1. 重离子碰撞与核物质状态方程 陈列文 ( 上海交通大学物理系 lwchen@sjtu.edu.cn) 2009 威海高能物理暑期论坛暨 “RHIC 物理和 CSR 强 子物理 ” 研讨会， 2009 年 8 月 8 日 -14 日，威海，山东

2. 目录 非对称核物质的状态方程与对称能 同位旋相关的重离子碰撞微观输运模型 重离子碰撞：对称核物质的状态方程 重离子碰撞：对称能的低密行为 重离子碰撞：对称能的高密行为 对称能对其他物理量的约束 总结和展望 Main References: L.W. Chen, C.M. Ko, B.A. Li, and G.C. Yong, Front. Phys. China 2(3), 327 (2007) B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113-281 (2008)

3. 一、非对称核物质的状态方程与对称能 极端条件下的核物理 Energy, Temperature, Density: Phase Transitions: Liquid-Gas; Chiral Restoration QGP-Hadrons …… Isospin: Nuclei far from Beta- Stability Line …… Spin: High Spin State and, SD (Super- Deformed), HD (Hyper- Deformed) …… A and Z: SHE (Super-Heavy Elements) …… HIC’s

4. 同位旋核物理 : 核结构 18 12 3 Isospin asymmetry δ=(ρ n -ρ p )/ρ ρ=ρn+ρpρ=ρn+ρp density Symmetric matter ρn=ρpρn=ρp Studied extensively A new dimension Very small isospin asymmetry ???

5. 同位旋核物理 : 中能重离子碰撞 Density Dependence of the Nuclear Symmetry Energy HIC’s induced by neutron-rich nuclei (CSR/Lanzho u,FRIB,GSI, RIKEN……) Most uncertain property of an asymmetric nuclear matter Isospin Nuclear Physics What is the isospin dependence of the in-medium nuclear effective interactions??? Neutron Stars … Structures of Radioactive Nuclei, SHE … Isospin Effects in HIC’s … Many-Body Theory Transport Theory General Relativity Nuclear Force EOS for Asymmetric Nuclear Matter On Earth!!! In Heaven!!!

6. 同位旋核物理 : 对称能 The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005). Isospin physics Isospin physics

8. 重离子加速器装置 1.HIRFL, CSR/HIRFL (China) 2.GANIL (France) 3.GSI (Germany) 4.NSCL/MSU,FRIB/MSU 5.RIKEN (Japan) 1 50 100 150 200 250 300 10 10 2 10 3 10 4 A E (MeV/u) HIRFL GANIL NSCL/MSU CSR/HIRFL RIKEN GSI Dubna, LBL, ORNL, TAMU, INFN, KVI,… AGS,RHIC/BNL SPS,LHC/CERN 中能重离子加速器装置

9. Radioactive beam facilities are being built around the world IMP CIAE Providing new opportunities for both nuclear physics and astrophysics World status of Rare Isotope Accelerators

10. 核物质的状态方程

11. Liquid-drop model 核的对称能 Symmetry energy term W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,…… Symmetry energy including surface diffusion effects (y s =S v /S s )

12. 核物质的对称能 EOS of Asymmetric Nuclear Matter (Parabolic law) The Nuclear Symmetry Energy Symmetry energy term Symmetric Nuclear Matter

13. Nuclear Matter EOS: Many-Body Approaches Microscopic Many-Body Approaches Non-relativistic Brueckner-Bethe-Goldstone (BBG) Theory Relativistic Dirac-Brueckner-Hartree-Fock (DBHF) approach Self-consistent Green’s Function (SCGF) Theory Variational Many-Body (VMB) approach …… Effective Field Theory Density Functional Theory (DFT) Chiral Perturbation Theory (ChPT) …… Phenomenological Approaches Relativistic mean-field (RMF) theory Relativistic Hartree-Fock (RHF) Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock) Thomas-Fermi (TF) approximations Phenomenological potential models ……

