi><mi>r</mi><mn>2</mn></msubsup></mrow></math> denotes the width parameter of the wave packet. <math id="M14"><mrow><msubsup><mi>σ</mi><mi>r</mi><mn>2</mn></msubsup></mrow></math> = 2 fm 2 is used in this study. k represents the nucleons of the same type around the outgoing nucleon i (or j ). The following two criteria were simultaneously considered: <math display="block" id="M15"><mrow><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac><msubsup><mi>r</mi><mrow><mi>i</mi><mi>k</mi></mrow><mn>3</mn></msubsup><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac><msubsup><mi>p</mi><mrow><mi>i</mi><mi>k</mi></mrow><mn>3</mn></msubsup><mo>≥</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>h</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mo>;</mo></mrow></math> <math display="block" id="M16"><mrow><msub><mi>P</mi><mrow><mtext>block</mtext></mrow></msub><mo>=</mo><mn>1</mn><mo>−</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>f</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>f</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow><mo><</mo><mi>ξ</mi><mn>.</mn></mrow></math> Here, r ik and p ik denote the relative distance and momentum between the nucleons i and k . The above conditions were also considered for nucleon j . The symbol ξ denotes a random number between 0 and 1. In addition, the isospin-dependent minimum spanning tree (iso-MST) algorithm was used to construct the clusters [ 36 36 - 38 38 36-38 ]. Nucleon pairs with relative distances smaller than R 0 and relative momenta smaller than P 0 = 0.25 GeV/ c are considered to be bound in a fragment. Here, <math id="M17"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mtext>pp</mtext></mrow></msubsup></mrow></math> = 2.8 fm and <math id="M18"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mtext>np</mtext></mrow></msubsup></mrow></math> = <math id="M19"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mtext>nn</mtext></mrow></msubsup></mrow></math> = 3.8 fm were used for proton-proton and neutron-neutron (neutron-proton) pairs, respectively. A fairly good agreement between the recently published experimental data (including collective flows, nuclear stopping power, and Hanbury–Brown–Twiss interferometry) and UrQMD model calculations was achieved with appropriate choices of the above parameters [ 45 45 , 35 35 , 42 42 , 51 51 , 52 52 ]. Statistical model: GEMINI++ 2 The statistical GEMINI++ decay model [ 30 30 , 53 53 ] is a Monte Carlo method used to simulate the decay of excited fragments, which includes light-particle evaporation, symmetric and asymmetric fission, and all possible binary decay modes. It has been widely used to treat the de-excitation of the primary fragments in studies of nuclear reactions at low and intermediate [ 23 23 , 25 25 , 54 54 - 60 60 54-60 ] and high energies [ 61 61 - 63 63 61-63 ]. The parameters of the shell-smoothed level density in GEMINI++ were set to their default values, which were k 0 = 7.3 MeV and <math id="M20"><mrow><msub><mi>k</mi><mi>∞</mi></msub></mrow></math> = 12 MeV. A more detailed discussion of the GEMINI++ parameters can be found in Refs. [ 60 60 , 53 53 , 64 64 ]. There are four inputs for the GEMINI++ code (with default parameter settings): the mass number A , charge number Z , excitation energy E * , and angular moment <math id="M21"><mover accent="true"><mi>L</mi><mo>→</mo></mover></math> of the primary fragment. After the UrQMD transport and iso-MST coalescence processes, the excitation energy of a fragment is calculated as follows: <math display="block" id="M22"><mrow><msup><mi>E</mi><mo>*</mo></msup><mo>=</mo><msubsup><mi>E</mi><mrow><mtext>bind</mtext></mrow><mrow><mtext>excited</mtext></mrow></msubsup><mo>−</mo><msubsup><mi>E</mi><mrow><mtext>bind</mtext></mrow><mrow><mtext>ground</mtext></mrow></msubsup><mn>,</mn></mrow></math> where <math id="M23"><mrow><msubsup><mi>E</mi><mrow><mtext>bind</mtext></mrow><mrow><mtext>excited</mtext></mrow></msubsup></mrow></math> is the binding energy of the primary fragment, that is, the sum of the potential and kinetic energies of all nucleons that belong to a fragment. <math id="M24"><mrow><msubsup><mi>E</mi><mrow><mtext>bind</mtext></mrow><mrow><mtext>ground</mtext></mrow></msubsup></mrow></math> is the binding energy of the ground state obtained from nuclear data table AME2020 [ 65 65 , 66 66 ]. When the excitation energy of the fragment was less than or equal to zero, it was regarded as a bound state and did not decay further. The angular momentum of the primary fragments was calculated using classical mechanics [ 56 56 ]. <math display="block" id="M25"><mrow><mover accent="true"><mi>L</mi><mo>→</mo></mover><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mi>i</mi></msub><mo>×</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>i</mi></msub></mrow></mstyle><mn>,</mn></mrow></math> where <math id="M26"><mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mi>i</mi></msub></mrow></math> and <math id="M27"><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>i</mi></msub></mrow></math> are the coordinate and momentum vectors, respectively, of the i -th nucleon in the primary fragments in the c.m. frame of the fragment. The total angular momentum is the sum of all the nucleons in the primary fragments. We verified that the results remained almost unchanged if the contribution of the angular momentum in GEMINI++ was not included. Observables 2 In this study, we focus on collective flows and nuclear stopping power, which are commonly used observables, and propose investigating the properties of dense nuclear matter. The directed ( v 1 ) and elliptic ( v 2 ) flows are defined as [ 67 67 ] <math display="block" id="M28"><mrow><msub><mi>v</mi><mn>1</mn></msub><mo>=</mo><mrow><mo>〈</mo><mrow><mfrac><mrow><msub><mi>p</mi><mi>x</mi></msub></mrow><mrow><msqrt><mrow><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></mfrac></mrow><mo>〉</mo></mrow><mn>,</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><msub><mi>v</mi><mn>2</mn></msub><mo>=</mo><mrow><mo>〈</mo><mrow><mfrac><mrow><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mrow><msqrt><mrow><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></mfrac></mrow><mo>〉</mo></mrow><mn>.