1. Constraining the properties of dense matter A.What is the EOS   1. Theoretical approaches   2. Example:T=0 with Skyrme   3. Present status   a) symmetric matter b) asymmetric matter and symmetry term. 4. Astrophysical relevance   B. Summary of first lecture   C. What observables are sensitive to the EOS and at what densities? 1. Binding energies   2. Radii of neutron and proton matter in nuclei   3. Giant resonances   4. Particle flow and particle production: symmetric EOS   5. Particle flow and particle production: symmetry energy   D Summary   A.What is the EOS   1. Theoretical approaches   2. Example:T=0 with Skyrme   3. Present status   a) symmetric matter b) asymmetric matter and symmetry term. 4. Astrophysical relevance   B. Summary of first lecture   C. What observables are sensitive to the EOS and at what densities? 1. Binding energies   2. Radii of neutron and proton matter in nuclei   3. Giant resonances   4. Particle flow and particle production: symmetric EOS   5. Particle flow and particle production: symmetry energy   D Summary   William Lynch, Michigan State University

5. O

6. Theoretical Approaches Variational and Bruekner model calculations with realistic two-body nucleon- nucleon interactions: (see Akmal et al., PRC 58, 1804 (1998) and refs therein.) –Variational minimizes with elaborate grounds state wavefunction that includes nucleon-nucleon correlations. –Incorporate three-body interactions. Some are "fundamental" Others model relativistic effects. Relativistic mean field calculations using relativistic effective interactions, (see Lalasissis et al., PRC 55, 540 (1997), Peter Ring lectures) –Well defined transformations under Lorentz boosts –Parameterization can be adjusted to incorporate new data. Skyrme parameterizations: ( Vautherin and Brink, PRC 5, 626 (1972).) –Requires transformation to local rest frame –Computationally straightforward - exampleexample Variational and Bruekner model calculations with realistic two-body nucleon- nucleon interactions: (see Akmal et al., PRC 58, 1804 (1998) and refs therein.) –Variational minimizes with elaborate grounds state wavefunction that includes nucleon-nucleon correlations. –Incorporate three-body interactions. Some are "fundamental" Others model relativistic effects. Relativistic mean field calculations using relativistic effective interactions, (see Lalasissis et al., PRC 55, 540 (1997), Peter Ring lectures) –Well defined transformations under Lorentz boosts –Parameterization can be adjusted to incorporate new data. Skyrme parameterizations: ( Vautherin and Brink, PRC 5, 626 (1972).) –Requires transformation to local rest frame –Computationally straightforward - exampleexample

7. Example: Skyrme interaction.

8. Hint: use the expressions for the differential increases in potential energy per unit volume above and do a parametric integration over from zero to one.

11. Brown, Phys. Rev. Lett. 85, 5296 (2001) The density dependence of symmetry energy is largely unconstrained. Unlike symmetric matter, the potential energy of neutron matter is repulsive. O

12. Constraints come mainly from collective flow measurements. Know pressure is zero at  =  0. Results from variational calculations and Relativistic mean field theory with density dependent couplings lie within the allowed boundaries. Constraints come mainly from collective flow measurements. Know pressure is zero at  =  0. Results from variational calculations and Relativistic mean field theory with density dependent couplings lie within the allowed boundaries. Constraints on symmetric and asymmetric matter EOS E/A ( ,  ) = E/A ( ,0) +  2  S(  )  = (  n -  p )/ (  n +  p ) = (N-Z)/A  1 Neutron matter EOS also includes the poorly constrained pressure from the symmetry energy. The uncertainty from the symmetry energy is larger than that from the symmetric matter EOS. Neutron matter EOS also includes the poorly constrained pressure from the symmetry energy. The uncertainty from the symmetry energy is larger than that from the symmetric matter EOS. O Danielewicz et al., Science 298,1592 (2002). Danielewicz et al., Science 298,1592 (2002).