14. Nuclear Matter EOS: Many-Body Approaches Saturation Pionts of Symmetric Nuclear Matter from Microscopic Many-Body Approaches: The Coester Line

15. Nuclear Matter EOS: Isospin Dependence I. Tanihata Preprint RIKEN-AF-NP-229, July, 1996. N/Z=0,0.2,0.4,0.6,0.8,1 (From top to bottom) Z.H. Li, (2007) Van Dalen/Fuchs/Faessler EPJA31, 29 (2007) SIIITM1

16. 核物质的对称能 Chen/Ko/Li, PRC72, 064309(2005) Chen/Ko/Li, PRC76, 054316(2007) Z.H. Li et al., PRC74, 047304(2006)Dieperink et al., PRC68, 064307(2003) BHF

17. 二、同位旋相关的重离子碰撞微观输运模型 Broad applications of transport models in astrophyics, plasma physics, electron transport in semiconductor and nanostructures, particle and nuclear physics, nuclear stockpile stewardship Transport Models for HIC’s at intermediate energies: N-body approaches CMD, QMD,IQMD,IDQMD, ImQMD,ImIQMD,AMD,FMD One-body approaches BUU, BNV, LV, IBL RBUU,RVUU,… Transport Models

18. Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin- (momentum-) dependent mean field potential Isospin-dependent N-N cross sections a. Experimental free space N-N cross section σ exp b. In-medium N-N cross section from the Dirac-Brueckner approach based on Bonn A potential σ in-medium c. Mean-field consistent cross section due to m* Isospin-dependent Pauli Blocking Isospin-dependent BUU (IBUU) model Transport model for HIC’s EOS

19. Isospin- and momentum-dependent potential (MDI) Chen/Ko/Li, PRL94,032701 (2005) Li/Chen, PRC72, 064611 (2005) Das/Das Gupta/Gale/Li, PRC67,034611 (2003) Transport model: IBUU04

20. in neutron-rich matter is the reduced mass of the colliding pair NN in medium J.W. Negele and K. Yazaki, PRL 47, 71 (1981) V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992) M. Kohno et al., PRC 57, 3495 (1998) D. Persram and C. Gale, PRC65, 064611 (2002). 1.In-medium cross sections are reduced 2.nn and pp cross sections are splitted due to the neutron-proton effective mass slitting in neutron-rich matter Li/Chen, PRC72 (2005)064611 Medium effects: effective mass on the incoming current in initial state and level density of the final state Neglecting medium effects on the transition matrix In-medium Nucleon-nucleon cross sections: Effective mass scaling model

21. 三、重离子碰撞：对称核物质的状态方程 Why Heavy-Ion Collisions? It is very difficult to obtain information on the nuclear matter EOS at higher densities from nuclear properties around normal density which can be extracted from nuclear structure of finite nuclei and nuclear excitation! LW Chen et al., PRC80, 014322 (2009)

22. (1) EOS of symmetric matter around the saturation density ρ 0 Giant Monopole Resonance K 0 =231±5 MeV PRL82, 691 (1999) Recent results: K 0 =240±10 MeV G. Colo et al. U. Garg et al. __ 对称核物质的状态方程

23. (2) EOS of symmetric matter for 1ρ 0 < ρ < 3ρ 0 from K + production in HIC’s J. Aichelin and C.M. Ko, PRL55, (1985) 2661 C. Fuchs, Prog. Part. Nucl. Phys. 56, (2006) 1 Transport calculations indicate that “results for the K + excitation function in Au + Au over C + C reactions as measured by the KaoS Collaboration strongly support the scenario with a soft EOS.” C. Fuchs et al, PRL86, (2001) 1974 See also: C. Hartnack, H. Oeschler, and J. Aichelin, PRL96, 012302 (2006)