</mn></mrow></math> Here, p x and p y are the two components of the transverse momentum p t = <math id="M29"><mrow><msqrt><mrow><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></math> . Angle brackets indicate the average of all considered particles from all events. The nuclear stopping power R E , which characterizes the transparency of the colliding nuclei, can be defined as [ 68 68 ] <math display="block" id="M30"><mrow><msub><mi>R</mi><mtext>E</mtext></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mo>∑</mo></mstyle><msub><mi>E</mi><mo>⊥</mo></msub></mrow><mrow><mn>2</mn><mstyle displaystyle="true"><mo>∑</mo></mstyle><msub><mi>E</mi><mo>∥</mo></msub></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M31"><mrow><msub><mi>E</mi><mo>⊥</mo></msub></mrow></math> ( <math id="M32"><mrow><msub><mi>E</mi><mo>∥</mo></msub></mrow></math> ) is the c.m. transverse (parallel) energy, and the sum runs over all considered particles. Another quantity vartl , which was proposed by the FOPI Collaboration [ 69 69 ], is widely used to measure stopping power. It is defined as the ratio of the variances of the transverse to those of the longitudinal rapidity distribution and is expressed as <math display="block" id="M33"><mrow><mi>v</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>l</mi><mo>=</mo><mfrac><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>y</mi><mi>x</mi><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow></mrow><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>y</mi><mi>z</mi><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M34"><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>y</mi><mi>x</mi><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow></mrow></math> and <math id="M35"><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>y</mi><mi>z</mi><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow></mrow></math> are the variances in the rapidity distributions of the particles in the x and z directions, respectively. Results and discussions 1 Rapidity distribution 2 The rapidity distributions of hydrogen ( Z = 1) and helium isotopes ( 3 He and 4 He) in 197 Au+ 197 Au collisions at E lab = 40 MeV/nucleon and impact parameter b = 5.5–7.5 fm are shown in Fig. 1 Fig. 1 . The left, middle, and right panels show the calculated results when the switching time of the UrQMD model was set to 200, 250, and 300 fm/ c , respectively. When considering the decay of the excited primary fragments, the yields of Z = 1 particles are slightly increased, whereas those of 3 He and 4 He clusters are clearly increased. The yields of 3 He and 4 He are approximately 1.5 and 6 times greater than those obtained from the simulations without considering sequential decay. It is known that the yields of light clusters such as 3 He and 4 He obtained using QMD-type codes are significantly smaller than experimentally measured yields. As discussed in our previous work [ 44 44 ], the yield of 3 He calculated using the UrQMD model was approximately three times lower than that of the experimental data. The inclusion of GEMINI++ improved the description of light cluster production to some extent; however, the yields of light clusters were still underestimated. Additionally, the inclusion of GEMINI++ may have led to an overestimation of the yield of free nucleons. To solve these problems further, additional issues should be considered, such as the spin degree of freedom and the dynamical production process [ 7 7 ]. Collective flows 2 The collective flows of Z = 1 particles as functions of y 0 in 197 Au+ 197 Au collisions at E lab = 40 MeV/nucleon are shown in Fig. 2 Fig. 2 (directed flow v 1 [left panel] and elliptic flow v 2 [right panel]). The simulations were terminated at 200 fm/ c . Clear differences are observed in the distributions of the collective flows between simulations with and without considering the decay of the primary fragments, especially in the case of elliptic flow v 2 from peripheral ( b = 5.5–7.5 fm) collisions. From Fig. 1 Fig. 1 , the difference in the yields of Z = 1 particles between simulations with and without considering sequential decay is due to the particles produced from the decay of excited primary fragments, which have larger mass. These heavier excited primary fragments are usually produced at the target/projectile rapidity; thus, the directed flow at the target/projectile rapidity is significantly affected by sequential decay. Increasing the impact parameter results in more large-mass primary fragments being produced and the effects of sequential decay on the collective flows becoming more evident. To quantitatively evaluate the influence of sequential decay on the collective flows, as shown in Fig. 3 Fig. 3 , the v 1 slope (top panels) and v 2 (bottom panels) at mid-rapidity for Z = 1 particles (left panels) and free protons (right panels) are calculated at different stopping times and compared with the experimental data. The slope of directed flow v 11 and elliptic