Progress of Quantum Molecular Dynamics model and its applications in Heavy Ion Collisions Yingxun Zhang,1, 2, ∗ Ning Wang,3, 2, y Qingfeng Li,4, 5, z Li Ou,3, 2, x Junlong Tian,6, 2, { Min Liu,3, 2, ∗∗ Kai Zhao,1, yy Xizhen Wu,1, 2, zz and Zhuxia Li1, 2, xx 1China Institute of Atomic Energy, Beijing, 102413, China 2Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin, 541004, China 3Guangxi Normal University, Guilin, 541004, China 4School of Science, Huzhou University, Huzhou, 313000, China 5Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China 6School of Physics and Electrical Engineering, Anyang Normal University, Anyang, 455002, China (Dated: May 14, 2020) In this review article, we first briefly introduce the transport theory and quantum molecular dy- namics model applied in the study of the heavy ion collisions from low to intermediate energies. The developments of improved quantum molecular dynamics model (ImQMD) and ultra-relativistic quantum molecular dynamics model (UrQMD), are reviewed. The reaction mechanism and phe- nomena related to the fusion, multinucleon transfer, fragmentation, collective flow and particle production are reviewed and discussed within the framework of the two models. The constraints on the isospin asymmetric nuclear equation of state and in-medium nucleon-nucleon cross sections by comparing the heavy ion collision data with transport models calculations in last decades are also discussed, and the uncertainties of these constraints are analyzed as well. Finally, we discuss the future direction of the development of the transport models for improving the understanding of the reaction mechanism, the descriptions of various observables, the constraint on the nuclear equation of state, as well as for the constraint on in-medium nucleon-nucleon cross sections. Contents A. Heavy ion fusion reactions 14 1. Dynamical nucleus-nucleus potential 15 I. Introduction 2 2. Neck Dynamics 16 3. Fusion cross sections 16 II. Transport theory and the Quantum B. Multi-nucleon transfer reactions 17 molecular dynamics model 4 C. Low-intermediate and intermediate-high A. Transport theory for N-body system 4 energy heavy ion collisions 20 B. Quantum Molecular Dynamics approach 5 1. Fragmentation mechansim 21 C. Improved Quantum Molecular Dynamics 2. Fluctuation and chaoticity in liquid-gas Model 8 phase transition 23 1. Nucleonic mean field 8 3. Collective flow 24 2. Collision part 10 D. Spallation reactions 26 3. Pauli blocking 11 IV. Study of in-medium NN cross sections 27 4. Initialization 12 A. In-medium NN cross sections from microscopic 5. Cluster recognization 13 approach 28 D. Ultra-relativistic Quantum Molecular B. In-medium NN cross sections from heavy ion Dynamics Model 14 collisions 29 C. In-medium NN cross sections from spallation III. Study of the phenomena and the reactions 31 mechanism in low-intermediate energy heavy ion reactions 14 V. Symmetry energy at different densities and temperatures and the constraints 32 A. Symmetry energy at subsaturation density 33 1. Constraints from n/p ratios and isospin ∗Electronic address: [email protected] yElectronic address: [email protected] diffusion 33 zElectronic address: [email protected] 2. Novel probes of symmetry energy at xElectronic address: [email protected] subsaturation denisty 35 {Electronic address: [email protected] B. Symmetry energy at suprasaturation density 36 ∗∗Electronic address: [email protected] yyElectronic address: [email protected] C. EOS and symmetry energy at finite zzElectronic address: [email protected] temperature 39 xxElectronic address: [email protected] D. Uncertainties in symmetry energy constraints 40 2 1. Uncertainties of the symmetry energy pears and provides the opportunity to study the equa- associated parameters 40 tion of state at sub-saturated densities [22, 23, 62{65] and 2. Influence of effective mass splitting on Ri, mechanism of liquid-gas phase transition in the finite sys- Ri(y), and n=p ratios 41 tem. The phenomena relating to the multifragmentation, ∗ 3. Influence of K0, S0, L, ms, and fI on such as isospin distillation, neutron-rich neck emission, isospin diffusion 43 bimodality, ..., evolved with the isospin asymmetry and the size of reaction system and the beam energy as well, VI. Discussion and Prospect 44 are very rich and complicated, and it provides us with the hints of equation of state (EOS), mechanism of frag- Acknowledgments 45 mentation and information of liquid-gas phase transition for two components finite system. References 46 When the beam energies increase from around 100 MeV per nucleon up to few GeV per nucleon, the col- lective motion of emitted nucleons and light particles I. INTRODUCTION appears, and it was named as collective flow [3, 9, 11{ 14, 16]. The measurement of collective flow also provides Heavy ion collisions (HICs) provide an unique way to a possibility for the study of the nuclear EOS at supersat- investigate the basic nuclear physics problems in