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Nuclear Symmetry energy and Intermediate heavy ion reactions R. Wada, M. Huang, W. Lin, X. Liu IMP, CAS

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Symmetry Energy of nuclear matter What is important for Symmetry energy at T>>1 and dense or diluted nuclear matter? 1. Heavy ion collisions -- Symmetry energy is one of key factors to determine isotope distribution 2. Astrophysics -- Radius and cooling of neutron stars E sym (ρ) = E(N=A,N=0) – E(N=Z,A) : Energy difference between the symmetric nuclear matter (N=Z) and neutron matter E(ρ,δ)/A = {E(ρ,δ=0) + S(ρ) δ 2 }/A : δ = (N-Z)/A asymmetry

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A BE=17.87 ̶ 8.95A ̶ a sym I2/A BE=Mass Formula BE=17.87 ̶ 8.95A N=Z Weizsäcker-Bethe mass formula M(Z,N) = a v A – a s A 2/3 – a c Z(Z-1)/A 1/3 – a sym (N-Z) 2 /A – δ(N,Z)

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According to the Modified Fisher Model, the yield is given by Y(N,Z) = y 0 A ̶ τ · exp(-F/T) (ρ n ) N (ρ p ) Z = y 0 A ̶ τ · exp[(W(N,Z)+μ n N+ μ p Z)/T], W(N,Z) can be given by the following generalized Weizsäcker-Bethe semi-classical mass formulation at a given T and density ρ, W(N,Z) = a v (ρ,T)A – a s (ρ,T)A 2/3 – a c (ρ,T)Z(Z-1)/A 1/3 – a sym (ρ,T) I 2 /A – δ(N,Z) I=N-Z and δ is the paring energy, given by = δ 0 (for odd-odd nucleus) δ(N,Z) = 0 (for even-odd nucleus) = -δ 0 (for even-even nucleus). 1. Isotope yield and Symmetry energy Related fragments For convenience, Y(N,Z) = Y(I,A) ; I = N-Z, A = N+Z (M. Huang et al., PRC 81, 044620 (2010) ) Related reaction systems N=Z

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Y(I+2,A) = C A −τ exp{ [a v A – a s A 2/3 – a c (Z-1)(Z-2)/A 1/3 – a sym (I+2) 2 /A– δ(N+1,Z-1) +μ n (N+1)+μ p (Z-1)]/T } Y(I,A) = C A −τ exp{ [ a v A – a s A 2/3 – a c Z(Z-1)/A 1/3 – a sym I 2 /A – δ(N,Z) +μ n N+μ p Z]/T} R(I+2,I,A) = exp{ [2a c ·(Z-1)/A 1/3 – a sym ·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T } · exp[(μ n - μ p )/T] 1.Yield 2. Ratio cancel out When we focused on isotopes with same A in a given reaction, then For I= –1, drop out For even-odd, drop out Reaction system

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Symmetry energy term: R(I+2,I,A) = exp{[2a c ·(Z-1)/A 1/3 – a sym ·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T}·exp[(μ n - μ p )/T] When I = 1, N – Z = odd, so (N,Z) is even-odd or odd-even. – δ(N+1,Z-1) + δ(N,Z) = 0. ln[R(3,1,A)] = [2a c ·(Z-1)/A 1/3 – 8a sym /A]/T + (μ n - μ p )/T Exp. Isobaric ratio Isoscaling variance { Cal : AMD+Gemini Cal : AMD (primary) Primary Secondary Reactions used are : 64 Zn, 64 Ni + 58,64 Ni, 112,124 Sm, Au, 232 Th @ 40 A MeV

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Two issues: 1. Sequential decay in the cooling process drastically change the isotope distributions and causes a significant mass dependence of the symmetry energy. 2. Temperature (and density) cannot be determine uniquely. Y(N,Z) = y 0 A ̶ τ · exp[(W(N,Z)+μ n N+ μ p Z)/T], T relates all parameters by in a/T terms. No density information. 64 Zn+ 112 Sn @ 40 A MeV ( 40 Ca+ 40 Ca @ 35 A MeV ) have been studied in detail. Thermometers a. Kinematic energy slope b. Excited-state population c. Double isotope ratio d. Kinematic fluctuation

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1. Sequential decay issues : Kinematical focusing Correlated LP (M. Rodrigues et al., PRC 88, 034605 (2013) ) Uncorrelated LP v