flow v 20 at mid-rapidity are extracted by assuming <math id="M36"><mrow><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi><mrow><mn>10</mn></mrow></msub><mo>+</mo><msub><mi>v</mi><mrow><mn>11</mn></mrow></msub><mo>⋅</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>v</mi><mrow><mn>13</mn></mrow></msub><mo>⋅</mo><msubsup><mi>y</mi><mn>0</mn><mn>3</mn></msubsup></mrow></math> and <math id="M37"><mrow><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi><mrow><mn>20</mn></mrow></msub><mo>+</mo><msub><mi>v</mi><mrow><mn>22</mn></mrow></msub><mo>⋅</mo><msubsup><mi>y</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>v</mi><mrow><mn>24</mn></mrow></msub><mo>⋅</mo><msubsup><mi>y</mi><mn>0</mn><mn>4</mn></msubsup></mrow></math> in the range of <math id="M38"><mrow><mo>|</mo><msub><mi>y</mi><mn>0</mn></msub><mo>|</mo><mo><</mo><mn>0.4</mn></mrow></math> , in the same manner as that in the experimental report [ 69 69 - 71 71 69-71 ]. The left panels show the results from the simulations at E lab = 50 MeV/nucleon, whereas the right panels show the results from the simulations at E lab = 400 MeV/nucleon. There is an obvious difference in v 11 and v 20 between the simulations with and without considering the decay of primary fragments at E lab = 50 MeV/nucleon. By considering the decay of the primary fragments, the value of v 11 decreases, while that of v 20 increases, and both are closer to the experimental data. This can be understood from the fact that both directed and elliptic flows are stronger for large-mass fragments [ 70 70 , 69 69 , 33 33 ]. The decay of fragments is isotropic in the statistical model. Thus, the light nuclei produced from the decay of primary fragments retain information about the primary fragments. Furthermore, at E lab = 50 MeV/nucleon, v 11 first decreases with time and then saturates (increases) after 200 fm/ c in the case without (with) the sequential decay effect. This can be explained by the competition between sequential decay and dynamical effects. At approximately t = 150 fm/ c , the nucleons are located in a low-density environment, and the final state interactions are attractive; thus, a decrease v 11 is observed. When considering GEMINI++ at t = 300 fm/ c , there are only a few large-mass fragments that can emit hydrogen isotopes, and the effect of sequential decay on v 11 gradually weakens. A comparison of the sequential decay effects at E lab = 50 MeV/nucleon with those at E lab = 400 MeV/nucleon shows that the sequential decay effects are more pronounced at lower beam energies. This is understandable because more large-mass fragments, which contribute to the production of light clusters, are formed at lower beam energies. Figure 4 Figure 4 shows the beam energy dependence of v 11 (top panel) and v 20 (bottom panel) for semi-central 197 Au+ 197 Au collisions. The value of v 11 from the simulations without considering sequential decay is higher than that of the experimental data at lower energies. When sequential decay is included, v 11 is driven down, and v 20 is pulled up at low energies, which is consistent with the results shown in Fig. 3 Fig. 3 . Again, both v 11 and v 20 are considerably closer to the experimental data for all investigated energies. Moreover, the effects of sequential decay on the collective flows decrease with increasing beam energy. With increasing beam energy, the reaction becomes more violent, and the multiplicities of large-mass primary fragments decrease. For a systematic investigation of the effects of sequential decay on the collective flows, the transverse 4-velocities u t0 dependence of the v 1 of light charged particles in semi-central 197 Au+ 197 Au collisions at E lab = 120, 150, and 250 MeV/nucleon are calculated with and without considering the decay of primary fragments and compared with the experimental data [ 69 69 ], as shown in Fig. 5 Fig. 5 . The effects of sequential decay on v 1 ( u t0 ) of particles with mass number A lt;4 are relatively weak; it only affects v 1 at low u t0 . By considering the sequential decay effect, v 1 of α particles is obviously influenced and can reproduce the experimental data reasonably well at higher u t0 . This is because after the decay of excited primary fragments is considered, the multiplicity of α particles is enhanced, as shown in Fig. 1 Fig. 1 . The α particles produced by the decay of the excited primary fragments have a large flow effect because they inherit the flow information of the excited primary fragments. The remaining discrepancies in u t0 -dependent v 1 of light clusters may be due to simplifications in the initial wave function of particles (nucleons and possible clusters) and quantum effects in two-body collisions, as well as the lack of a dynamical production process, which requires further study [ 7 7 ]. Nuclear stopping power 2 The nuclear stopping power characterizes the transparency of the colliding nuclei and can provide insight into the rate of equilibration of the colliding system. Thus, it is meaningful to explore the effects of sequential decay on the nuclear stopping power. The yield distributions of deuterons and 4 He clusters as functions of the reduced longitudinal ( <math id="M39"><mrow><msub><mi>y</mi><mi>z</mi></msub><mo>/</mo><msub><mi>y</mi><mrow><mtext>pro</mtext></mrow></msub></mrow></math> ) and transverse ( <math id="M40"><mrow><msub><mi>y</mi><mi>x</mi></msub><mo>/</mo><msub><mi>y</mi><mrow><mtext>pro</mtext></mrow></msub></mrow></math> ) rapidities for central 197 Au+ 197 Au collisions at E lab = 40 MeV/nucleon with and without considering sequential decay are