24. (3) Present constraints on the EOS of symmetric nuclear matter for 2ρ 0 < ρ < 5ρ 0 using flow data from BEVALAC, SIS/GSI and AGS Use constrained mean fields to predict the EOS for symmetric matter Width of pressure domain reflects uncertainties in comparison and of assumed momentum dependence. P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002) The highest pressure recorded under laboratory controlled conditions in nucleus-nucleus collisions High density nuclear matter 2 to 5ρ 0 对称核物质的状态方程

25. Promising Probes of the E sym (ρ) in Nuclear Reactions (an incomplete list !) 四、重离子碰撞：对称能的低密行为

26. 对称能探针：同位旋弥散 How to measure Isospin Diffusion? PRL84, 1120 (2000) ______________________________________ A+A,B+B,A+B X: isospin tracer Isospin Diffusion/Transport

27. 对称能探针：同位旋弥散 int Isospin diffusion in 124 Sn+ 112 Sn@E/A=50 MeV and b=6 fm The theoretical analysis (BUU) did NOT include Momentum Dependence for nuclear potential ! If complete isospin mixing/ equilibrium

28. 对称能探针：同位旋弥散 Chen/Ko/Li PRL94, (2005) 032701 Momentum-independent Momentum-dependent Comparing momentum-dependent IBUU04 clculations with data on isospin transport All have the same Esym (ρ)=31.6 (ρ/ρ 0 ) 1.05

29. 对称能探针：同位旋弥散 Li/ Chen, PRC72, 064611(2005) Symmetry energy, isospin diffusion, in-medium cross section Isospin Diffusion Data  K asy =-500±50 MeV L=86±25 MeV Chen/Ko/Li, PRC72,064309 (2005)

30. 对称能探针：同位旋标度 Isoscaling in HIC’s Isoscaling observed in many reactions M.B. Tsang et al. PRL86, 5023 (2001)

31. 对称能探针：同位旋标度 Consistent with isospin diffusion data! Constraining Symmetry Energy by Isocaling: TAMU Data Shetty et al. PRC75(07);PRC76(07)

32. IBUU04 : S~31.6(  /  o )    ImQMD: n/p ratios and two isospin diffusion measurements Consistent constraints from the  2 analysis of three observables S=12.5(  /  o ) 2/3 + 17.6 (  /  o )  i  i   i   i  ii 对称能探针：同位旋弥散和双 n/p 比率 Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009)

33. (IBUU04) (ImQMD) X=-1 Tsang et al., PRL 102, 122701 (2009) 对称能的亚饱和密度行为 Chen/Ko/Li, PRL 94, 032701 (2005) ( 亚饱和密度： 0.2-0.3<ρ/ρ 0 <1.2)

34. 极端低密时的对称能：结团效应 S. Kowalski, et al., PRC 75 (2007) 014601. Horowitz and Schwenk, Nucl. Phys. A 776 (2006) 55

35. 对称能探针：轻粒子产生 The coalescence model Chen/Ko/Li, PRC68, 014605 (2003); NPA729, 809 (2003)  Depends on constituents’ space-time structure at freeze-out  Neglecting the binding energy effect (T>>E binding ), Coalescence probability: Wigner phase-space density in the rest-frame of the cluster.  Rare process has been assumed (the coalescence process can be treated perturbatively). Higher energy collisions and higher energy cluster production! Light Cluster Production and Coalescence Model

36. 对称能探针：轻粒子产生 Isospin symmetric collisions at E/A≈100 MeV  Deuteron energy spectra reproduced  Low energy tritons slightly underestimated  Inverse slope parameter of 3 He underestimated; probably due to neglect of larger binding effect stronger Coulomb effect wave function Data are taken from INDRA Collaboration (P. Pawlowski, EPJA9) Try Coalescence model at intermediate energies! Chen/Ko/Li, NPA729, 809 (2003)

37. 对称能探针：轻粒子产生 Symmetry Energy Effects on t/ 3 He ratio Stiffer symmetry energy gives smaller t/ 3 He ratio

