Betty Tsang Subal Das Gupta Festschrift McGill University, Montreal Dec 4, 2004

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1 Betty Tsang Subal Das Gupta Festschrift McGill University, Montreal Dec 4, 2004 The National Superconducting Cyclotron Laboratory Michigan State University

2 Subal Das Gupta Festschrift McGill University, Montreal Dec 4, 2004 The National Superconducting Cyclotron Laboratory Michigan State University

3 Isoscaling in Nuclear Reactions Betty Tsang Subal Das Gupta Festschrift McGill University, Montreal Dec 4, 2004 The National Superconducting Cyclotron State University

4 Outline What is isoscaling? Where is it observed? From multifragmentation to binary reactions What is the physics of isoscaling? Subal D. What can we learn from it? Density dependence of symmetry energy.

5 Isoscaling constructed from Measured Isotopic yields T.X Liu et al. PRC 69, P T

6 Isoscaling from Relative Isotope Ratios e R 21 =Y 2 / Y 1 N µ / T + Z µ n p / T MB Tsang et al. PRC 64,054615

7 112 Sn+ 58 Ni and 124 Sn+ 64 Ni at 35 AMeV; Central collisions, CHIMERA-REVERSE experiment E. Geraci et al., Nucl. Phys. A732 (2004) 173

8 Simple derivation of the isoscaling law Basic trends from Grand Canonical ensemble: Yields term with exponential dependence on the chemical potentials. Ratios to reduce sensitivity to secondary decays: Scaling parameters C, [ ] ( ) ( ) ), ( ), ( ), ( ) / exp( 1 2 ), ( / ), ( exp ), ( * int int Z N f Z N Y Z N Y T E J where Z Z N Z T Z N B Z N Z N Y HOT COLD i i i p n HOT = + = + + µ µ feeding correction ( ) T Z T N C Z N Y Z N Y Z N R / / p n e ), ( ), (, µ µ + = T T p n /, / µ β µ α = =

9 Isoscaling in statistical models : PRC 64, (2001) 1. Temperature dependence 2. Density dependence 3. Size and charge dependence 4. α δ 1 δ 2 δ = (N-Z)/(N+Z) Aspects of statistical model for multifragmentation P. Bhattacharyya, S. Das Gupta, and A. Z. Mekjian PRC 60, (1999)

10 Isoscaling in statistical models Primary distributions show good isoscaling A 2 =186, Z 2 =75; A 1 =168, Z 1 =75 WCI statistical model working group (2004)

11 Isoscaling in Antisymmetrized Molecular Dynamical model A. Ono et al. PRC 68, (2003)

12 Isoscaling observed in many reactions Y 2 / Y 1 e ( N µ + Z µ n p ) / T More Data 58 Ni+ 58 Ni 58 Fe+ 58 Fe E/A=30,40,47 Shetty et al (2003) p,4 He+ 116 Sn p,4 He+ 124 Sn E/A>1 GeV Botvina,Trautmann (2002) 86 Kr+ 116 Sn, 124 Sn 86 Kr+ 58 Ni, 64 Ni E/A=35 MeV Souliotis et al(2003) PRL, 86, 5023 (2001)

13 P b T Q Value, Sep. E E Coul E sym R 21 exp[(- S n N- S p Z)/T] P T Separation Energy E Coul E sym R 21 exp[((- S n + f n* ) N+(- S p + f p* + Φ) Z)/T] P T Chemical Potentials R 21 exp[(- µ n N- µ p Z)/T] E Coul E sym ρ p ρ n

14 Origin of isoscaling

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16 Origin of isoscaling Isoscaling disappears when the symmetry energy is set to zero Provides an observable to study symmetry energy

17 Symmetry energy from AMD A. Ono et al. PRC 68, (2003) α depends on symmetry term interactions

18 Boltzmann equation for heavy ion collisions G. F. Bertsch and H. Kruse, S. Das Gupta PRC 29, 673 (1984) Sn+Sn; E/A=50 MeV b=6.6 fm Micha Kilburn, REU 2003

19 Isospin diffusion in the projectile-like region Basic ideas: Peripheral reactions Asymmetric collisions 124 Sn+ 112 Sn, 112 Sn+ 124 Sn -- diffusion Lijun Shi Projectile Target δ = ( N Z) /( N + Z)

20 Isospin diffusion in the projectile-like region Basic ideas: Peripheral reactions Asymmetric collisions 124 Sn+ 112 Sn, 112 Sn+ 124 Sn -- diffusion Symmetric Collisions 124 Sn+ 124 Sn, 112 Sn+ 112 Sn -- no diffusion Relative change between target and projectile is the diffusion effect Target δ = ( N Z) /( N + Z)

21 Isoscaling of mixed systems Y 21 exp(αn+βz)

22 Experimental: isoscaling;y 21 exp(αn+βz) Theoretical : δ = (N-Z)/(N+Z) α δ 1 δ 2 (Subal s SMM) Isospin Transport Ratio R i = 2x x x x x x=experimental or theoretical isospin observable x=x R i = 1. x=x R i = -1. Rami et al., PRL, 84, 1120 (2000)

23 BUU predictions Lijun Shi Ε(ρ, δ) = Ε(ρ, 0)+S sym (ρ) δ 2 S sym (ρ) ρ γ Experimental results are in better agreement with predictions using hard symmetry terms

24 BUU predictions Ε(ρ, δ) = Ε(ρ, 0)+S sym (ρ) δ 2 S sym (ρ) (ρ) γ Including the momentum dependence in the mean-field in BUU changes the agreement Need more experimental constraints B.-A. Li, C. B. Das, S. Das Gupta, and C. Gale Phys. Rev. C 69, (2004)

25 Ln(Y( 7 Li)/Y( 7 Be)) & R i ( 7 Li, 7 Be) Mirror nuclei ratios provide another observable: R = Ln ) µ µ 7 ( Y7 Be / Y7 Li p n previous results using α Rapidity dependence is weak except at mid rapidity May provide better constraints on the momentum dependence of the mean field

26 Summary A lot of work has been done on isoscaling. Robust observable Seen in many different reactions Promising tool to study symmetry energy with heavy ion collisions Isospin Diffusion

27 Acknowledgements Subal Das Gupta P. Danielew icz, C.K. Gelbke, T.X. Liu, X.D. Liu, W.G. Lynch, L.J. Shi, R. Shomin, M.B. Tsang, W.P. Tan, M.J. Van Goethem, G. Verde, A. Wagner, H.F. Xi, H.S. Xu, Akira Ono, Bao-An Li, B. Davin, Y. Larochelle, R.T. de Souza, R.J. Charity, L.G. Sobotka, S.R. Souza, R. Donangelo

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