shown in Fig. 6 Fig. 6 . The values of the calculated R E are shown in each panel, and they slightly decrease; that is, the nuclear stopping power is depressed when considering the sequential decay. Notably, the values of R E are slightly influenced by sequential decay; however, the yield spectrum of each particle is obviously influenced, especially for 4 He clusters. In conjunction with the results presented in Fig. 1 Fig. 1 , it can be concluded that sequential decay has a strong effect on the particle yield and yield distribution; however, the nuclear stopping power is only weakly influenced because the de-excitation of the primary fragments in the GEMINI++ code is isotropic. The nuclear stopping power R E ( E lab = 40 MeV/nucleon, left panel) and vartl ( E lab = 400 MeV/nucleon, right panel) for free protons from central ( b = 0–2 fm) 197 Au+ 197 Au collisions calculated at different stopping times are shown in Fig. 7 Fig. 7 . Similar to the collective flows shown in Fig. 3 Fig. 3 , the effects of sequential decay on the nuclear stopping power at lower energies are more obvious than those at higher energies. In all cases, the results were saturated and were close to or covered by the experimental data above 200 fm/ c . Figure 8 Figure 8 shows the degree of nuclear stopping ( R E or vartl ) in the central 197 Au+ 197 Au collisions as a function of the beam energy. It is clear that both R E and vartl decrease and are much closer to the experimental data when the GEMINI++ code is applied, and the difference in the values of the nuclear stopping power gradually disappears with increasing beam energy. With the same impact parameter (centrality), more violent reactions occurred at higher beam energies, and less heavily excited primary fragments were produced. The difference with free protons is that they are produced from the decay of heavy excited primary fragments, which usually have a weak stopping power (smaller values of R E and vartl ) [ 74 74 , 73 73 ]. As a result, the nuclear stopping power was slightly reduced by including the GEMINI++ code. Summary and outlook 1 In summary, the effects of the sequential decay of excited primary fragments on the rapidity distribution, collective flows of light nuclei, and nuclear stopping power in Au+Au collisions at intermediate energies were investigated. Primary fragments were produced using the UrQMD model, and sequential decay of the excited primary fragments was performed using the GEMINI++ code. It was observed that the sequential decay of the excited primary fragments have an obvious influence on the rapidity distribution and collective flows but relatively weakly affect the stopping power. This is because of the memory effect; that is, the light particles produced from the sequential decay of the excited primary fragments inherit the collective properties of the excited primary fragments. Furthermore, the sequential decay effects gradually decreased with increasing beam energy because fewer large-mass fragments that can emit light particles were produced at higher beam energies. More importantly, the ability to reproduce the relevant experimental data at the investigated energies was improved to a certain extent by the inclusion of sequential decay effects. These results are meaningful for extracting the properties of nuclear matter by comparing transport model calculations with experimental data. The effects of secondary sequential decay on the observables, such as the N/Z ratios and elliptic flow ratios of neutrons vs. hydrogen isotopes, will be discussed in future publications. References G.F. Bertsch , S. Das Gupta , A Guide to microscopic models for intermediate-energy heavy ion collisions . Phys. Rept. 160 , 189 - 233 ( 1988 ). doi: 10.1016/0370-1573(88)90170-6 http://doi.org/10.1016/0370-1573(88)90170-6 A Guide to microscopic models for intermediate-energy heavy ion collisions J. Aichelin , ’Quantum’ molecular dynamics: A Dynamical microscopic n body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions . Phys. Rept. 202 , 233 - 360 ( 1991 ). doi: 10.1016/0370-1573(91)90094-3 http://doi.org/10.1016/0370-1573(91)90094-3 ’Quantum’ molecular dynamics: A Dynamical microscopic n body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions P. Danielewicz , R. Lacey , W.G. Lynch , Determination of the equation of state of dense matter . Science 298 , 1592 - 1596 ( 2002 ). doi: 10.1126/science.1078070 http://doi.org/10.1126/science.1078070 Determination of the equation of state of dense matter B.A. Li , L.W. Chen , C.M. Ko , Recent progress and new challenges in isospin physics with heavy-ion reactions . Physics Reports 464 , 113 - 281 ( 2008 ). doi: 10.1016/j.physrep.2008.04.005 http://doi.org/10.1016/j.physrep.2008.04.005 Recent progress and new challenges in isospin physics with heavy-ion reactions J. Xu , Transport approaches for the description of intermediate-energy heavy-ion collisions . Progress in Particle and Nuclear Physics 106 , 312 - 359 ( 2019 ). doi: 10.1016/j.ppnp.2019.02.009 http://doi.org/10.1016/j.ppnp.2019.02.009 Transport approaches for the description of intermediate-energy heavy-ion collisions D. Guo , et al. , An FPGA-based trigger system for CSHINE . Nucl. Sci. Tech. 33 , 162 ( 2022 ). doi: 10.1007/s41365-022-01149-0 http://doi.org/10.1007/s41365-022-01149-0 An FPGA-based trigger system for CSHINE A. Ono , Dynamics of clusters and fragments in heavy-ion collisions . Prog. Part. Nucl. Phys. 105 , 139 - 179 ( 2019 ). doi: 10.1016/j.ppnp.2018.11.001 http://doi.org/10.1016/j.ppnp.2018.11.001 Dynamics of clusters and fragments in heavy-ion collisions