Collective flow and nuclear stopping in heavy ion collisions in Fermi energy domain

Collective flow and nuclear stopping in heavy ion collisions in Fermi energy domain 1 2 first-author 2 corresp wangyongjia@zjhu.edu.cn; wangyongjia@zjhu.edu.cn; wangyongjia@zjhu.edu.cn 2 3 corresp liqf@zjhu.edu.cn liqf@zjhu.edu.cn liqf@zjhu.edu.cn 1 3 School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China School of Science, Huzhou University, Huzhou 313000, China School of Science, Huzhou University, Huzhou 313000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China The effects of the in-medium nucleon-nucleon ( NN ) elastic cross section on the observables in heavy ion collisions in the Fermi energy domain are investigated within the framework of the ultrarelativistic quantum molecular dynamics model. The results simulated using medium correction factors of <math id="M1"><mrow><mi mathvariant="script">F</mi><mo>=</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>/</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup><mo>=</mo><mn>0.2,</mn><mi> </mi><mn>0.3,</mn><mi> </mi><mn>0.5</mn></mrow></math> and the density- and momentum-dependent factor obtained from the FU3FP1 parametrization are compared with the FOPI and INDRA experimental data. It is found that the calculations using the correction factors <math id="M2"><mi mathvariant="script">F</mi></math> = 0.2 and 0.5 reproduce the experimental data (i.e., collective flow and nuclear stopping) at 40 and 150 MeV/nucleon, respectively. Calculations with the FU3FP1 parametrization can best fit these experimental data. These conclusions can be confirmed in both 197 Au+ 197 Au and 129 Xe+ 120 Sn. Nucleon-nucleon elastic cross section Heavy ion collisions Nuclear stopping Collective flow. 1 Introduction 1 The properties of hot, dense nuclear matter are usually extracted by comparing the observables in heavy ion collision (HIC) experiments with the corresponding results from transport model simulations. The most popular transport methods for studying HICs at low and intermediate energies are the quantum molecular dynamics (QMD) model [ 1 1 ], the Boltzmann (Vlasov)–Uehling–Uhlenbeck (BUU or VUU) model [ 2 2 ], and their improved versions [ 3 3 - 5 5 3-5 ]. The mean-field potential and nucleon-nucleon cross section (NNCS) are two essential components of these models. The mean-field potential in transport models has been studied extensively [ 6 6 - 8 8 6-8 ]. Regarding the NNCS, it is we ll-known that compared with the free NNCS, the in-medium NNCS is suppressed [ 9 9 - 24 24 9-24 ]; however, the degree of suppression is still uncertain and requires further improvement. Thus, the in-medium NNCS has been studied by many groups using different methods [ 25 25 - 34 34 25-34 ]. Recently, a systematic experimental study of nuclear stopping in central collisions at intermediate energies pointed out that the correction factor <math id="M3"><mrow><mi mathvariant="script">F</mi><mo>=</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>/</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup></mrow></math> is deduced to be 0.16±0.04 at 40 MeV/nucleon and 0.5±0.06 at 100 MeV/nucleon [ 35 35 ]. New opportunities emerged to study the in-medium NNCS using transport models based on these stopping data [ 36 36 - 39 39 36-39 ]. There are many parameterized in-medium NNCSs adopted by various models. For example, <math id="M4"><mrow><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mi>ρ</mi><mo>/</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup></mrow></math> with η =0.2 has been adopted in the BUU, relativistic BUU, and improved QMD models [ 10 10 , 18 18 ]. In the pBUU model, the in-medium NNCS is independent of the density, energy, and isospin, and reads as <math id="M5"><mrow><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>=</mo><mn>0.85</mn><msup><mi>ρ</mi><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>/</mo><mi>tanh</mi><mo stretchy="false">(</mo><mfrac><mrow><msup><mi>σ</mi><mrow><mtext>free</mtext></mrow></msup></mrow><mrow><mn>0.85</mn><msup><mi>ρ</mi><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></mrow></math> [ 21 21 ]. In the isospin-dependent BUU model and the Lanzhou QMD model [ 40 40 ], the NNCS in the nuclear medium is corrected by a factor of <math id="M6"><mrow><mi mathvariant="script">F</mi><mo>=</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>/</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mo>*</mo></msubsup><mo>/</mo><msub><mi>μ</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></math> . Here μ NN and <math id="M7"><mrow><msubsup><mi>μ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mo>*</mo></msubsup></mrow></math> are the κ masses of the colliding nucleon partners in free space and in the nuclear medium, respectively [ 16 16 , 22 22 , 41 41 ]. In our previous works, we found that the collective flow and nuclear stopping can be reproduced using the parameterized medium correction factor <math id="M8"><mi mathvariant="script">F</mi></math> (which depends on the density and momentum) for the free NNCS within the ultrarelativistic QMD (UrQMD) model [ 42 42 - 44 44 42-44 ]. It is desirable to evaluate the differences between the <math id="M9"><mi mathvariant="script">F</mi></math> values extracted from experimental data [ 35 35 ] and the parameterized one adopted currently in the UrQMD model. This paper is organized as follows. In Sec. 2, we review the UrQMD model with the improved potential. Sec. 3 presents an investigation of the effects of the in-medium NNCS on the observables of free protons and hydrogen isotopes in HICs in the Fermi energy domain. Finally, a summary is given in Sec. 4. 