38. 对称能探针：核子 - 核子关联函数 The two-particle correlation function is obtained by convoluting the emission function g(p,x), i.e., the probability of emitting a particle with momentum p from space-time point x=(r,t), with the relative wave function of the two particle, i.e., The two-particle correlation function is a sensitive probe to the space-time structure of particle emission source by final state interaction and quantum statistical effects ( φ (q,r)) Correlation After Burner (Crab): including final-state nuclear and Coulomb interactions (S. Pratt, NPA 566, 103 (1994)) How to detect the space-time structure of nucleon emission experimentally? Two-Nucleon Correlation Functions

39. 对称能探针：核子 - 核子关联函数  High momentum nucleons emitted earlier than low momentum ones  Earlier emissions for stiffer symmetry energy  Larger separation in neutron and proton emission times for softer symmetry energy Space-time structure of nucleon emission Emission times from IBUU Chen/Ko/Li, PRL90, 162701 (2003)

40. 对称能探针：核子 - 核子关联函数 MDI Das, Das Gupta, Gale and Li PRC67, (2003) Effects of momentum-dependence of nuclear potential Pairs with P>500 MeV: n-p CF: 11% The isospin effects on two-particle correlation functions are really observed in recent experimental data !!! R. Ghetti et al., PRC69 (2004) 031605 肖志刚等, PLB, (2006) Chen/Ko/Li, PRC69, 054606 (2004)

41. 五、重离子碰撞：对称能的高密行为 n/p ratio of the high density region Li/Yong/Zuo, PRC 71, 014608 (2005) Isospin fractionation! Heavy-Ion Collisions at Higher Energies

42. 对称能高密探针 :pion 比率

43. IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102, 062502(2009) 对称能高密探针 :pion 比率 A Quite Soft Esym at supra-saturation densities !!! M. Zhang et al., arXiv:0904.0447

44. Subthreshold K 0 /K + yield may be a sensitive probe of the symmetry energy at high densities Aichelin/Ko, PRL55, 2661 (1985): Subthreshold kaon yield is a sensitive probe of the EOS of nuclear matter at high densities Theory: Famiano et al., PRL97, 052701 (2006)Exp.: Lopez et al. FOPI, PRC75, 011901(R) (2007) K 0 /K + yield is not so sensitive to the symmetry energy! Lower energy and more neutron-rich system??? 对称能高密探针 :Kaon 比率

45. 对称能高密探针：挤出 n/p 比率 In the squeeze-out direction: nucleons emitted from the high density participant region have a better chance to escape without being hindered by the spectators. These nucleons thus carry more direct information about the high density phase of the reaction. Yong/Li/Chen, nucl-th/0703042 PLB650, 344 (2007) The effect can be 40% at higher p T !

46. 对称能高密探针 : n-p 差分流 Neutron-proton differential flow pxpx y symmetry potential is generally repulsive for neutrons and attractive for protons for n and p Bao-An Li, PRL 85, 4221 (2000). Yong/Li/Zuo, High energy physics and nuclear physics (2005). Isospin asymmetry of free nucleons

47. 六、对称能对其他物理量的约束 Chen/Ko/Li, PRC72,064309 (2005) (1) Symmetry Energy from SHF and RMF Chen/Ko/Li, PRC76, 054316(2007) Only 4/21! Only 5/23!

48. 对称能：中子皮 (2) Neutron-Skin of 208 Pb from SHF Good linear Correlation: S-L Chen/Ko/Li, PRC72,064309 (2005)

49. Lattimer/Prakash, Science 304, 536 (2004) Neutron star has solid crust over liquid core. Rotational glitches: small changes in period from sudden unpinning of superfluid vortices. Evidence for solid crust. 1.4% of Vela moment of inertia glitches. Needs to know the transition density to calculate the fractional moment of inertia of the crust Link et al., PRL83,3362 (99) core-crust transition 对称能：中子星的壳芯转变密度