labora- urated densities as well as the information of in-medium tory, such as where is the end of nuclear chart? how can nucleon-nucleon (NN) cross sections [24], because the we reach it? what is the properties of neutron-rich matter changing of the momentum of emitted nucleons are re- in cosmos and on the earth? and the origin of elements lated to the gradient of pressure between the participant heavier than Fe? In the past 50 years, a large number of and spectator of the reaction system, which is caused experimental data for HICs have been accumulated and by the nucleonic mean field and nucleon-nucleon colli- lots of interesting results have been obtained [1{24]. It sions. The sub-threshold and above threshold production has been found that the phenomena and the physics pro- of mesons take place in addition to the multifragmenta- cesses in HICs at various energy domains are very rich tion and the collective flow effects. The yields of mesons and complicated. Thus, understanding the mechanism and their yield ratios between different charge states as behind them is very helpful for us to obtain the knowl- well as the flow effects evolve with energy and isospin edge of related nuclear phenomena and physics. asymmetry of the system [66{71, 73{84, 135], contain a For low energy heavy ion reactions (in this review pa- lot information of the isospin asymmetric EOS of nu- clear matter, the isospin dependent in-medium NN cross per, we consider this energy domain to be Ebeam ≤ 10 MeV/nucleon), the synthesis of superheavy nuclei and sections and the reaction dynamics. The study of the new neutron (proton)-rich isotopes are the hot topics [25{ phenomena at the beam energy from low to intermediate 61]. Related to them, there are also a lot of interesting energies, such as the production of new isotopes, multi- problems in the reactions, such as the role of the dy- fragmentation, and flow effects closely relating to equa- namical effect, the nuclear structure effect and the in- tion of state stimulate the building of next generation of terplay between them, the dependence of the barrier on rare isotope facilities, such as the Intensity Heavy ion Ac- the reaction systems and incident energies, etc., need to celerator Facility (HIAF) at Institute of Modern Physics be clarified. Multi-nucleon transfer (MNT) reactions be- (IMP) in China [85{87], Facility for rare isotope beams tween very heavy nuclei at near or above barrier energies, (FRIB)[88] in USA, Rare isotope Accelerator complex attracted a lot of attention as well [52{54] since it could for ON-line experiment (RAON)[89] in Korea, RI beam lead to production of new heavy and possibly superheavy factory (RIBF/RIKEN) in Japan[90], SPIRAL2/GANIL neutron-rich isotopes, especially those nuclei at waiting in France[91], Facility for Antiproton and Ion Research point for r-process which are important to the nuclear (FAIR) at GSI in Germany[92], Selective Production astrophysics. In addition, the study of MNT reaction is of Exotic Species (SPES/LNL) in Italy[93], Nuclotron- highly required due to the new facilities of High Inten- based Ion Collider facility at the Joint Institute for Nu- sity heavy ion Accelerator Facility (HIAF) and Argonne clear Research(NICA/JINR) in Russia[94], and the pro- Tandem Linac Accelerator System (ATLAS). posed Beijing Isotope-Separation-On-Line Neutron-Rich As the beam energy increases, the interplay of one- Beam Facility (BISOL)[95] in China. body mean field and two-body scattering makes the phe- In order to describe the phenomena and processes in nomena in this energy domain fascinating and compli- HICs at each energy domain, kinds of successful mod- cated. The ternary and four-fragment breakup process els have been developed for explaining and investigat- may take place accompanying the binary process. At the ing the related physics problems. For instance, for low low-intermediate energies region, i.e. ∼20 MeV/nucleon beam energy reactions, both the diffusion model by us- to ∼300 MeV/nucleon (the definition of energy domain ing Langevin equation and the di-nuclear model can de- of intermediate energies, usually differs among authors), scribe fusion cross sections [46, 52, 53, 53, 54, 96{117]; multifragmentation process which was thought as a sig- for the intermediate energy region, the statistical mul- nal of liquid-gas phase transition in finite systems, ap- tifragmentation model (SMM) is quite successful in de- 3 scribing the charge distribution of multifragmentations will need to introduce the fluctuation which actually ex- at intermediate energy reactions [118{125], etc. How- ist in low-intermediate energy heavy ion collisions. There ever, heavy ion reaction or collision process is essentially are some effort to do it through the Boltzmann-Langevin a non-equilibrium process. It is highly demanded to de- equation (BLE) which adds a fluctuation term on the velop a model or theory which can describe the time right hand side of Eq.