M. Bleicher , E. Bratkovskaya , Modelling relativistic heavy-ion collisions with dynamical transport approaches . Progress in Particle and Nuclear Physics 122 , 103920 ( 2022 ). doi: 10.1016/j.ppnp.2021.103920 http://doi.org/10.1016/j.ppnp.2021.103920 Modelling relativistic heavy-ion collisions with dynamical transport approaches M. Colonna , Collision dynamics at medium and relativistic energies . Progress in Particle and Nuclear Physics 113 , 103775 ( 2020 ). doi: 10.1016/j.ppnp.2020.103775 http://doi.org/10.1016/j.ppnp.2020.103775 Collision dynamics at medium and relativistic energies S. Huth , P.T. Pang , I. Tews , et al. , Constraining neutron-star matter with microscopic and macroscopic collisions . Nature 606 , 276 - 280 ( 2022 ). doi: 10.1038/s41586-022-04750-w http://doi.org/10.1038/s41586-022-04750-w Constraining neutron-star matter with microscopic and macroscopic collisions S.W. Lan , S.S. Shi , Anisotropic flow in high baryon density region . Nucl. Sci. Tech. 33 , 21 ( 2022 ). doi: 10.1007/s41365-022-01006-0 http://doi.org/10.1007/s41365-022-01006-0 Anisotropic flow in high baryon density region C.Z. Shi , Y.G. Ma , α -clustering effect on flows of direct photons in heavy-ion collisions . Nucl. Sci. Tech. 32 , 66 ( 2021 ). doi: 10.1007/s41365-021-00897-9 http://doi.org/10.1007/s41365-021-00897-9 α-clustering effect on flows of direct photons in heavy-ion collisions Z.W. Lin , L. Zheng , Further developments of a multi-phase transport model for relativistic nuclear collisions . Nuclear Science and Techniques 32 , 113 ( 2021 ). doi: 10.1007/s41365-021-00944-5 http://doi.org/10.1007/s41365-021-00944-5 Further developments of a multi-phase transport model for relativistic nuclear collisions H. Wang , J.H. Chen , Study on open charm hadron production and angular correlation in high-energy nuclear collisions . Nucl. Sci. Tech. 32 , 2 ( 2021 ). doi: 10.1007/s41365-020-00839-x http://doi.org/10.1007/s41365-020-00839-x Study on open charm hadron production and angular correlation in high-energy nuclear collisions Y.J. Wang , et al. , CSHINE for studies of HBT correlation in Heavy Ion Reactions . Nucl. Sci. Tech. 32 , 4 ( 2021 ). doi: 10.1007/s41365-020-00842-2 http://doi.org/10.1007/s41365-020-00842-2 CSHINE for studies of HBT correlation in Heavy Ion Reactions C. Liu , X.G. Deng , Y.G. Ma , Density fluctuations in intermediate-energy heavy-ion collisions . Nucl. Sci. Tech. 33 , 52 ( 2022 ). doi: 10.1007/s41365-022-01040-y http://doi.org/10.1007/s41365-022-01040-y Density fluctuations in intermediate-energy heavy-ion collisions R.S. Wang , L. Ou , Z.G. Xiao , Production of high-energy neutron beam from deuteron breakup . Nucl. Sci. Tech. 33 , 92 ( 2022 ). doi: 10.1007/s41365-022-01075-1 http://doi.org/10.1007/s41365-022-01075-1 Production of high-energy neutron beam from deuteron breakup J. Xu , et al. , Understanding transport simulations of heavy-ion collisions at 100A and 400A MeV: Comparison of heavy-ion transport codes under controlled conditions . Phys. Rev. C 93 , 044609 ( 2016 ). doi: 10.1103/PhysRevC.93.044609 http://doi.org/10.1103/PhysRevC.93.044609 Understanding transport simulations of heavy-ion collisions at 100A and 400A MeV: Comparison of heavy-ion transport codes under controlled conditions H. Wolter , et al. , Transport model comparison studies of intermediate-energy heavy-ion collisions . Prog. Part. Nucl. Phys. 125 , 103962 ( 2022 ). doi: 10.1016/j.ppnp.2022.103962 http://doi.org/10.1016/j.ppnp.2022.103962 Transport model comparison studies of intermediate-energy heavy-ion collisions M. Colonna , et al. , Comparison of heavy-ion transport simulations: Mean-field dynamics in a box . Phys. Rev. C 104 , 024603 ( 2021 ). doi: 10.1103/PhysRevC.104.024603 http://doi.org/10.1103/PhysRevC.104.024603 Comparison of heavy-ion transport simulations: Mean-field dynamics in a box A. Ono , et al. , Comparison of heavy-ion transport simulations: Collision integral with pions and Δ resonances in a box . Phys. Rev. C 100 , 044617 ( 2019 ). doi: 10.1103/PhysRevC.100.044617 http://doi.org/10.1103/PhysRevC.100.044617 Comparison of heavy-ion transport simulations: Collision integral with pions and Δ resonances in a box Y.X. Zhang , et al. , Comparison of heavy-ion transport simulations: Collision integral in a box . Phys. Rev. C 97 , 034625 ( 2018 ). doi: 10.1103/PhysRevC.97.034625 http://doi.org/10.1103/PhysRevC.97.034625 Comparison of heavy-ion transport simulations: Collision integral in a box W.D. Tian , et al. , Dynamical and sequential decay effects on isoscaling and density dependence of the symmetry energy . Int. J. Mod. Phys. E 17 , 1705 - 1719 ( 2008 ). doi: 10.1142/S0218301308010714 http://doi.org/10.1142/S0218301308010714 Dynamical and sequential decay effects on isoscaling and density dependence of the symmetry energy J. Su , W. Trautmann , L. Zhu , et al. , Dynamical properties and secondary decay effects of projectile fragmentations in Sn124,Sn107+Sn120 collisions at 600 MeV/nucleon . Phys. Rev. C 98 , 014610 ( 2018 ). doi: 10.1103/PhysRevC.98.014610 http://doi.org/10.1103/PhysRevC.98.014610 Dynamical properties and secondary decay effects of projectile fragmentations in Sn124,Sn107+Sn120 collisions at 600 MeV/nucleon C.C. Guo , J. Su , L. Zhu , Secondary decay effects of the isospin fractionation in the projectile fragmentation at GeV/nucleon . Nucl. Sci. Tech. 31 , 123 ( 2020 ). doi: 10.1007/s41365-020-00832-4 http://doi.org/10.1007/s41365-020-00832-4 Secondary decay effects of the isospin fractionation in the projectile fragmentation at GeV/nucleon M.B. Tsang , et al. , Comparisons of Statistical Multifragmentation