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4.5≤V IMF <5.5 cm/ns 3.5≤V IMF <4.5 cm/ns 5.5≤V IMF <6.5 cm/ns data Total Uncorr(kM n (Li)) Corr(M n ( 23 Na)) θ IMF-n 4.5≤V IMF <5.5 cm/ns 35o35o 15 o 25 o 45 o

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Extracted Multiplicities Neutrons Secondary Reconstructed isotope Primary

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Reconstructed multiplicity distribution and multiplicity distribution of the AMD primary fragments Exp(cold) Reconstructed AMD primary (W. Lin et al., PR C 90, 044603 (2014) )

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2. Temperature, density and symmetry energy ( ) BE=17.87 ̶ 8.95A ̶ a sym I2/A BE=Mass Formula BE=17.87 ̶ 8.95A (X. Liu et al., NPA in Press, 2014) 3.Extract a sym /T values using all available isotopes with parameters defined in step (1) and (2).

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Δμ/T = 0.6 a c /T = 0.18 ) For I = –1 ln [R(1, -1, A)] A

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For N=Z=A/2 (I = 0) () BE=17.87 ̶ 8.95A ̶ a sym I2/A BE=Mass Formula BE=17.87 ̶ 8.95A Exp. Cal.

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Symbols Exp. Lines: Cal. Exp. AMD g0 g0AS g0ASS

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Exp. AMD g0 g0AS g0ASS g0/g0ASS g0/g0AS g0/Exp

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Temperature Extraction 40 Zn+ 112 Sn @ 40 AMeV AMD : 40 Ca+ 40 Ca @ 35 AMeV a sym (MeV) (X. Liu et al., PR C 90, 014605 (2014) )

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40 Zn+ 112 Sn @ 40 AMeV All Fragments have the same T : Modified Fisher Model assume thermal and chemical equilibriums in the fragmenting system

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T=T 0 (1- kA) k is determined iteratively; I.) In the first round k=k 1 =0. II.) Use step 1-3 to calculate all parameters and extract ρ, T and a sym values. III.) Determine a new slope parameter k’ from the temperature distribution. if k’=0, then stop and extract T 0. Otherwise set a new k as k 2 =k 1 +1/2(k’). Repeat the procedure II.) 1. For I = ─ 1 2. For N = Z 3. For N ≠ Z

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AMD : 40 Ca+ 40 Ca @ 35 AMeV 40 Zn+ 112 Sn @ 40 AMeV T=5.0MeV k=0.0 k=0.007

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T

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Mass Dependent apparent Temperature 1.Distribute the thermal energy to each fragemt by a Maxwellian distribution as 2. Require the momentum conservation. larger fragment has less probable to have large momentum larger fragments have smaller temperature Fluctuation thermometer results in a flat temperature AMD : 40 Ca+ 40 Ca @ 35 AMeV 3.A smaller system has a larger mass dependence 40 Zn+ 112 Sn @ 40 AMeV

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Summary: 1. Sequential decay modified the isotope distribution of the final products. This makes difficult to extract the properties of the fragmenting system at the freeze-out density. 5. The mass-dependent apparent temperature originates from the momentum conservation in the fragmentation system and it is system-mass dependent. Smaller systems show a larger mass dependence. 4. Modified Fisher Model is extended with an apparent mass-dependent temperature and ρ/ρ0 = 0.65 +/- 0.2, a sym = 23.1 +/- 0.6 MeV, T=5.0 +/- 0.4 MeV are determined. 3. Modified Fisher Model and a self-consistent method is applied for the reconstructed isotope distributions and AMD primary isotope distributions with different interactions in the symmetry energy density dependence. a. Density, temperature and symmetry energy values are extracted. b. Extracted temperature values show a mass dependence. 2. Kinematical focusing technique is applied to re construct the primary isotope distribution at the time of the fragment formation.

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Thank you for your attention !

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It is interesting to see how the density and temperature change as a function of incident energy. Unfortunately there are no available experimental data of the reconstructed isotope yield distributions in different incident energies. We performed AMD simulations at 35 - 300 AMeV for a 40 Ca+ 40 Ca system. 10,000 events were generated for b=0 fm at 35, 50, 80, 100, 140, 300 with g0,g0AS and g0ASS. Analyzed by the self-consistent method. (X. Liu et al., PRC in submission Oct. 2014) AMD with MFM and self-consistent method

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0 fm/c 10 fm/c 30 fm/c 60 fm/c 100 fm/c -10 0 10 Z cm Density Temperature 50 A MeV