2 Model description and observables 1 The UrQMD model is based on the same principles as the QMD model. In the UrQMD model, more than 55 different baryon and 32 different meson species, as well as the corresponding antiparticle and isospin-projected states, are considered. The model has been extensively and successfully used to study the nuclear reactions of p + p , p + A, and A + A systems within a large range of beam energies, from the low-energy regime of the INDRA/GSI experiments up to the highest energies presently available at LHC/CERN. In addition, the other two main features unique to the collision term of the UrQMD model are the unique collision time for each individual collision and the two-step particle production process. More discussions can be found in recent works on the transport model comparison project, i.e., Refs. [ 3 3 , 4 4 ]. At intermediate energies, the potential used in the UrQMD model depends on the momentum and density [ 1 1 , 42 42 , 45 45 ], and reads as <math display="block" id="M10"><mrow><mi>U</mi><mo>=</mo><mi>α</mi><mo>⋅</mo><mo stretchy="false">(</mo><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo>⋅</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mi>γ</mi></msup><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><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><msub><mi>a</mi><mrow><mtext>md</mtext></mrow></msub><msup><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mi>i</mi></msub><mo>−</mo><msub><mi>p</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mn>.</mn></mrow></math> Here α = -393 MeV, β = 320 MeV, γ = 1.14, t md = 1.57 MeV, and a md =0.0005 MeV -2 ; these values yield an incompressibility K 0 of 200 MeV for isospin-symmetric nuclear matter. In order to better describe the recent experimental data, the surface, surface asymmetry energy, and symmetry energy terms obtained from the Skyrme potential energy density functional have been further introduced into the present version [ 44 44 , 46 46 , 47 47 ]: <math display="block" id="M11"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>u</mi><mrow><mtext>Skyrme</mtext></mrow></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>u</mi><mrow><mtext>sur</mtext></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><mtext>sur,iso</mtext></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><mtext>sym</mtext></mrow></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><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 stretchy="false">(</mo><mo>∇</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><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 stretchy="false">[</mo><mo>∇</mo><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>n</mi></msub><mo>−</mo><msub><mi>ρ</mi><mi>p</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mo stretchy="false">(</mo><msub><mi>A</mi><mrow><mtext>sym</mtext></mrow></msub><mfrac><mrow><msup><mi>ρ</mi><mn>2</mn></msup></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>+</mo><msub><mi>B</mi><mrow><mtext>sym</mtext></mrow></msub><mfrac><mrow><msup><mi>ρ</mi><mrow><mi>η</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><msubsup><mi>ρ</mi><mn>0</mn><mi>η</mi></msubsup></mrow></mfrac><mo>+</mo><msub><mi>C</mi><mrow><mtext>sym</mtext></mrow></msub><mfrac><mrow><msup><mi>ρ</mi><mrow><mn>8</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msubsup><mi>ρ</mi><mn>0</mn><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">)</mo><msup><mi>δ</mi><mn>2</mn></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> <math display="block" id="M12"><mrow><mfrac><mrow><msub><mi>g</mi><mrow><mtext>sur</mtext></mrow></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>64</mn></mrow></mfrac><mo stretchy="false">(</mo><mn>9</mn><msub><mi>t</mi><mn>1</mn></msub><mo>−</mo><mn>5</mn><msub><mi>t</mi><mn>2</mn></msub><mo>−</mo><mn>4</mn><msub><mi>x</mi><mn>2</mn></msub><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>ρ</mi><mn>0</mn></msub><mn>,</mn></mrow></math> <math display="block" id="M13"><mrow><mfrac><mrow><msub><mi>g</mi><mrow><mtext>sur,iso</mtext></mrow></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>64</mn></mrow></mfrac><mo stretchy="false">[</mo><mn>3</mn><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>ρ</mi><mn>0</mn></msub><mn>,</mn></mrow></math> <math display="block" id="M14"><mrow><msub><mi>A</mi><mrow><mtext>sym</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow><mn>4</mn></mfrac><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msub><mi>ρ</mi><mn>0</mn></msub><mn>,</mn></mrow></math> <math display="block" id="M15"><mrow><msub><mi>B</mi><mrow><mtext>sym</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>t</mi><mn>3</mn></msub></mrow><mrow><mn>24</mn></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msubsup><mi>ρ</mi><mn>0</mn><mi>η</mi></msubsup><mn>,</mn></mrow></math> <math display="block" id="M16"><mrow><msub><mi>C</mi><mrow><mtext>sym</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>24</mn></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><mn>3</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msubsup><mi>ρ</mi><mn>0</mn><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msubsup><msub><mi>Θ</mi><mrow><mtext>sym</mtext></mrow></msub><mn>,</mn></mrow></math> where Θ sym =3 t 1 x 1 - t 2 (4+5 x 2 ) [ 44 44 ]. In this work, the SV-sym34 force, which