13. Type II supernova: (collapse of 20 solar mass star) Supernovae scenario: (Bethe Reference) –Nuclei H  He  C ...  Si  Fe –Fe stable, Fe shell cools and the star collapses –Matter compresses to  >4  s and then expands Relevant densities and matter properties –Compressed matter inside shock radius  0 <  <10  0,  0.4–0.9 What densities are achieved? What is the stored energy in the shock? What is the neutrino emission from the proto-neutron star? –Clustered matter outside shock radius – mixed phase of nucleons and nuclear drops - nuclei:  <  0,  0.3–0.5 How much energy is dissipated in vaporizing the drops during the explosion? What is the nature of the matter that interacts and traps the neutrinos? What are the seed nuclei that are present at the beginning of r-process which makes roughly half of the elements? Supernovae scenario: (Bethe Reference) –Nuclei H  He  C ...  Si  Fe –Fe stable, Fe shell cools and the star collapses –Matter compresses to  >4  s and then expands Relevant densities and matter properties –Compressed matter inside shock radius  0 <  <10  0,  0.4–0.9 What densities are achieved? What is the stored energy in the shock? What is the neutrino emission from the proto-neutron star? –Clustered matter outside shock radius – mixed phase of nucleons and nuclear drops - nuclei:  <  0,  0.3–0.5 How much energy is dissipated in vaporizing the drops during the explosion? What is the nature of the matter that interacts and traps the neutrinos? What are the seed nuclei that are present at the beginning of r-process which makes roughly half of the elements? O

14. Neutron Stars Neutron Star stability against gravitational collapse Stellar density profile Internal structure: occurrence of various phases. Observational consequences:   –Stellar masses, radii and moments of inertia. –Cooling rates of proto-neutron stars –Cooling rates for X-ray bursters. Neutron Star stability against gravitational collapse Stellar density profile Internal structure: occurrence of various phases. Observational consequences:   –Stellar masses, radii and moments of inertia. –Cooling rates of proto-neutron stars –Cooling rates for X-ray bursters. Neutron Star Structure: Pethick and Ravenhall, Ann. Rev. Nucl. Part. Sci. 45, 429 (1995) O

15. Some examples These equations of state differ only in their density dependent symmetry terms. Clear sensitivity to the density dependence of the symmetry terms These equations of state differ only in their density dependent symmetry terms. Clear sensitivity to the density dependence of the symmetry terms Neutrino signal from collapse. Feasibility of URCA processes for proto-neutron star cooling if f p > 0.1. This occurs if S(  ) is strongly density dependent. p+e -  n+ n  p+e - + Neutron star radii:Cooling of proto-neutron stars: O 0 0.5 1.5 2 Lattimer, Ap. J., 550, 426 (2001).

16. Summary of last lecture The EOS describes the macroscopic response of nuclear matter and finite nuclei. –It can be calculated by various techniques. Skyrme parameterizations are a relatively easy and flexible way to do so.. –The high density behavior and the behavior at large isospin asymmetries of the EOS are not well constrained. The behavior at large isospin asymmetries is described by the symmetry energy. –The symmetry energy has a profound influence on neutron star properties: stellar radii, maximum masses, cooling of proto-neutron stars, phases in the stellar interior, etc. The EOS describes the macroscopic response of nuclear matter and finite nuclei. –It can be calculated by various techniques. Skyrme parameterizations are a relatively easy and flexible way to do so.. –The high density behavior and the behavior at large isospin asymmetries of the EOS are not well constrained. The behavior at large isospin asymmetries is described by the symmetry energy. –The symmetry energy has a profound influence on neutron star properties: stellar radii, maximum masses, cooling of proto-neutron stars, phases in the stellar interior, etc.  ( ,0,  ) =  ( ,0,0) +  2  S(  ) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A O

17. Binding energies as probes of the EOS Fits of the liquid drop binding energy formula experimental masses can provide values for a v, a s, a c, b 1, b 2, C d and  A, Z. Relationship to EOS –a v =  (  s,0,0); a v b 1 =S(  s ) –a s and a s b 2 provide information about the density dependence of  (  s,0,0) and S(  s ) at subsaturation densities   1/2  s. (See Danielewicz, Nucl. Phys. A 727 (2003) 233.) –The various parameters are correlated. Coulomb and symmetry energy terms are strongly correlated. Shell effects make masses differ from LDM. Measurement techniques: –Penning traps:  =qB/m –Time of flight: TOF=distance/v B  =mv/q –Transfer reactions: A(b,c)D Q=(m A +m b -m c -m D )c 2 Mass compilations exist: e.g. Audi et al,.,NPA 595, (1995) 409. Fits of the liquid drop binding energy formula experimental masses can provide values for a v, a s, a c, b 1, b 2, C d and  A, Z. Relationship to EOS –a v =  (  s,0,0); a v b 1 =S(  s ) –a s and a s b 2 provide information about the density dependence of  (  s,0,0) and S(  s ) at subsaturation densities   1/2  s. (See Danielewicz, Nucl. Phys. A 727 (2003) 233.) –The various parameters are correlated. Coulomb and symmetry energy terms are strongly correlated. Shell effects make masses differ from LDM. Measurement techniques: –Penning traps:  =qB/m –Time of flight: TOF=distance/v B  =mv/q –Transfer reactions: A(b,c)D Q=(m A +m b -m c -m D )c 2 Mass compilations exist: e.g. Audi et al,.,NPA 595, (1995) 409. B A,Z = a v [1-b 1 ((N-Z)/A)²]A - a s [1-b 2 ((N-Z)/A)²]A 2/3 - a c Z²/A 1/3 + δ A,Z A -1/2 + C d Z²/A, O