50. Kazuhiro OyamatsuKazuhiro Oyamatsu, Kei IidaKei Iida Phys. Rev. C75 (2007) 015801 pasta Xu/Chen/Li/Ma, PRC79, 035802 (2009) Xu/Chen/Li/Ma, ApJ 697, 1547 (2009), arXiv:0901.2309 Parabolic Approximation has been assumed !!! Significantly less than their fiducial values: ρ t =0.07-0.08 fm -3 and P t =0.65 MeV/fm 3 对称能：中子星的壳芯转变密度

51. The softest symmetry energy that the TOV is still stable is x=0.93 giving M_max=0.11 solar mass and R=>28 km For pure nucleonic matter? Hyperon? Quark? New Physics??? Soft symmetry energy at HD ? K 0 =211 MeV is used, higher incompressibility for symmetric matter will lead to higher masses systematically ? 对称能的高密行为：中子星？ Medium effects on pion production??? Xu/Ko/Oh, arXiv:0906.1602

52. 七、总结和展望 The isospin diffusion data, Isoscaling and double n/p ratio seem to give a stringent constraint for the sub-normal density behavior of the symmetry energy (L=86±25 MeV and K asy =-500±50 MeV) (Crosscheck is definitely needed !!!) Pion ratio data from FOPI favors a supersoft symmetry energy at super-saturation densities. Probing the high density behavior of the symmetry energy remains a big challenge and CSR/Lanzhou can make important contribution. Experiments: detectors (n,p,fragments,pi,kaons,photons… ) The transport model provides an important tool to study equation of state of asymmetric nuclear matter, especially for the nuclear symmetry energy: How to treat fragments? Clustering effects? Medium effects? Many-body collisions? Lorentz Covariance? Self-consistency?……

53. 谢谢！

54. Development and Challenges of Transport Models 1. Development of transport models Transport codes often implement extra physical assumptions and dynamical mechanisms which go beyond the equations used to motivate their designs. These algorithms often undergo evolutions with time as we make progresses in our R&D efforts and also as our needs for including new processes arise. They may involve many phenomenological parameters which are not all well experimentally constrained yet because of the lack of the relevant experimental data, and some of them are exactly what we want to learn. 2. Challenges of transport models for reactions involving radioactive beams Develop practically implementable quantum transport theories ( the de Broglie wavelength may be comparable to the nucleon mean free path in energetic central reactions, and the uncertainty principle imposes a strong correlation between delocalization of nucleons and their momentum distribution) Include more structure information in the initial state especially for peripheral reactions. Use consistently all inputs (initial state, mean field and б NN ) from the same interactions 3. Self-consistent Isospin dependent Lorentz Covariant transport models for HIC’s at higher energies 4. Many-body collisions and clustering effects

55. 放射性核束装置 Cooling Storage Ring (CSR) Facility at HIRFL in China up to 500 MeV/u for 238 U http://www.impcas.ac.cn/zhuye/en/htm/247.htm. Radioactive Ion Beam (RIB) Factory at RIKEN in Japan http://www.riken.jp/engn/index.html FAIR/GSI in Germany up to 2 GeV/u for 132 Sn http://www.gsi.de/fair/index_e.html. SPIRAL2/GANIL in France http://ganinfo.in2p3.fr/research/developments /spiral2 Facility for Rare Isotope Beams (FRIB) in USA up to 200 MeV/u for 132Sn http://www.frib.msu.edu

56. 对称能探针：轻粒子产生 Hulthen wave function Wigner phase-space density for Deuteron Wigner transformation

57. 对称能探针：轻粒子产生 t/ 3 He Wigner phase-space density and root-mean-square radius: Wigner phase-space density for t/ 3 He Assume nucleon wave function in t/ 3 He can be described by the harmonic oscillator wave function, i.e.,

58. 对称能探针：轻粒子产生 t/ 3 He ratio Still sensitive to the symmetry energy Effects of momentum-dependence of nuclear potential Chen/Ko/Li, PRC69, 054606 (2004)