gives g sur =18.2 MeV fm 2 , g sur,iso =8.9 MeV fm 2 , A sym =20.3 MeV, B sym =14.4 MeV, C sym =-9.2 MeV, and the slope parameter of the symmetry energy L =80.95 MeV, is chosen. In the UrQMD model, the in-medium NN elastic cross section is the product of the free cross sections with a medium correction factor <math id="M17"><mi mathvariant="script">F</mi></math> ( ρ , p ), which is momentum- and density-dependent. In the UrQMD model, the free proton-neutron cross sections are larger than the free neutron-neutron (proton-proton) cross sections at low beam energy, which is consistent with experimental data. In our previous work [ 17 17 , 33 33 , 43 43 ], we found that the medium correction factor <math id="M18"><mi mathvariant="script">F</mi></math> depends on the density, momentum, and isospin. However, the isospin dependence is relatively weak; see, e.g., [ 48 48 , 49 49 ]. <math display="block" id="M19"><mrow><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>=</mo><mi mathvariant="script">F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>*</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup><mn>,</mn></mrow></math> <math display="block" id="M20"><mrow><mi mathvariant="script">F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><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><mi>ρ</mi><mo>/</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo>/</mo><mi>ζ</mi></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><mi>N</mi><mi>N</mi></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></mrow></math> Here λ =1/3, ζ =1/3, f 0 =1, p 0 =0.425 GeV/ c , and κ =5, which corresponds to the FU3FP1 parametrization used in Ref. [ 43 43 ]. Moreover, <math id="M21"><mi mathvariant="script">F</mi></math> ( ρ , p ) is set to unity when p NN is larger than 1 GeV/ c . The in-medium correction factor <math id="M22"><mi mathvariant="script">F</mi></math> ( ρ , p ) obtained from the FU3FP1 parametrization as functions of both the reduced density ρ / ρ 0 and momentum p = p NN is shown in Fig.1 Fig.1 . It is seen that the values deduced from the experiment [ 35 35 ] can be covered by the values of the in-medium correction factor <math id="M23"><mi mathvariant="script">F</mi></math> ( ρ , p ) obtained from the FU3FP1 parametrization. In order to compare the effect of the in-medium NN cross section on the collective flow and nuclear stopping, three fixed values, <math id="M24"><mi mathvariant="script">F</mi></math> = 0.2, 0.3, and 0.5, are further adopted in this work. 3 Results and discussion 1 3.1 Charge distribution 2 In this work, fragments are recognized by the isospin-dependent minimum spanning tree (iso-MST) method. Nucleons with relative distances smaller than <math id="M26"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mi>p</mi><mi>p</mi></mrow></msubsup><mo>=</mo><mn>2.8</mn></mrow></math> fm, <math id="M27"><mrow><msubsup><mi>R</mi><mn>0</mn><mrow><mi>n</mi><mi>n</mi></mrow></msubsup><mo>=</mo><msubsup><mi>R</mi><mn>0</mn><mrow><mi>n</mi><mi>p</mi></mrow></msubsup><mo>=</mo><mn>3.8</mn></mrow></math> fm and relative momenta smaller than P 0 =0.25 GeV/ c are considered to belong to the same cluster. The charge distribution of Au + Au collisions at a beam energy of 150 MeV/nucleon is displayed in Fig.2 Fig.2 . The distribution of charged fragments decreased exponentially with increasing charge number, and this behavior can be reproduced by both the simulations with FU3FP1 and those with the fixed in-medium correction factors. For large fragments (Z ≥ 6), calculations with <math id="M28"><mi mathvariant="script">F</mi></math> =0.5 seem to be closer to the experimental data. However, one cannot conclude that <math id="M29"><mi mathvariant="script">F</mi></math> =0.5 is preferable to the others because the charge distribution will also be affected by unconstrained (or less constrained) parameters in the transport model; e.g., the chosen coalescence model parameters will strongly influence the charge distribution but not the collective flow and nuclear stopping [ 44 44 , 50 50 - 57 57 50-57 ]. 3.2 Collective flow 2 One can obtain the directed ( v 1 ) and elliptic ( v 2 ) flows from the Fourier expansion of the azimuthal distribution of detected particles [ 59 59 , 60 60 ]; they read as <math display="block" id="M33"><mrow><msub><mi>v</mi><mn>1</mn></msub><mo>≡</mo><mo>〈</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>=</mo><mrow><mo>〈</mo><mrow><mfrac><mrow><msub><mi>p</mi><mi>x</mi></msub></mrow><mrow><msub><mi>p</mi><mtext>t</mtext></msub></mrow></mfrac></mrow><mo>〉</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M34"><mrow><msub><mi>v</mi><mn>2</mn></msub><mo>≡</mo><mo>〈</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><mi>ϕ</mi><mo stretchy="false">)</mo><mo>〉</mo><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><msubsup><mi>p</mi><mtext>t</mtext><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>〉</mo></mrow><mn>,</mn><mi> </mi><msub><mi>p</mi><mtext>t</mtext></msub><mo>=</mo><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><mn>.