18. Neutron and proton matter radii A simple approximation to the density profile is a Fermi function  (r)=  0 /(1+exp(r- R)/a). For stable nuclei, R p has been measured by electron scattering to about 0.02 fm accuracy. –(see G. Fricke et al., At. Data Nucl. Data Tables 60, 177 (1995).) A simple approximation to the density profile is a Fermi function  (r)=  0 /(1+exp(r- R)/a). For stable nuclei, R p has been measured by electron scattering to about 0.02 fm accuracy. –(see G. Fricke et al., At. Data Nucl. Data Tables 60, 177 (1995).) Neutron radii can be measured by hadronic scattering, which is more model dependent and less accurate (  R n  0.2 fm) because the interaction is mainly on the surface. a  0.5 – 0.6 fm for stable spherical nuclei, but near the neutron dripline, a n can be much larger. –Strong interaction radius for 11 Li is about the same as that for 208 Pb. Neutron radii can be measured by hadronic scattering, which is more model dependent and less accurate (  R n  0.2 fm) because the interaction is mainly on the surface. a  0.5 – 0.6 fm for stable spherical nuclei, but near the neutron dripline, a n can be much larger. –Strong interaction radius for 11 Li is about the same as that for 208 Pb. 208 Pb  (r) (fm -3 ) r (fm)

19. Comparison of R n and R p The asymmetry in the nuclear surface can be larger when S(  ) is strongly density dependent because S(  ) vanishes.more rapidly at low density when S(  ) is stiff. –Stiff symmetry energy  larger neutron skins. (See Danielewicz lecture.) Measurements of 208 Pb using parity violating electron scattering are expected to provide strong constraints on 1/2 - 1/2 and on S(  ) for  <  s. Uncertainties are of order 0.06 fm. (see Horowitz et al., 63, 025501(2001).) The upper figure shows how the predicted neutron skins depend on P sym =  2 dS(  )/d  Analyses of 1/2 - 1/2 for Na isotopes have placed some constraints on S(  ) for  <  s, (see Danielewicz, NPA 727, 203 (2003).   The asymmetry in the nuclear surface can be larger when S(  ) is strongly density dependent because S(  ) vanishes.more rapidly at low density when S(  ) is stiff. –Stiff symmetry energy  larger neutron skins. (See Danielewicz lecture.) Measurements of 208 Pb using parity violating electron scattering are expected to provide strong constraints on 1/2 - 1/2 and on S(  ) for  <  s. Uncertainties are of order 0.06 fm. (see Horowitz et al., 63, 025501(2001).) The upper figure shows how the predicted neutron skins depend on P sym =  2 dS(  )/d  Analyses of 1/2 - 1/2 for Na isotopes have placed some constraints on S(  ) for  <  s, (see Danielewicz, NPA 727, 203 (2003).   Brown, Phys. Rev. Lett. 85, 5296 (2001) at  =0.1 fm O stiffer softer

20. Radii of Na isotopes The relationship between cross- section and Na interaction radius is: –Getting the actual neutron radius is model dependent. The relationship between cross- section and Na interaction radius is: –Getting the actual neutron radius is model dependent. Proton radii are determined by measuring atomic transitions in Na, which has a 3s g.s. orbit. Neutron radii increase faster than R=r 0 A 1/3, reflecting the thickness of neutron skin, e.g. RMF calculation.  1/2 ~ 0.1 fm O Suzuki, et al., PRL 75, 3241 (1995)