59. 对称能探针：核子 - 核子关联函数 Pairs with P>500 MeV: n-n CF: 20% p-p CF: 20% n-p CF: 30% Symmetry Energy Effects on Two-Nucleon Correlation Functions Effects are very small for both isoscalar potential and N-N cross sections Chen/Ko/Li, PRL90, 162701 (2003)

60. 对称能探针：同位旋标度 Li/Chen, PRC74,034610 (2006) Xu/Chen/Li/Ma, PRC75,014607 (2007)

61. 四、对称能对其他物理量的约束 Chen/Ko/Li, PRC72,064309 (2005) Nuclear Matter Symmetry Energy from SHF

62. 四、对称能对其他物理量的约束 Non-linear RMF(10): NL1,NL2,NL3,NL-SH, TM1,PK1,FSU-Gold. HA,NLρ,NLρ Density-dependent RMF(7): TW99,DD-ME1,DD-ME2 PKDD,DD,DD-F DDRH-corr Point-Coupling RMF(6): PC-F1,PC-F2,PC-F3, PC-F4PC-LA FKVW Chen/Ko/Li, PRC76, 054316(2007) Nuclear Matter Symmetry Energy from RMF

63. 四、对称能对其他物理量的约束 Neutron-Skin of 208 Pb from SHF Good linear Correlation: S-L Chen/Ko/Li, PRC72,064309 (2005)

64. 四、对称能对其他物理量的约束 Chen/Ko/Li PRC72, 064309 (05) Constrain N-Skin from obtained Esym

65. Lattimer/Prakash, Science 304, 536 (2004) Neutron star has solid crust over liquid core. Rotational glitches: small changes in period from sudden unpinning of superfluid vortices. Evidence for solid crust. 1.4% of Vela moment of inertia glitches. Needs to know the transition density to calculate the fractional moment of inertia of the crust Link et al., PRL83,3362 (99) core-crust transition 四、对称能对其他物理量的约束

66. Thermodynamic approach Or, similarly one can use the RPA If one uses the parabolic approximation (PA) Then the stability condition is: >0 均匀电中性 beta 稳定物质的不稳定性 Dynamical approach k  0 (neglecting Coul.) Stability condition:

67. 壳芯 (core-crust) 转变密度：抛物线近似失效 ! (1)It is NOT enough to know the symmetry energy, one almost has to know the exact EOS of n-rich matter Why? Because it is the determinant of the curvature matrix that determines the stability condition Example: Not so surprise: Zhang/Chen, CPL 18 (2000) 142 Steiner, Phys.Rev. C74 (2006) 045808 Higher-order term effects on direct URCA Xu/Chen/Li/Ma, PRC79, 035802 (2009)

68. (2) Locating the inner edge of neutron star crust Kazuhiro OyamatsuKazuhiro Oyamatsu, Kei IidaKei Iida Phys. Rev. C75 (2007) 015801 pasta Xu/Chen/Li/Ma, PRC79, 035802 (2009) Xu/Chen/Li/Ma, ApJ, in press (2009), arXiv:0901.2309 Parabolic Approximation has been assumed !!! Significantly less than their fiducial values: ρ t =0.07-0.08 fm -3 and P t =0.065 MeV/fm 3

69. (3) Constraints on M-R relation of NS (Isospin Diff) (Empirical estimate Link et al., PRL83,3362(99))

70. (4) Properties of neutron star crusts Xu/Chen/Li/Ma, ApJ, in press (2009), arXiv:0901.2309 Larger L leads to thicker neutron-skin, but thinner neutron star crust !!!

71. The softest symmetry energy that the TOV is still stable is x=0.93 giving M_max=0.11 solar mass and R=>28 km For pure nucleonic matter??? Soft symmetry energy at HD ? K 0 =211 MeV is used, higher incompressibility for symmetric matter will lead to higher masses systematically ? (5) HD Esym and properties of neutron stars