</mn></mrow></math> The reduced rapidity ( y z / y pro ) dependence of the directed ( v 1 ) and elliptic ( v 2 ) flows of free protons from 197 Au + 197 Au collisions at 40, 100, and 150 MeV/nucleon with different <math id="M35"><mi mathvariant="script">F</mi></math> is shown in Fig.3 Fig.3 . One can see that both v 1 and v 2 obtained with different <math id="M36"><mi mathvariant="script">F</mi></math> are different, and the differences become more evident with increasing beam energy. The reason is that with increasing beam energy, the collision term plays a more important role. The v 1 and v 2 values obtained with the FU3FP1 and <math id="M37"><mi mathvariant="script">F</mi></math> =0.3 are very close to each other. In addition, with a larger <math id="M38"><mi mathvariant="script">F</mi></math> , which means a smaller reduction in the cross section, one can obtain a larger slope for v 1 and a more negative v 2 at mid-rapidity, y z / y pro = 0. Nucleons are known to be more likely to undergo a bounce-off (positive v 1 slope) motion and squeeze-out (negative v 2 ) pattern because of the increasing collision number. Figure 4 Figure 4 shows the v 1 slope and v 2 at y z / y pro = 0 for hydrogen isotopes calculated with different <math id="M39"><mi mathvariant="script">F</mi></math> values in comparison with the FOPI and INDRA experimental data. The chosen impact parameters are b = 2–5.5 fm and b = 5.5–7.5 fm for v 1 and v 2 , respectively. The v 1 slope is obtained assuming <math id="M40"><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>11</mn></mrow></msub><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>v</mi><mrow><mn>13</mn></mrow></msub><msubsup><mi>y</mi><mn>0</mn><mn>3</mn></msubsup><mo>+</mo><mi>c</mi></mrow></math> in the range of | y 0 |=| y z / y pro | < 0.4. With increasing E lab , the value of the v 1 slope increases, and the value of v 2 decreases. The results obtained with FU3FP1 and <math id="M41"><mi mathvariant="script">F</mi></math> =0.3 are very close to each other. Furthermore, the correction factors <math id="M42"><mi mathvariant="script">F</mi></math> = 0.2 and 0.5 are required to fit the experimental data at beam energies of 40 and 150 MeV/nucleon, respectively. In addition, calculations with the FU3FP1 parametrization can simultaneously reproduce both v 1 and v 2 well. To understand more clearly the influence of the medium correction factor on the NNCS, Fig.5 Fig.5 shows the proton energy spectra of 197 Au+ 197 Au collisions at beam energies of 40 and 150 MeV/nucleon. Overall, the energy spectra drop exponentially with energy and can also be affected by changing the medium correction factor. The energy spectrum obtained with <math id="M43"><mrow><mi mathvariant="script">F</mi><mo>=</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>/</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup><mo>=</mo><mn>0.2</mn></mrow></math> has a steeper slope than the others because the number of collisions is smallest in this case. 3.3 Nuclear stopping 2 The nuclear stopping power is another important observable that characterizes the transparency of colliding nuclei [ 15 15 , 18 18 , 19 19 43 43 ]. In this work, we calculated the two quantities R E and vartl for the same reaction. The former was proposed by the FOPI collaboration [ 59 59 ] and is defined as the ratio of the variances of the transverse and longitudinal rapidity distributions: <math display="block" id="M44"><mrow><mi>v</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>l</mi><mo>=</mo><mfrac><mrow><mo>〈</mo><msubsup><mi>y</mi><mi>x</mi><mn>2</mn></msubsup><mo>〉</mo></mrow><mrow><mo>〈</mo><msubsup><mi>y</mi><mi>z</mi><mn>2</mn></msubsup><mo>〉</mo></mrow></mfrac><mn>,</mn><mi> </mi><mi> </mi><mo>〈</mo><msubsup><mi>y</mi><mrow><mi>x</mi><mn>,</mn><mi>z</mi></mrow><mn>2</mn></msubsup><mo>〉</mo><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mo>∑</mo></mstyle><mo stretchy="false">(</mo><msubsup><mi>y</mi><mrow><mi>x</mi><mn>,</mn><mi>z</mi></mrow><mn>2</mn></msubsup><msub><mi>N</mi><mrow><msub><mi>y</mi><mrow><mi>x</mi><mn>,</mn><mi>z</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><mo>∑</mo></mstyle><msub><mi>N</mi><mrow><msub><mi>y</mi><mrow><mi>x</mi><mn>,</mn><mi>z</mi></mrow></msub></mrow></msub></mrow></mfrac><mn>.</mn></mrow></math> The other quantity, R E , was proposed by the INDRA collaboration and is defined as the ratio of the transverse and parallel energies: <math display="block" id="M45"><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 E ⊥ ( E ‖ ) is the transverse (parallel) kinetic energy of particles in the center-of-mass system [ 61 61 ]. The beam energy dependence of the degree of nuclear stopping for free protons in central HICs is displayed in Fig.6 Fig.6 . Similar to the results shown in Fig.4 Fig.4 , the degree of nuclear stopping can be reproduced well with <math id="M46"><mi mathvariant="script">F</mi></math> ≈0.3 at E lab = 40 MeV/nucleon. However, <math id="M47"><mi mathvariant="script">F</mi></math> ≈0.5 is needed at E lab = 150 MeV/nucleon. With increasing beam energy, a nuclear medium with higher density and momentum can be created; thus, it is reasonable that the reduction factors at different beam energies are different. With increasing beam energy, the difference in R E between <math id="M48"><mi mathvariant="script">F</mi></math> =0.3 and FU3FP1 increases steadily. The experimental data and the trend in stopping power with increasing beam energy can be reproduced reasonably well by the calculations with FU3FP1. In contrast, the other calculations remain almost constant because the reduced factor in the FU3FP1 depends on both the density and momentum. In addition, these conclusions can be confirmed in both 197 Au+ 197 Au and 129 Xe+ 120 Sn. 4 Summary 1 The effects of the in-medium nucleon-nucleon cross section on the collective flow and nuclear stopping in the Fermi energy domain are investigated within the UrQMD model, in which medium correction factors of <math id="M49"><mrow><mi mathvariant="script">F</mi><mo>=</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>in</mtext><mo>−</mo><mtext>medium</mtext></mrow></msubsup><mo>/</mo><msubsup><mi>σ</mi><mrow><mi>N</mi><mi>N</mi></mrow><mrow><mtext>free</mtext></mrow></msubsup><mo>=</mo><mn>0.2,</mn><mi> </mi><mn>0.3,</mn><mi> </mi><mn>0.5</mn></mrow></math> and the parametrization factor FU3FP1 (which is density- and momentum- dependent) are considered. It is demonstrated that to best fit both the collective flow and nuclear stopping data, <math id="M50"><mi mathvariant="script">F</mi></math> ≈ 0.2 and 0.5 are required at beam energies of 40 and 150 MeV/nucleon, respectively. These results are consistent with the results obtained by the INDRA collaboration. The calculations with the FU3FP1 parametrization reproduce both the directed and elliptic flow data reasonably well. The slightly increased stopping power with increasing beam energy can also be found in the calculations with the FU3FP1 parametrization, whereas the calculations with <math id="M51"><mi mathvariant="script">F</mi></math> = 0.2, 0.3, and 0.5 yield roughly flat energy-dependent. References 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 ( 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 G. F. Bertsch and S. Das Gupta , A guide to microscopic models for intermediate energy heavy ion collisions . Phys. Rept . 160 , 189 ( 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. Xu , L. W. Chen , M. B. Tsang 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 Y. X. Zhang , Y. J. Wang , M. Colonna 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 Z. F. Zhang , D. Q. Fang , Y. G. Ma , Decay modes of highly excited nuclei . Nucl. Sci. Tech . 29 78 ( 2018 ). doi: 10.1007/s41365-018-0427-8 http://doi.org/10.1007/s41365-018-0427-8 Decay modes of highly excited nuclei P. Danielewicz , R. Lacey and W. G. Lynch , Determination of the equation of state of dense matter . Science 298 , 1592 ( 2002 ). doi: 10.1126/science.1078070 http://doi.org/10.1126/science.1078070 Determination of the equation of state of dense matter X. G. Deng , Y. G. Ma , Electromagnetic field effects on nucleon transverse momentum for heavy ion collisions around 100?A?MeV . Nucl. Sci. Tech . 28 182 ( 2017 ). doi: 10.1007/s41365-017-0337-1 http://doi.org/10.1007/s41365-017-0337-1 Electromagnetic field effects on nucleon transverse momentum for heavy ion collisions around 100?A?MeV Y. J. Wang , C. C. Guo , Q. F. Li , A. Le Fèvre , Y. Leifels and W. Trautmann , 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 ( 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 H. M. Xu , Disappearance of flow and the in-medium nucleon-nucleon cross section . Phys. Rev. C 46 , R389 ( 1992 ). doi: 10.1103/PhysRevC.46.R389 http://doi.org/10.1103/PhysRevC.46.R389 Disappearance of flow and the in-medium nucleon-nucleon cross section G. D. Westfall , W. Bauer , D. Craig et al ., Mass dependence of the disappearance of flow in nuclear collisions . Phys. Rev. Lett . 71 , 1986 ( 1993 ). doi: 10.1103/PhysRevLett.71.1986 http://doi.org/10.1103/PhysRevLett.71.1986 Mass dependence of the disappearance of flow in nuclear collisions B. A. Li and A. T. Sustich , Differential Flow in Heavy-Ion Collisions at Balance Energies . Phys. Rev. Lett . 82 , 5004 ( 1999 ). doi: 10.1103/PhysRevLett.82.5004 http://doi.org/10.1103/PhysRevLett.82.5004 Differential Flow in Heavy-Ion Collisions at Balance Energies D. J. Magestro , W. Bauer and G. D. Westfall , Isolation of the nuclear compressibility with the balance energy . Phys. Rev. C 62 , 041603 ( 2000 ). doi: 10.1103/PhysRevC.62.041603 http://doi.org/10.1103/PhysRevC.62.041603 Isolation of the nuclear compressibility with the balance energy J. Y. Liu , W. J. Guo , S. J. Wang et al ., Nuclear Stopping as a Probe for In-Medium Nucleon-Nucleon Cross Sections in Intermediate Energy Heavy Ion Collisions . Phys. Rev. Lett . 86 , 975 ( 2001 ). doi: 10.1103/PhysRevLett.86.975 http://doi.org/10.1103/PhysRevLett.86.975 Nuclear Stopping as a Probe for In-Medium Nucleon-Nucleon Cross Sections in Intermediate Energy Heavy Ion Collisions D. Persram and C. Gale , Elliptic flow in intermediate energy heavy ion collisions and in-medium effects . Phys. Rev. C 65 , 064611 ( 2002 ). doi: 10.1103/PhysRevC.65.064611 http://doi.org/10.1103/PhysRevC.65.064611 Elliptic flow in intermediate energy heavy ion collisions and in-medium effects T. Gaitanos , C. Fuchs , and H. H. Wolter , Nuclear stopping and flow in heavy-ion collisions and the in-medium NN cross section . Phys. Lett. B 609 , 241 ( 2005 ). doi: 10.1016/j.physletb.2005.01.069 http://doi.org/10.1016/j.physletb.2005.01.069 Nuclear stopping and flow in heavy-ion collisions and the in-medium NN cross section B. A. Li , P. Danielewicz and W. G. Lynch , Probing the isospin dependence of the in-medium nucleon-nucleon cross sections with radioactive beams . Phys. Rev. C 71 , 054603 ( 2005 ). doi: 10.1103/PhysRevC.71.054603; http://doi.org/10.1103/PhysRevC.71.054603; Probing the isospin dependence of the in-medium nucleon-nucleon cross sections with radioactive beams B. A. Li and L. W. Chen , Nucleon-nucleon cross sections in neutron-rich matter and isospin transport in heavy-ion reactions at intermediate energies . Phys. Rev. C 72 , 064611 ( 2005 ). doi: 10.1103/PhysRevC.72.064611 http://doi.org/10.1103/PhysRevC.72.064611 Nucleon-nucleon cross sections in neutron-rich matter and isospin transport in heavy-ion reactions at intermediate energies Q. F. Li , Z. X. Li , S. Soff et al ., Medium modifications of the nucleon-nucleon elastic cross section in neutron-rich intermediate energy HICs . J. Phys. G: Nucl. Part. Phys . 