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Statistical nature in AMD 3. Mass distribution can be described by an statistical ensemble. (Furuta et al., PRC79, 014608 (2009)) Statistical ensembles are made by AMD to enclose 36 nucleons in a spherical volume with given T and ρ (Volume). Mass distributions are evaluated as a long time average. Same code is used for AMD simulations and the statistical ensemble generation. 40 Ca+ 40 Ca @ 35 A MeV

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AMD: 64 Zn+ 197 Au : b ~ 2fm AMD Gemini

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N.Marie et al., PRC 58, 256, 1998 S.Hudan et al., PRC 67, 064613, 2003 Gemini Exp p d t h α 32 A MeV 39 A MeV 45 A MeV 50 A MeV

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20 0 64 Zn 47 A MeV Experiment IMF 20 o 129-300-1000-1000 μm Projectiles: 64 Zn, 64 Ni, 70 Zn at 40 A MeV Target : 58,64 Ni, 112,124 Sn, 197 Au, 232 Th 64 Zn+ 112 Sn at 40 A MeV

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Exp. vs AMD-Gemini Semi-violent collisions 16 O “Multi-fragmentation” Time v PLF v IV v TLF

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64 Ni+ 124 Sn at 40A MeV E (MeV) Z=14 15 16 17 18 19 20 7 8 9 10 11 12 13

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Black Histogram: Exp. Red: individual isotope Green : linear BG Blue: total Isotope Identification and yield evaluation

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I = N – Z = ̶ 1 : even-odd : R(1,-1,A) = exp{ 2a c ·(Z-1)/A 1/3 /T } · exp[(μ n - μ p )/T] ln[R(1,-1,A)] ̶ Δμ/T = 2a c ·(Z-1)/A 1/3 /T + (μ n - μ p )/T 1. Coulomb term and Chemical potential (μ n - μ p )/T= 0.71 : averaged values over all A = + : same for all reactions 1. Chemical potential between different reactions (μ n - μ p )/T= [(μ n - μ p )/T] 0 + Δμ (Z/A)/T Δμ (Z/A)/T=c 1 (Z/A)+c 2 (c 1 =-13.0 c 2 =8.7) a c /T = 0.35 (Z/A) sys R(I+2,I,A) = exp{ [2a c ·(Z-1)/A 1/3 – a sym ·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T } · exp[(μ n - μ p )/T] Reactions used are : 64 Zn, 64 Ni + 58,64 Ni, 112,124 Sm, Au, 232 Th @ 40 A MeV

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Symmetry energy term: R(I+2,I,A) = exp{[2a c ·(Z-1)/A 1/3 – a sym ·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T}·exp[(μ n - μ p )/T] When I = 1, N – Z = odd, so (N,Z) is even-odd or odd-even. – δ(N+1,Z-1) + δ(N,Z) = 0. : Fixed ln[R(3,1,A)] = [2a c ·(Z-1)/A 1/3 – 8a sym /A]/T + (μ n - μ p )/T Exp. Isobaric ratio Isoscaling variance { Cal : AMD+Gemini Cal : AMD (primary) Y(N,Z) = y 0 A ̶ τ · exp[(W(N,Z) +μ n N+ μ p Z)/T], W(N,Z) = a v A – a s A 2/3 – a c Z(Z-1)/A 1/3 –a sym I 2 /A – δ(N,Z)

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Published works: A. Isoscaling and Symmetry energy 1. Z. Chen et al., “Isocaling and the symmetry energy in the multifragmentation regime of heavy-ion collisions”, Phys. Rev. C 81, 064613 (2010) 2. M. Huang et al., “A novel approach to Isoscaling: the role of the order parameter m=(N_f-Z_f)/A_f”, Nuclear Physics, A 847, 233 (2010) B. Isobaric yield ratio and Symmetry energy 3. M. Huang et al., “Isobaric yield ratios and the symmetry energy in heavy-ion reactions near the Fermi energy”, Phys. Rev. C 81, 044620 (2010) C. Landau formulation of isotope yield and critical phenomena 4. A. Bonasera et al., “Phys. Rev. Lett. 101,122702 (2008), 5. M. Huang et al., “Isospin dependence of the nuclear equation of state near the critical point”, Phys. Rev. C 81, 044618 (2010) D.Power Law distribution and critical phenomena 6. M. Huang et al., “Power law behavior of isotope yield distribution in the multifragmentation regime of the heavy ion reactions”, Phys. Rev. C82,054602 (2010)

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