21. Giant resonances Imagine a macroscopic, i.e. classical excitation of the matter in the nucleus. –e.g. Isoscaler Giant Monopole (GMR) resonance GMR provides information about the curvature of  ( ,0,0) about minimum. Inelastic  particle scattering e.g. 90 Zr( ,  ) 90 Zr* can excite the GMR. (see Youngblood et al., PRL 92, 691 (1999).) –Peak is strongest at 0  Imagine a macroscopic, i.e. classical excitation of the matter in the nucleus. –e.g. Isoscaler Giant Monopole (GMR) resonance GMR provides information about the curvature of  ( ,0,0) about minimum. Inelastic  particle scattering e.g. 90 Zr( ,  ) 90 Zr* can excite the GMR. (see Youngblood et al., PRL 92, 691 (1999).) –Peak is strongest at 0 

22. Giant resonances 2 HW 3: Assume that we can approximate a nucleus as having a sharp surface at radius R and ignore the surface, Coulomb and symmetry energy contributions to the nuclear energy. –In the adiabatic approximation show that –Show that In practice there are surface, Coulomb and symmetry energy corrections to the GMR energy. (see Harakeh and van der Woude, “Giant Resonances” Oxford Science...) –Leptodermous expansion: HW 3: Assume that we can approximate a nucleus as having a sharp surface at radius R and ignore the surface, Coulomb and symmetry energy contributions to the nuclear energy. –In the adiabatic approximation show that –Show that In practice there are surface, Coulomb and symmetry energy corrections to the GMR energy. (see Harakeh and van der Woude, “Giant Resonances” Oxford Science...) –Leptodermous expansion: O

23. Giant Resonances 3 Isovector Giant Dipole Resonance: neutrons and protons oscillate against each other. The restoring force is the surface energy of the nucleus. Danielewicz has shown that E GDR depends on the surface symmetry energy but not on the volume symmetry energy. (Danielewicz, NP A 727 (2003) 233.) Isovector Giant Dipole Resonance: neutrons and protons oscillate against each other. The restoring force is the surface energy of the nucleus. Danielewicz has shown that E GDR depends on the surface symmetry energy but not on the volume symmetry energy. (Danielewicz, NP A 727 (2003) 233.) NP O

24. Probes of the symmetric matter EOS Nuclear collisions are the only way to make variations in nuclear density under “experimentally controlled” conditions and obtain information about the EOS. Theoretical tool: transport theory: –Example Boltzmann-Uehling-Uhlenbeck eq. (Bertsch Phys. Rep. 160, 189 (1988).): –Describes the time evolution of the Wigner transform of the one-body density matrix: (quantum analogue to classical phase space distribution) –classically, f= ( the number of nucleons/d 3 rd 3 p at ). –Semiclassical: “time dependent Thomas-Fermi theory” –Each nucleon is represented by ~1000 test particles that propogate classically under the influence of the mean field U and subject to collisions due to the residual interaction. The mean field is self consistent, at each time step, one: propogates nucleons, etc. subject to the mean field and collisions, and recalculates the mean field potential according to the new positions. Nuclear collisions are the only way to make variations in nuclear density under “experimentally controlled” conditions and obtain information about the EOS. Theoretical tool: transport theory: –Example Boltzmann-Uehling-Uhlenbeck eq. (Bertsch Phys. Rep. 160, 189 (1988).): –Describes the time evolution of the Wigner transform of the one-body density matrix: (quantum analogue to classical phase space distribution) –classically, f= ( the number of nucleons/d 3 rd 3 p at ). –Semiclassical: “time dependent Thomas-Fermi theory” –Each nucleon is represented by ~1000 test particles that propogate classically under the influence of the mean field U and subject to collisions due to the residual interaction. The mean field is self consistent, at each time step, one: propogates nucleons, etc. subject to the mean field and collisions, and recalculates the mean field potential according to the new positions.

25. Constraining the EOS at high densities by laboratory collisions Two observable consequences of the high pressures that are formed: –Nucleons deflected sideways in the reaction plane. –Nucleons are “squeezed out” above and below the reaction plane.. Two observable consequences of the high pressures that are formed: –Nucleons deflected sideways in the reaction plane. –Nucleons are “squeezed out” above and below the reaction plane.. pressure contours density contours Au+Au collisions E/A = 1 GeV)