32 407 ( 2006 ). doi: 10.1088/0954-3899/32/4/001 http://doi.org/10.1088/0954-3899/32/4/001 Medium modifications of the nucleon-nucleon elastic cross section in neutron-rich intermediate energy HICs Y. X. Zhang , Z. X. Li , Elliptic flow and system size dependence of transition energies at intermediate energies . Phys. Rev. C 74 , 014602 ( 2006 ). doi: 10.1103/PhysRevC.74.014602; http://doi.org/10.1103/PhysRevC.74.014602; Elliptic flow and system size dependence of transition energies at intermediate energies Y. X. Zhang , Z. X. Li and P. Danielewicz , In-medium NN cross sections determined from the nuclear stopping and collective flow in heavy-ion collisions at intermediate energies . Phys. Rev. C 75 , 034615 ( 2007 ). doi: 10.1103/PhysRevC.75.034615 http://doi.org/10.1103/PhysRevC.75.034615 In-medium NN cross sections determined from the nuclear stopping and collective flow in heavy-ion collisions at intermediate energies Y. Yuan , Q. F. Li , Z. X. Li et al ., Transport model study of nuclear stopping in heavy-ion collisions over the energy range from 0.09A to 160A GeV . Phys. Rev. C 81 , 034913 ( 2010 ). doi: 10.1103/PhysRevC.81.034913 http://doi.org/10.1103/PhysRevC.81.034913 Transport model study of nuclear stopping in heavy-ion collisions over the energy range from 0.09A to 160A GeV G. Q. Zhang , Y. G. Ma , X. G. Cao et al ., Unified description of nuclear stopping in central heavy-ion collisions from 10A MeV to 1.2A GeV . Phys. Rev. C 84 , 034612 ( 2011 ). doi: 10.1103/PhysRevC.84.034612 http://doi.org/10.1103/PhysRevC.84.034612 Unified description of nuclear stopping in central heavy-ion collisions from 10A MeV to 1.2A GeV D. D. S. Coupland , W. G. Lynch , M. B. Tsang et al ., Influence of transport variables on isospin transport ratios . Phys. Rev. C 84 , 054603 ( 2011 ). doi: 10.1103/PhysRevC.84.054603 http://doi.org/10.1103/PhysRevC.84.054603 Influence of transport variables on isospin transport ratios Z. Q. Feng , Nuclear in-medium effects and collective flows in heavy-ion collisions at intermediate energies . Phys. Rev. C 85 , 014604 ( 2012 ). doi: 10.1103/PhysRevC.85.014604 http://doi.org/10.1103/PhysRevC.85.014604 Nuclear in-medium effects and collective flows in heavy-ion collisions at intermediate energies Y. X. Zhang , D. D. S. Coupland , P. Danielewicz et al ., Influence of in-medium NN cross sections, symmetry potential, and impact parameter on isospin observables . Phys. Rev. C 85 , 024602 ( 2012 ). doi: 10.1103/PhysRevC.85.024602 http://doi.org/10.1103/PhysRevC.85.024602 Influence of in-medium NN cross sections, symmetry potential, and impact parameter on isospin observables J. Su , Ch. Y. Huang , W. J. Xie et al ., Effects of in-medium nucleon-nucleon cross sections on stopping observable and ratio of free protons in heavy-ion collisions at 400 MeV/nucleon . Eur. Phys. J. A 52 , 207 ( 2016 ). doi: 10.1140/epja/i2016-16207-x http://doi.org/10.1140/epja/i2016-16207-x Effects of in-medium nucleon-nucleon cross sections on stopping observable and ratio of free protons in heavy-ion collisions at 400 MeV/nucleon W. G. Love and M. A. Franey , Effective nucleon-nucleon interaction for scattering at intermediate energies . Phys. Rev. C 24 , 3 ( 1981 ). doi: 10.1103/PhysRevC.24.1073 http://doi.org/10.1103/PhysRevC.24.1073 Effective nucleon-nucleon interaction for scattering at intermediate energies G. Q. Li and R. Machleidt , Microscopic calculation of in-medium nucleon-nucleon cross sections . Phys. Rev. C 48 , 1702 ( 1993 ). doi: 10.1103/PhysRevC.48.1702; http://doi.org/10.1103/PhysRevC.48.1702; Microscopic calculation of in-medium nucleon-nucleon cross sections G. Q. Li and R. Machleidt , Microscopic calculation of in-medium proton-proton cross sections . Phys. Rev. C 49 , 566 ( 1994 ). doi: 10.1103/PhysRevC.49.566 http://doi.org/10.1103/PhysRevC.49.566 Microscopic calculation of in-medium proton-proton cross sections G. J. Mao , Z. X. Li , Y. Z. Zhuo et al ., Medium effects on the NN inelastic cross section in relativistic heavy-ion collisions . Phys. Lett. B 327 , 183 ( 1994 ). doi: 10.1016/0370-2693(94)90715-3; http://doi.org/10.1016/0370-2693(94)90715-3; Medium effects on the NN inelastic cross section in relativistic heavy-ion collisions G. J. Mao , Z. X. Li , Y. Z. Zhuo et al ., Study of in-medium NN inelastic cross section from relativistic Boltzmann-Uehling-Uhlenbeck approach . Phys. Rev. C 49 , 3137 ( 1994 ). doi: 10.1103/PhysRevC.49.3137 http://doi.org/10.1103/PhysRevC.49.3137 Study of in-medium NN inelastic cross section from relativistic Boltzmann-Uehling-Uhlenbeck approach T. Alm , G. Röpke , W. Bauer et al ., The in-medium nucleon-nucleon cross section and BUU simulations of heavy-ion reactions . Nucl. Phys. A 587 , 815 ( 1995 ). doi: 10.1016/0375-9474(95)00026-W http://doi.org/10.1016/0375-9474(95)00026-W The in-medium nucleon-nucleon cross section and BUU simulations of heavy-ion reactions H. J. Schulze , A. Schnell , G. Ropke et al ., Nucleon-nucleon cross sections in nuclear matter . Phys. Rev. C 55 , 3006 ( 1997 ). doi: 10.1103/PhysRevC.55.3006 http://doi.org/10.1103/PhysRevC.55.3006 Nucleon-nucleon cross sections in nuclear matter C. Fuchs , A. Faessler , and M. El-Shabshiry , Off-shell behavior of the in-medium nucleon-nucleon cross section . Phys. Rev. C 64 , 024003 ( 2001 ). doi: 10.1103/PhysRevC.64.024003 http://doi.org/10.1103/PhysRevC.64.024003 Off-shell