and Evaporation Models for Heavy Ion Collisions . Eur. Phys. J. A 30 , 129 - 139 ( 2006 ). [Erratum: Eur.Phys.J.A 32, 243 (2007)] . doi: 10.1140/epja/i2007-10389-2 http://doi.org/10.1140/epja/i2007-10389-2 Comparisons of Statistical Multifragmentation and Evaporation Models for Heavy Ion Collisions J.P. Bondorf , R. Donangelo , I.N. Mishustin , et al. , Statistical multifragmentation of nuclei . Nucl. Phys. A 443 , 321 - 347 ( 1985 ). doi: 10.1016/0375-9474(85)90266-0 http://doi.org/10.1016/0375-9474(85)90266-0 Statistical multifragmentation of nuclei J.P. Bondorf , A.S. Botvina , A.S. Ilinov , et al. , Statistical multifragmentation of nuclei . Phys. Rept. 257 , 133 - 221 ( 1995 ). doi: 10.1016/0370-1573(94)00097-M http://doi.org/10.1016/0370-1573(94)00097-M Statistical multifragmentation of nuclei W. Reisdorf , Analysis of fissionability data at high excitation energies . Zeitschrift für Physik A Atoms and Nuclei 300 , 227 - 238 ( 1981 ). doi: 10.1007/BF01412298 http://doi.org/10.1007/BF01412298 Analysis of fissionability data at high excitation energies R.J. Charity , et al. , Systematics of complex fragment emission in niobium-induced reactions . Nucl. Phys. A 483 , 371 - 405 ( 1988 ). doi: 10.1016/0375-9474(88)90542-8 http://doi.org/10.1016/0375-9474(88)90542-8 Systematics of complex fragment emission in niobium-induced reactions R.K. Puri , C. Hartnack , J. Aichelin , Early fragment formation in heavy ion collisions . Phys. Rev. C 54 , R28 - R31 ( 1996 ). doi: 10.1103/PhysRevC.54.R28 http://doi.org/10.1103/PhysRevC.54.R28 Early fragment formation in heavy ion collisions R.K. Puri , J. Aichelin , Simulated annealing clusterization algorithm for studying the multifragmentation . J. Comput. Phys. 162 , 245 - 266 ( 2000 ). doi: 10.1006/jcph.2000.6534 http://doi.org/10.1006/jcph.2000.6534 Simulated annealing clusterization algorithm for studying the multifragmentation Y. Wang , C. Guo , Q. Li , et al. , Collective flows of light particles in the Au+Au collisions at intermediate energies . Phys. Rev. C 89 , 034606 ( 2014 ). doi: 10.1103/PhysRevC.89.034606 http://doi.org/10.1103/PhysRevC.89.034606 Collective flows of light particles in the Au+Au collisions at intermediate energies P.C. Li , Y.J. Wang , Q.F. Li , et al. , Collective flow and nuclear stopping in heavy ion collisions in Fermi energy domain . Nucl. Sci. Tech. 29 , 177 ( 2018 ). doi: 10.1007/s41365-018-0510-1 http://doi.org/10.1007/s41365-018-0510-1 Collective flow and nuclear stopping in heavy ion collisions in Fermi energy domain P. Li , Y. Wang , Q. Li , et al. , Accessing the in-medium effects on nucleon-nucleon elastic cross section with collective flows and nuclear stopping . Phys. Lett. B 828 , 137019 ( 2022 ). doi: 10.1016/j.physletb.2022.137019 http://doi.org/10.1016/j.physletb.2022.137019 Accessing the in-medium effects on nucleon-nucleon elastic cross section with collective flows and nuclear stopping Q. Li , Y. Wang , X. Wang , et al. , Influence of coalescence parameters on the production of protons and Helium-3 fragments . Sci. China Phys. Mech. Astron. 59 , 672013 ( 2016 ). doi: 10.1007/s11433-016-0120-3 http://doi.org/10.1007/s11433-016-0120-3 Influence of coalescence parameters on the production of protons and Helium-3 fragments Q. Li , Y. Wang , X. Wang , et al. , Helium-3 production from Pb+Pb collisions at SPS energies with the UrQMD model and the traditional coalescence afterburner . Sci. China Phys. Mech. Astron. 59 , 632002 ( 2016 ). doi: 10.1007/s11433-015-5775-3 http://doi.org/10.1007/s11433-015-5775-3 Helium-3 production from Pb+Pb collisions at SPS energies with the UrQMD model and the traditional coalescence afterburner Q. Li , Y. Wang , X. Wang , et al. , Rapidity distribution of protons from the potential version of UrQMD model and the traditional coalescence afterburner . Sci. China Phys. Mech. Astron. 59 , 622001 ( 2016 ). doi: 10.1007/s11433-015-5768-2 http://doi.org/10.1007/s11433-015-5768-2 Rapidity distribution of protons from the potential version of UrQMD model and the traditional coalescence afterburner S.A. Bass , et al. , Microscopic models for ultrarelativistic heavy ion collisions . Prog. Part. Nucl. Phys. 41 , 255 - 369 ( 1998 ). doi: 10.1016/S0146-6410(98)00058-1 http://doi.org/10.1016/S0146-6410(98)00058-1 Microscopic models for ultrarelativistic heavy ion collisions M. Bleicher , et al. , Relativistic hadron hadron collisions in the ultrarelativistic quantum molecular dynamics model . J. Phys. G 25 , 1859 - 1896 ( 1999 ). doi: 10.1088/0954-3899/25/9/308 http://doi.org/10.1088/0954-3899/25/9/308 Relativistic hadron hadron collisions in the ultrarelativistic quantum molecular dynamics model Q. Li , C. Shen , C. Guo , et al. , Nonequilibrium dynamics in heavy-ion collisions at low energies available at the GSI Schwerionen Synchrotron . Phys. Rev. C 83 , 044617 ( 2011 ). doi: 10.1103/PhysRevC.83.044617 http://doi.org/10.1103/PhysRevC.83.044617 Nonequilibrium dynamics in heavy-ion collisions at low energies available at the GSI Schwerionen Synchrotron Y.J. Wang , Q.F. Li , Application of microscopic transport model in the study of nuclear equation of state from heavy ion collisions at intermediate energies . Front. Phys. (Beijing) 15 , 44302 ( 2020 ). doi: 10.1007/s11467-020-0964-6 http://doi.org/10.1007/s11467-020-0964-6 Application of microscopic transport model in the study of nuclear equation of state from heavy ion collisions at intermediate energies Y. Zhang , N. Wang , Q.F. Li , et al. , Progress of quantum molecular dynamics model and its applications in heavy ion collisions . Front. Phys. (Beijing) 15 , 54301 ( 2020 ). doi: 10.1007/s11467-020-0961-9 http://doi.org/10.1007/s11467-020-0961-9 Progress of quantum molecular dynamics model and its applications in heavy ion collisions Y. Wang , C. Guo , Q. Li , et al. , 3 H/ 3 He ratio as a probe of the nuclear symmetry energy at sub-saturation densities . Eur. Phys. J. A 51 , 37 ( 2015 ). doi: 10.1140/epja/i2015-15037-8 http://doi.org/10.1140/epja/i2015-15037-8 3H/3He ratio as a probe of the nuclear symmetry energy at sub-saturation densities Y. Wang , C. Guo , Q. Li , et al. , Determination of the nuclear incompressibility from the rapidity-dependent elliptic flow in heavy-ion collisions at beam energies 0.4 A –1.0 A GeV . Phys. Lett. B 778 , 207 - 212 ( 2018 ). doi: 10.1016/j.physletb.2018.01.035 http://doi.org/10.1016/j.physletb.2018.01.035 Determination of the nuclear incompressibility from the rapidity-dependent elliptic flow in heavy-ion collisions at beam energies 0.4 A –1.0 A GeV J.F. Xu , C.J. Xia , Z.Y. Lu , et al. , Symmetry energy of strange quark matter and tidal deformability of strange quark stars . Nucl. Sci. Tech. 33 , 143 ( 2022 ). doi: 10.1007/s41365-022-01130-x http://doi.org/10.1007/s41365-022-01130-x Symmetry energy of strange quark matter and tidal deformability of strange quark stars G.F. Wei , X. Huang , Q.J. Zhi , et al. , Effects of the momentum dependence of nuclear symmetry potential on pion observables in Sn + Sn collisions at 270 MeV/nucleon . Nucl. Sci. Tech. 33 , 163 ( 2022 ). doi: 10.1007/s41365-022-01146-3 http://doi.org/10.1007/s41365-022-01146-3 Effects of the momentum dependence of nuclear symmetry potential on pion observables in Sn + Sn collisions at 270 MeV/nucleon J. Liu , C. Gao , N. Wan , et al. , Basic quantities of the equation of state in isospin asymmetric nuclear matter . Nucl. Sci. Tech. 32 , 117 ( 2021 ). doi: 10.1007/s41365-021-00955-2 http://doi.org/10.1007/s41365-021-00955-2 Basic quantities of the equation of state in isospin asymmetric nuclear matter Y. Wang , Q. Li , Y. Leifels , et al. , Study of the nuclear symmetry energy from the rapidity-dependent elliptic flow in heavy-ion collisions around 1 GeV/nucleon regime . Physics Letters B 802 , 135249 ( 2020 ). doi: 10.1016/j.physletb.2020.135249 http://doi.org/10.1016/j.physletb.2020.135249 Study of the nuclear symmetry energy from the rapidity-dependent elliptic flow in heavy-ion collisions around 1 GeV/nucleon regime P. Li , Y. Wang , Q. Li , et al. , Effects of the in-medium nucleon-nucleon cross section on collective flow and nuclear stopping in heavy-ion collisions in the Fermi-energy domain . Phys. Rev. C 97 , 044620 ( 2018 ). doi: 10.1103/PhysRevC.97.044620 http://doi.org/10.1103/PhysRevC.97.044620 Effects of the in-medium nucleon-nucleon cross section on collective flow and nuclear stopping in heavy-ion collisions in the Fermi-energy domain P. Li , Y. Wang , Q. Li , et al. , Transport model analysis of the pion interferometry in au+au collisions at e beam =1.23 gev/nucleon . SCIENCE CHINA Physics, Mechanics & Astronomy 66 , 222011 ( 2023 ). doi: 10.1007/s11433-022-2026-5 http://doi.org/10.1007/s11433-022-2026-5 Transport model analysis of the pion interferometry in au+au collisions at ebeam=1.23 gev/nucleon P. Li , J. Steinheimer , T. Reichert , et al. , Effects of a phase transition on two-pion interferometry in heavy ion collisions at <math id="M41"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.4</mn></mrow></math> –7.7 gev . SCIENCE CHINA Physics, Mechanics & Astronomy 66 , 232011 ( 2023 ). doi: 10.1007/s11433-022-2041-8 http://doi.org/10.1007/s11433-022-2041-8 Effects of a phase transition on two-pion interferometry in heavy ion collisions at sNN=2.4 –7.7 gev R.J. Charity , A Systematic description of evaporation spectra for light and heavy compound nuclei . Phys. Rev. C 82 , 014610 ( 2010 ). doi: 10.1103/PhysRevC.82.014610 http://doi.org/10.1103/PhysRevC.82.014610 A Systematic description of evaporation spectra for light and heavy compound nuclei K. Hagel , et al. , Violent collisions and multifragment final states in the Ca-40 + Ca-40 reaction at 35 MeV/nucleon . Phys. Rev. C 50 , 2017 - 2034 ( 1994 ). doi: 10.1103/PhysRevC.50.2017 http://doi.org/10.1103/PhysRevC.50.2017 Violent collisions and multifragment final states in the Ca-40 + Ca-40 reaction at 35 MeV/nucleon J. Su , F.S. Zhang , Isotopic dependence of nuclear temperatures . Phys. Rev. C 84 , 037601 ( 2011 ). doi: 10.1103/PhysRevC.84.037601 http://doi.org/10.1103/PhysRevC.84.037601 Isotopic dependence of nuclear temperatures Z.T. Dai , D.Q. Fang , Y.G. Ma , et al. , Effect of neutron skin thickness on projectile fragmentation . Phys. Rev. C 91 , 034618 ( 2015 ). doi: 10.1103/PhysRevC.91.034618 http://doi.org/10.1103/PhysRevC.91.034618 Effect of neutron skin thickness on projectile fragmentation Y.G. Ma , R. Wada , K. Hagel , et al. , Sequential decay distortion of Goldhaber model widths for spectator fragments . Phys. Rev. C 65 , 051602 ( 2002 ). [Erratum: Phys.Rev.C 66, 029902 (2002)] . doi: 10.1103/PhysRevC.66.029902 http://doi.org/10.1103/PhysRevC.66.029902 Sequential decay distortion of Goldhaber model widths for spectator fragments M. Huang , et al. , Isobaric Yield Ratios and The Symmetry Energy In Fermi Energy Heavy Ion Reactions . Phys. Rev. C 81 , 044620 ( 2010 ). doi: 10.1103/PhysRevC.81.044620 http://doi.org/10.1103/PhysRevC.81.044620 Isobaric Yield Ratios and The Symmetry Energy In Fermi Energy Heavy Ion Reactions L. Li , F.Y. Wang , Y.X. Zhang , Isospin effects on intermediate mass fragments at intermediate energy-heavy ion collisions . Nucl. Sci. Tech. 33 , 58 ( 2022 ). doi: 10.1007/s41365-022-01050-w