26. Procedure to study high pressures Measure collisions Simulate collisions with BUU or other transport theory Identify observables that are sensitive to EOS (see Danielewicz et al., Science 298,1592 (2002). for flow observables) Directed transverse flow (in-plane) –“Elliptical flow” out of plane, e.g. “squeeze-out” –Kaon production. ( Schmah, PRC C 71, 064907 (2005)) Analyze data and model calculations to measured and calculated observable assuming some specific forms of the mean field potentials for neutrons and protons. At some energies, produced particles, like pions, etc. must be calculated as well. Find the mean field(s) that describes the data. If more than one mean field describes the data, resolve the ambiguity with additional data. Constrain the effective masses and in-medium cross sections by additional data. Use the mean field potentials to apply the EOS information to other contexts like neutron stars, etc. Measure collisions Simulate collisions with BUU or other transport theory Identify observables that are sensitive to EOS (see Danielewicz et al., Science 298,1592 (2002). for flow observables) Directed transverse flow (in-plane) –“Elliptical flow” out of plane, e.g. “squeeze-out” –Kaon production. ( Schmah, PRC C 71, 064907 (2005)) Analyze data and model calculations to measured and calculated observable assuming some specific forms of the mean field potentials for neutrons and protons. At some energies, produced particles, like pions, etc. must be calculated as well. Find the mean field(s) that describes the data. If more than one mean field describes the data, resolve the ambiguity with additional data. Constrain the effective masses and in-medium cross sections by additional data. Use the mean field potentials to apply the EOS information to other contexts like neutron stars, etc.

27. Directed transverse flow Event has “elliptical” shape in momentum space. The long axis lies in the reaction plane, perpendicular to the total angular momentum. Analysis procedure: –Find the reaction plane –Determine in this plane –note: Event has “elliptical” shape in momentum space. The long axis lies in the reaction plane, perpendicular to the total angular momentum. Analysis procedure: –Find the reaction plane –Determine in this plane –note: The data display the “s” shape characteristic of directed transverse flow. –The TPC has in-efficiencies at y/y beam < -0.2. –Slope is determined at –0.2<y/y beam <0.3 projectile target E beam /A Au+Au collisions EOS TPC data y/y beam (in C.M) Partlan, PRL 75, 2100 (1995).

28. Determination of symmetric matter EOS from nucleus-nucleus collisions The curves labeled by K nm represent calculations with parameterized Skyrme mean fields –They are adjusted to find the pressure that replicates the observed transverse flow. The curves labeled by K nm represent calculations with parameterized Skyrme mean fields –They are adjusted to find the pressure that replicates the observed transverse flow. The boundaries represent the range of pressures obtained for the mean fields that reproduce the data. They also reflect the uncertainties from the effective masses in in- medium cross sections. The boundaries represent the range of pressures obtained for the mean fields that reproduce the data. They also reflect the uncertainties from the effective masses in in- medium cross sections. Danielewicz et al., Science 298,1592 (2002). O

29. Probes of the symmetry energy Common features of some of these studies –Vary isospin of detected particle Sign in U asy is opposite for n vs. p. Shape is influenced by “stiffness”. –Vary isospin asymmetry  of reaction. Low densities (  <  0 ): –Isospin diffusion –Neutron/proton spectra and flows –Neutron, proton radii, E1 collective modes. High densities (  2  0 ) :   –Neutron/proton spectra and flows   –  + vs.  - production  Common features of some of these studies –Vary isospin of detected particle Sign in U asy is opposite for n vs. p. Shape is influenced by “stiffness”. –Vary isospin asymmetry  of reaction. Low densities (  <  0 ): –Isospin diffusion –Neutron/proton spectra and flows –Neutron, proton radii, E1 collective modes. High densities (  2  0 ) :   –Neutron/proton spectra and flows   –  + vs.  - production   ( ,0,  ) =  ( ,0,0) +  2  S(  ) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A F 1 =2u 2 /(1+u) F 2 =u F 3 =  u F1F1 F2F2 F3F3 u = stiff soft U asy (MeV)  =0.3

30. Constraining the density dependence of the symmetry energy Observable: Isospin diffusion in peripheral collisions In a reference frame where the matter is stationary: –D  the isospin diffusion coef. Two effect contribute to diffusion –Random walk –Potential (EOS) driven flows D  governs the relative flow of neutrons and protons –D  decreases with  np –D  increases with S int (  ) In a reference frame where the matter is stationary: –D  the isospin diffusion coef. Two effect contribute to diffusion –Random walk –Potential (EOS) driven flows D  governs the relative flow of neutrons and protons –D  decreases with  np –D  increases with S int (  ) Shi et al, C 68, 064604 (2003) softer stiffer R is the ratio between the diffusion coefficient with a symmetry potential and without a symmetry potential.