behavior of the in-medium nucleon-nucleon cross section F. Sammarruca and P. Krastev , Effective nucleon-nucleon cross sections in symmetric and asymmetric nuclear matter . Phys. Rev. C 73 , 014001 ( 2006 ). doi: 10.1103/PhysRevC.73.014001 http://doi.org/10.1103/PhysRevC.73.014001 Effective nucleon-nucleon cross sections in symmetric and asymmetric nuclear matter H. F. Zhang , Z. H. Li , U. Lombardo et al ., Nucleon-nucleon cross sections in dense nuclear matter . Phys. Rev. C 76 , 054001 ( 2007 ). doi: 10.1103/PhysRevC.76.054001; http://doi.org/10.1103/PhysRevC.76.054001; Nucleon-nucleon cross sections in dense nuclear matter H. F. Zhang , U. Lombardo , and W. Zuo , Transport parameters in neutron stars from in-medium NN cross sections , Phys. Rev. C 82 , 015805 ( 2010 ). doi: 10.1103/PhysRevC.82.015805 http://doi.org/10.1103/PhysRevC.82.015805 Transport parameters in neutron stars from in-medium NN cross sections Q. F. Li , Z. X. Li and G. J. Mao , Isospin dependence of nucleon-nucleon elastic cross section . Phys. Rev. C 62 , 014606 ( 2000 ). doi: 10.1103/PhysRevC.62.014606; http://doi.org/10.1103/PhysRevC.62.014606; Isospin dependence of nucleon-nucleon elastic cross section Q. F. Li , Z. X. Li and E. G. Zhao , Density and temperature dependence of nucleon-nucleon elastic cross section . Phys. Rev. C 69 , 017601 ( 2004 ). doi: 10.1103/PhysRevC.69.017601; http://doi.org/10.1103/PhysRevC.69.017601; Density and temperature dependence of nucleon-nucleon elastic cross section Q. F. Li and Z. X. Li , The isospin dependent nucleon-nucleon inelastic cross section in the nuclear medium . Phys. Lett. B 773 , 557 ( 2017 ). doi: 10.1016/j.physletb.2017.09.013 http://doi.org/10.1016/j.physletb.2017.09.013 The isospin dependent nucleon-nucleon inelastic cross section in the nuclear medium J. Chen , Z. Q. Feng , J. S. Wang , Nuclear in-medium effects on η dynamics in proton-nucleus collisions . Nucl. Sci. Tech . 27 73 ( 2016 ). doi: 10.1007/s41365-016-0069-7 http://doi.org/10.1007/s41365-016-0069-7 Nuclear in-medium effects on η dynamics in proton-nucleus collisions O. Lopez , D. Durand , G. Lehaut et al ., (INDRA Collaboration) , In-medium effects for nuclear matter in the Fermi-energy domain . Phys. Rev. C 90 , 064602 ( 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 B Rubina , K Mandeep and K Suneel , Correlation between directed transverse flow and nuclear stopping around the energy of vanishing flow . Indian J. Phys . 89 967 ( 2015 ). doi: 10.1007/s12648-015-0672-1 http://doi.org/10.1007/s12648-015-0672-1 Correlation between directed transverse flow and nuclear stopping around the energy of vanishing flow Z. Basrak , P. Eudes , and V. de la Mota , Aspects of the momentum dependence of the equation of state and of the residual NN cross section, and their effects on nuclear stopping . Phys. Rev. C 93 , 054609 ( 2016 ). doi: 10.1103/PhysRevC.93.054609 http://doi.org/10.1103/PhysRevC.93.054609 Aspects of the momentum dependence of the equation of state and of the residual NN cross section, and their effects on nuclear stopping H. L. Liu , Y. G. Ma , A. Bonasera et al ., Mean free path and shear viscosity in central 129Xe+119Sn collisions below 100 MeV/nucleon . Phys. Rev. C 96 , 064604 ( 2017 ). doi: 10.1103/PhysRevC.96.064604 http://doi.org/10.1103/PhysRevC.96.064604 Mean free path and shear viscosity in central 129Xe+119Sn collisions below 100 MeV/nucleon Y. Z. Xing , H. F. Zhang , X. B. Liu et al ., Pauli-blocking effect in two-body collisions dominates the in-medium effects in heavy-ion reactions near Fermi energy . Nucl. Phys. A 957 135 ( 2017 ). doi: 10.1016/j.nuclphysa.2016.08.006 http://doi.org/10.1016/j.nuclphysa.2016.08.006 Pauli-blocking effect in two-body collisions dominates the in-medium effects in heavy-ion reactions near Fermi energy Z. Q. Feng , Nuclear dynamics and particle production near threshold energies in heavy-ion collisions . Nucl. Sci. Tech . 29 , 40 ( 2018 ). doi: 10.1007/s41365-018-0379-z http://doi.org/10.1007/s41365-018-0379-z Nuclear dynamics and particle production near threshold energies in heavy-ion collisions B. A. Li , B. J. Cai , L. W. Chen et al ., Isospin dependence of nucleon effective masses in neutron-rich matter . Nucl. Sci. Tech . 27 , 141 ( 2016 ). doi: 10.1007/s41365-016-0140-4 http://doi.org/10.1007/s41365-016-0140-4 Isospin dependence of nucleon effective masses in neutron-rich matter Q. F. Li , Z. X. Li , S. Soff et al ., Probing the equation of state with pions . J. Phys. G 32 , 151 ( 2006 ). doi: 10.1088/0954-3899/32/2/007 http://doi.org/10.1088/0954-3899/32/2/007 Probing the equation of state with pions Q. F. Li , C. W. Shen , C. 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 , C. C. Guo , Q. F. Li et al ., Constraining the high-density nuclear symmetry energy with the transverse-momentum-dependent elliptic flow . Phys. Rev. C 89 , 044603 ( 2014 ). doi: 10.1103/PhysRevC.89.044603; http://doi.org/10.1103/PhysRevC.89.044603; Constraining the high-density nuclear symmetry energy with the transverse-momentum-dependent elliptic flow Y. J. Wang , C. C. Guo , Q. F. Li et al ., Collective flow of light particles in 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 flow of light particles in Au + Au collisions at intermediate energies Y. J. Wang , C. C. Guo , Q. F. Li et al ., Influence of differential elastic nucleon-nucleon cross section on stopping and collective flow in heavy-ion collisions at intermediate energies . Phys. Rev. C 94 , 024608 ( 2016 ). doi: 10.1103/PhysRevC.94.024608 