http://doi.org/10.1007/s41365-022-01050-w Isospin effects on intermediate mass fragments at intermediate energy-heavy ion collisions J. Su , L. Zhu , C. Guo , Influence of the nuclear level density on the odd-even staggering in Fe56+p spallation at energies from 300 to 1500 MeV/nucleon . Phys. Rev. C 97 , 054604 ( 2018 ). doi: 10.1103/PhysRevC.97.054604 http://doi.org/10.1103/PhysRevC.97.054604 Influence of the nuclear level density on the odd-even staggering in Fe56+p spallation at energies from 300 to 1500 MeV/nucleon L.M. Liu , C.J. Zhang , J. Xu , et al. , Free spectator nucleons in ultracentral relativistic heavy-ion collisions as a probe of neutron skin . Phys. Rev. C 106 , 034913 ( 2022 ). doi: 10.1103/PhysRevC.106.034913 http://doi.org/10.1103/PhysRevC.106.034913 Free spectator nucleons in ultracentral relativistic heavy-ion collisions as a probe of neutron skin X. Lei , E. Xiao , Y. Feng , et al. , Production of neutron-deficient nuclei around N= 126 by proton-induced spallation . Chin. Phys. C 47 , 014102 ( 2023 ). doi: 10.1088/1674-1137/ac9601 http://doi.org/10.1088/1674-1137/ac9601 Production of neutron-deficient nuclei around N= 126 by proton-induced spallation Q.F. Song , L. Zhu , J. Su , Target dependence of isotopic cross sections in the spallation reactions 238 U + p, d and 9 Be at 1 AGeV . Chin. Phys. C 46 , 074108 ( 2022 ). doi: 10.1088/1674-1137/ac6249 http://doi.org/10.1088/1674-1137/ac6249 Target dependence of isotopic cross sections in the spallation reactions 238U + p, d and 9Be at 1 AGeV D. Mancusi , R.J. Charity , J. Cugnon , Unified description of fission in fusion and spallation reactions . Phys. Rev. C 82 , 044610 ( 2010 ). doi: 10.1103/PhysRevC.82.044610 http://doi.org/10.1103/PhysRevC.82.044610 Unified description of fission in fusion and spallation reactions M. Wang , W.J. Huang , F.G. Kondev , et al. , The AME 2020 atomic mass evaluation (II) . Tables, graphs and references. Chin. Phys. C 45 , 030003 ( 2021 ). doi: 10.1088/1674-1137/abddaf http://doi.org/10.1088/1674-1137/abddaf The AME 2020 atomic mass evaluation (II) Z. Gao , Y. Wang , H. Lü , et al. , Machine learning the nuclear mass . Nuclear Science and Techniques 32 , 109 ( 2021 ). doi: 10.1007/s41365-021-00956-1 http://doi.org/10.1007/s41365-021-00956-1 Machine learning the nuclear mass W. Reisdorf , H.G. Ritter , Collective flow in heavy-ion collisions . Ann. Rev. Nucl. Part. Sci. 47 , 663 - 709 ( 1997 ). doi: 10.1146/annurev.nucl.47.1.663 http://doi.org/10.1146/annurev.nucl.47.1.663 Collective flow in heavy-ion collisions G. Lehaut , et al. , Study of Nuclear Stopping in Central Collisions at Intermediate Energies . Phys. Rev. Lett. 104 , 232701 ( 2010 ). doi: 10.1103/PhysRevLett.104.232701 http://doi.org/10.1103/PhysRevLett.104.232701 Study of Nuclear Stopping in Central Collisions at Intermediate Energies W. Reisdorf , et al. , Systematics of azimuthal asymmetries in heavy ion collisions in the 1 A GeV regime . Nucl. Phys. A 876 , 1 - 60 ( 2012 ). doi: 10.1016/j.nuclphysa.2011.12.006 http://doi.org/10.1016/j.nuclphysa.2011.12.006 Systematics of azimuthal asymmetries in heavy ion collisions in the 1 A GeV regime A. Andronic , J. Lukasik , W. Reisdorf , et al. , Systematics of Stopping and Flow in Au+Au Collisions . Eur. Phys. J. A 30 , 31 - 46 ( 2006 ). doi: 10.1140/epja/i2006-10101-2 http://doi.org/10.1140/epja/i2006-10101-2 Systematics of Stopping and Flow in Au+Au Collisions A. Le Fèvre , Y. Leifels , C. Hartnack , et al. , Origin of elliptic flow and its dependence on the equation of state in heavy ion reactions at intermediate energies . Phys. Rev. C 98 , 034901 ( 2018 ). doi: 10.1103/PhysRevC.98.034901 http://doi.org/10.1103/PhysRevC.98.034901 Origin of elliptic flow and its dependence on the equation of state in heavy ion reactions at intermediate energies O. Lopez , et al. , In-medium effects for nuclear matter in the Fermi energy domain . Phys. Rev. C 90 , 064602 ( 2014 ). [Erratum: Phys.Rev.C 90, 069903 (2014)] . doi: 10.1103/PhysRevC.90.064602 http://doi.org/10.1103/PhysRevC.90.064602 In-medium effects for nuclear matter in the Fermi energy domain W. Reisdorf , et al. , Systematics of central heavy ion collisions in the 1A GeV regime . Nucl. Phys. A 848 , 366 - 427 ( 2010 ). doi: 10.1016/j.nuclphysa.2010.09.008 http://doi.org/10.1016/j.nuclphysa.2010.09.008 Systematics of central heavy ion collisions in the 1A GeV regime W. Reisdorf , et al. , Nuclear stopping from 0.09-A-GeV to 1.93-A-GeV and its correlation to flow . Phys. Rev. Lett. 92 , 232301 ( 2004 ). doi: 10.1103/PhysRevLett.92.232301 http://doi.org/10.1103/PhysRevLett.92.232301 Nuclear stopping from 0.09-A-GeV to 1.93-A-GeV and its correlation to flow All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Kui Xiao and Peng-Cheng Li. The first draft of the manuscript was written by Kui XIao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors acknowledge support from computing server C3S2 at Huzhou University. 10.1007/s41365-023-01205-3 This work was partly supported by the National Natural Science Foundation of China (Nos. U2032145 and 11875125) and the National Key Research and Development Program of China (No. 2020YFE0202002). 2023-01-11 2023-02-20 2023-03-23 2023-04-27 2023-06-07 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-023-01205-3 http://dx.doi.org/10.1007/s41365-023-01205-3 . 2023 38408316 The data that support the findings of this study are openly available in Science Data Bank at https://www.doi.org/10.57760/sciencedb.07846 and http://resolve.pid21.cn/31253.11.sciencedb.07846 Data Availability Statement Nuclear Science and Techniques 4 34