Yingxun Zhang 张英逊 Constraints on the density dependence of symmetry energy 十三届全国中高能核物理大会暨第七届全国中高能核物理专题研讨会 USTC ， 合肥， 11/6/2009 合作者： Zhuxia Li （李祝霞）, M.B.Tsang, P.Danielewicz, W.G.Lynch China Institute of Atomic Energy, 中国原子能科学研究院 JINA, NSCL/MSU  Motivation and Strategies on Study the density dependence of symmetry energy  Brief introduction on ImQMD model  Constraints on the density dependence of symmetry energy from the ImQMD model  Summary and outlook Outline

 Motivation and strategies on study the density dependence of symmetry energy Symmetry energy and the density dependence of symmetry energy In asymmetric nuclear matter, the EOS E sym ( r ):the density dependence of symmetry energy E/A( ,  ) = E/A( ,0) + E sym (r)  2 ＋ O(d 4 ) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A E/A( ,  ) = E/A( ,0) + E sym (r)  2 ＋ O(d 4 ) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A It is a fundamental properties of nuclear matter, and is very important for understanding neutron-rich HICs process nuclear structure thickness of the neutron skins,.. properties of NS, …. (Lie-Wen’s Talk)

Danielewicz, Lacey, Lynch, Science 298,1592 (2002) E/A( ,  ) = E/A( ,0) +  2  E sym (r) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A Equation of State of Nuclear Matter 因此非对称核物质状态方程的 确定依赖于对称能密度依赖形 式的确定 集体流以及 Kaon 凝聚的实验数 据对于 1 ～ 5 r 0 的对称核物质状 态方程给出了约束 对称能密度依赖形式有很大的不确定性 ！

我们的工作主要是利用丰中子核中能重离子碰撞的方法对于 低密区对称能密度依赖形式的约束 Low densities (  <  0): –Isoscaling of fragments –Isospin diffusion –n/p spectra and flows; R(n/p), R(t/3He) –Fragment isotopic distributions, R(N/Z) –Correlation function, C(q) –Neutron, proton radii, E1 collective modes. High densities (  >  0)  –Neutron/proton, t/3He spectra and flows; C(q) –  + vs.  - production (Z.G.Xiao talk), k, hyperon production. Observables to probe E sym (  ) Why HICs ? (BA Li, Tsang, Danielewciz, LW Chen, QFLi, YX Zhang, ZG Xiao…… )

而对于中能重离子碰撞：包含各种复杂碎块的产生。 data ImQMD 能够较好的重现中能区重 离子碰撞的特点。 ImQMD T.X.Liu.et.al., PRC76, 利用 ImQMD 模型的计算结果与 MSU/NSCL 的 中 心以及擦边碰撞下的实验数据比较，获取低于饱和密度时对称能密度 依赖形式的信息；研究碎块发射对于观测量的影响

 Brief introduction on the ImQMD model  the mean fields acting on nucleon wavepackets are derived from Skyrme potential energy density functional Microscopic dynamics n-body transport model, nucleon is represented by a Gauss WF potential energy density functional: Change the symmetry potential in the transport model by varying the  主要改进与发展：

Parameters in V loc are obtained from standard Skyrme interactions parameters  P hase space constraint for considering Pauli principle The occupation number in cubic of h 3 should be small than 1(Pauli principle) Cugnon, et al., Nucl.Instr.Meth.Phys. B111, 215(1996)  Isospin dependent nucleon-nucleon cross sections are adopted, the medium corrections are Energy density functional from Momentum dependent interaction

成功的描述中能重离子碰撞的一些基本特点描述 1) multifragmentation Zhang, Li, PRC71(2005)24606 不同反应系统 不同入射能量下 （ <400AMeV)

2) Stopping and flow observables 集体流与直接流的激发函数 阻止能力 ImQMD Zhang, Li, PRC74,014602

 Constraints on the density dependence of symmetry energy with multi-observables from ImQMD model 1), the density dependence of symmetry energy with the interaction in ImQMD ImQMD 对称能密度依赖形式

2) The n/p ratios of emitted nucleons and the influence of cluster emission more pre-equilibrium neutrons get emitted from the neutron rich 124 Sn+ 124 Sn system relatively more pre-equilibrium neutrons are also emitted in the calculations with softer symmetry energy b=2fm YX Zhang, et al, PLB664(2008)145 对称势特点： 中子排斥 质子吸引 n/p ratio is related to the stiffness of symmetry potential at subsaturation density (BALi, PRL(1997))

Significant cluster and sequential decay effects are at low energy! Over the whole energy range for both free and coalescence- invariant DR(n/p) the data seem closer to the g i =0.5 calculation For very small gi, the system completely disintegrated, DR decreases and approaches the limit of reaction system. the double ratio values peak around g i ~0.7. By comparing with data, the g i =[0.4,1.05] within   <2  DR(n/p)=R n/p (124)/R n/p (112) Detailed comparison with data by varying g i MBTsang,YXZhang et.al., PRL102(2009) YX Zhang, P.Danielewicz, et al, PLB664(2008)145

3) The isospin transport ratio and isospin diffusion and the influence of cluster emission No isospin diffusion between symmetric systems Isospin diffusion occurs only in asymmetric systems A+B, and diffusion ability depends on the symmetry energy R i = 1 R i = -1 R i =(2X-X AA -X BB )/(X AA -X BB ) 同位旋扩散的能力由同位旋转移之比来量度 In absence of isospin diffusion R=1 or R=-1, R~0 for isospin equilibrium Theory: X is the , the isospin asymmetry of projectile residues Exp: X is the isoscaling parameter , There is linear relationship between  and , So, Ri(  )=Ri(  )

Tsang,YX Zhang, et.al., PRL102,122701(2009) Cluster emission effects make Ri larger for Ri ( d (Zmax>20)) The differences between Ri( d ) and Ri( d(Zmax>20) ) reflect the isospin diffusion mechanism for peripheral collisions. Isospin diffusion mainly occurs around neck region. X=  from heaviest fragments with Z>20 Fast equilibration occurs for soft symmetry energy within 2 s uncertainties, gamma_i=[0.4,1.0] X=  from projectile residues with relevant rapidity

4) The rapidity distribution of R 7 Ri increase with rapidity increasing with 2s uncertainties, gamma_i=[0.45,0.95] 利用 ImQMD 分析三种观测量，确定出在 2  的不确定度内  i 的范围是 [0.45,0.95], Cs=35.2MeV

Summary and Outlook The cluster emission play important roles on the isospin sensitive observables at intermediate energy HICs. The consistent constraints on the density dependence of symmetry energy at subsaturation density can be obtained from three observables, DR(n/p) ratios, Ri and R7 with ImQMD model. more observables and accurate data are needed in future in order to reduce the uncertainties on constraints on the density dependence of symmetry energy and reduce the uncertainties of the transport model.

与其他模型的比较 The results overlap with recent constraints obtained from giant dipole resonances, pygmy dipole resonance, and mass data. SMF, Wotler, arXiv:

中低能重离子输运理论中不确定性的来源： 平均场、动量相关势，核子有效质量 介质中核子－核子散射截面 泡里阻塞的处理 ……

International workshop on simulations of low and intermediate energy HICs ECT*, Italy, 2009 Same EOS, Same cross section different results!

输运模型计算上的一些技术细节同样会导致很大的不确定性  如： test part 波包宽度、如何抽取初始核， …… 需要更多、精确的实验数据来减少输运理论的不确定型。 输运模型本身还需进一步的发展。 谢谢！