http://doi.org/10.1103/PhysRevC.94.024608 Influence of differential elastic nucleon-nucleon cross section on stopping and collective flow in heavy-ion collisions at intermediate energies C. Hartnack , R. K. Puri , J. Aichelin et al ., Modelling the many-body dynamics of heavy ion collisions: Present status and future perspective . Eur. Phys. J. A 1 , 151 ( 1998 ). doi: 10.1007/s100500050045 http://doi.org/10.1007/s100500050045 Modelling the many-body dynamics of heavy ion collisions: Present status and future perspective Y. J. Wang , C. C. Guo , Q. F. Li et al ., 3H/3He 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 P. C. Li , Y. J. Wang , Q. F. Li , H. F. Zhang , 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 Q. F. Li , C. W. Shen and M. Di Toro , Probing the momentum-dependent medium modifications of the nucleon-nucleon elastic cross section , Mod. Phys. Lett. A 25 669 ( 2010 ). doi: 10.1142/S0217732310032846 http://doi.org/10.1142/S0217732310032846 Probing the momentum-dependent medium modifications of the nucleon-nucleon elastic cross section W. M. Guo and G. C. Yong and Y. J. Wang and Q. F. Li and H. F. Zhang and W. Zuo , Model dependence of isospin sensitive observables at high densities . Phys. Lett. B 726 , 211 ( 2013 ). doi: 10.1016/j.physletb.2013.07.056 http://doi.org/10.1016/j.physletb.2013.07.056 Model dependence of isospin sensitive observables at high densities H. Kruse , B. V. Jacak , J. J. Molitoris et al ., Vlasov-Uehling-Uhlenbeck theory of medium energy heavy ion reactions: Role of mean field dynamics and two body collisions . Phys. Rev. C 31 , 1770 ( 1985 ). doi: 10.1103/PhysRevC.31.1770 http://doi.org/10.1103/PhysRevC.31.1770 Vlasov-Uehling-Uhlenbeck theory of medium energy heavy ion reactions: Role of mean field dynamics and two body collisions P. Russotto , P. Z. Wu , M. Zoric et al ., Symmetry energy from elliptic flow in 197Au+197Au . Phys. Lett. B 697 , 471 ( 2011 ). doi: 10.1016/j.physletb.2011.02.033 http://doi.org/10.1016/j.physletb.2011.02.033 Symmetry energy from elliptic flow in 197Au+197Au Q. F. Li , Y. J. Wang , X. B. 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 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 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 , 622001 ( 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 Y. X. Zhang , Z. X. Li , C. S. Zhou et al ., Effect of isospin-dependent cluster recognition on the observables in heavy ion collisions . Phys. Rev. C 85 , 051602 ( 2012 ). doi: 10.1103/PhysRevC.85.051602 http://doi.org/10.1103/PhysRevC.85.051602 Effect of isospin-dependent cluster recognition on the observables in heavy ion collisions S. Kumar , Y. G. Ma , Directed flow of isospin sensitive fragments within a modified clusterization algorithm in heavy-ion collisions . Nucl. Sci. Tech . 24 , 050509 ( 2013 ). doi: 10.13538/j.1001-8042/nst.2013.05.009 http://doi.org/10.13538/j.1001-8042/nst.2013.05.009 Directed flow of isospin sensitive fragments within a modified clusterization algorithm in heavy-ion collisions T. T. Ding , C. W. Ma , An improved thermometer for intermediate-mass fragments . Nucl. Sci. Tech . 27 , 132 ( 2016 ). doi: 10.1007/s41365-016-0142-2 http://doi.org/10.1007/s41365-016-0142-2 An improved thermometer for intermediate-mass fragments C. W. Ma , C. Y. Qiao , T. T. Ding , Y. D. Song , Temperature of intermediate mass fragments in simulated 40Ca + 40Ca reactions around the Fermi energies by AMD model . Nucl. Sci. Tech . 27 , 111 ( 2016 ). doi: 10.1007/s41365-016-0112-8 http://doi.org/10.1007/s41365-016-0112-8 Temperature of intermediate mass fragments in simulated 40Ca + 40Ca reactions around the Fermi energies by AMD model H. L. Lao , F. H. Liu et al ., Kinetic freeze-out temperatures in central and peripheral collisions: which one is larger? Nucl. Sci. Tech . 29 , 82 ( 2018 ). doi: 10.1007/s41365-018-0425-x http://doi.org/10.1007/s41365-018-0425-x Kinetic freeze-out temperatures in central and peripheral collisions: which one is larger? A. Andronic , J. Lukasik , W. Reisdorf et al ., Systematics of stopping and flow in Au+ Au collisions . Eur. Phys. J. A 30 , 31 ( 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 W. Reisdorf et al ., Systematics of azimuthal asymmetries in heavy ion collisions in the 1A GeV regime . Nucl. Phys. A 876 , 1 ( 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 1A GeV regime H. C. Song , Y. Zhou , Katarína Gajdošová, Collective flow and hydrodynamics in large and small systems at the LHC . Nucl. Sci. Tech . 28 , 99 ( 2017 ). doi: 10.1007/s41365-017-0245-4 http://doi.org/10.1007/s41365-017-0245-4 Katarína Gajdošová, Collective flow and hydrodynamics in large and small systems at the LHC 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 ., (FOPI Collab orations), Systematics of central heavy ion collisions in the 1A GeV regime . Nucl. Phys. A 848 366 ( 2010 ). doi: 10.1016/j.nuclphysa.2010.09.008 http://doi.org/10.1016/j.nuclphysa.2010.09.008 orations), Systematics of central heavy ion collisions in the 1A GeV regime 10.1007/s41365-018-0510-1 This work was supported by the National Natural Science Foundation of China (Nos. 11875125, 11747312, 11675066, and 11505057) and the Zhejiang Provincial Natural Science Foundation of China (No. LY18A050002). 2018-08-31 2018-10-08 2018-10-09 2018-11-17 2018-12-01 2021-12-20 © 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. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0510-1 http://dx.doi.org/10.1007/s41365-018-0510-1 . 2018 Nuclear Science and Techniques 12 29