31. Vary isospin driving forces by changing the isospin of projectile and target. Probe the asymmetry  =(N-Z)/(N+Z) of the projectile spectator after the collision. The asymmetry of the spectator can change due to diffusion, but it also can changed due to pre-equilibrium emission. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: Useful limits for R i for 124 Sn+ 112 Sn collisions: –R i =±1:no diffusion –R i  0: Isospin equilibrium Vary isospin driving forces by changing the isospin of projectile and target. Probe the asymmetry  =(N-Z)/(N+Z) of the projectile spectator after the collision. The asymmetry of the spectator can change due to diffusion, but it also can changed due to pre-equilibrium emission. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: Useful limits for R i for 124 Sn+ 112 Sn collisions: –R i =±1:no diffusion –R i  0: Isospin equilibrium Probe: Isospin diffusion in peripheral collisions P N mixed 124 Sn+ 112 Sn n-rich 124 Sn+ 124 Sn p-rich 112 Sn+ 112 Sn mixed 124 Sn+ 112 Sn n-rich 124 Sn+ 124 Sn p-rich 112 Sn+ 112 Sn neutron-rich No isospin diffusion 1.0 0.0 Complete mixing proton-rich RiRi 

32. Sensitivity to symmetry energy Tsang et al., PRL92(2004) Stronger density dependence Weaker density dependence Lijun Shi, thesis The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre- equilibrium emission. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre- equilibrium emission. The use of the isospin transport ratio R i (  ) isolates the diffusion effects:

33. Probing the asymmetry of the Spectators The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay The the shift can be compactly described by the isoscaling parameters  and  obtained by taking ratios of the isotopic distributions: Liu et al., (2006) Tsang et. al.,PRL 92, 062701 (2004 )

34. Determining  R i (  ) Statistical theory provides: Consider the isoscaling ratio R i (X), where X =  or  When X depends linearly on  2 : By direct substitution: –true for known production models –linear dependences confirmed by data.  

35. Probing the asymmetry of the Spectators The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay The the shift can be compactly described by the isoscaling parameters  and  obtained by taking ratios of the isotopic distributions: Liu et al., (2006) Tsang et. al.,PRL 92, 062701 (2004 ) no diffusion 1.0 0.33 -0.33 Ri()Ri()

36. Constraints from Isospin Diffusion Data M.B. Tsang et. al., PRL 92, 062701 (2004) L.W. Chen, C.M. Ko, and B.A. Li, PRL 94, 032701 (2005) C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001) B.A. Li and A.W. Steiner, nucl-th/0511064 Approximate representation of the various asymmetry terms used in BUU calcuations: E sym (  ) ~ 32(  /  0 )  [(  n -  p ) /(  n +  p )] 2  for cases A, B, C) A B C 124 Sn+ 112 Sn data Interpretation requires assumptions about isospin dependence of in-medium cross sections and effective masses O

37. Final Summary The EOS describes the macroscopic response of nuclear matter and finite nuclei. –It can be calculated by various techniques. Skyrme parameterizations are relatively easy. –The high density behavior and the behavior at large isospin asymmetries of the EOS are not well constrained. The behavior at large isospin asymmetries is described by the symmetry energy. –It influences many nuclear physics quantities: binding energies, neutron skin thicknesses, isovector giant resonances, isospin diffusion, etc. Measurements of these quantities can constrain the symmetry energy. –The symmetry energy has a profound influence on neutron star properties: stellar radii, maximum masses, cooling of proto-neutron stars, phases in the stellar interior, etc. Constraints on the symmetry energy and on the EOS will be improved by planned experiments. Some of the best ideas have not yet been discovered. The EOS describes the macroscopic response of nuclear matter and finite nuclei. –It can be calculated by various techniques. Skyrme parameterizations are relatively easy. –The high density behavior and the behavior at large isospin asymmetries of the EOS are not well constrained. The behavior at large isospin asymmetries is described by the symmetry energy. –It influences many nuclear physics quantities: binding energies, neutron skin thicknesses, isovector giant resonances, isospin diffusion, etc. Measurements of these quantities can constrain the symmetry energy. –The symmetry energy has a profound influence on neutron star properties: stellar radii, maximum masses, cooling of proto-neutron stars, phases in the stellar interior, etc. Constraints on the symmetry energy and on the EOS will be improved by planned experiments. Some of the best ideas have not yet been discovered.  ( ,0,  ) =  ( ,0,0) +  2  S(  ) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A

38. Influence of production mechanism on isoscaling parameters Statistical theory: –Final isoscaling parameters are often similar to those of the primary distribution –Both depend linearly on   R(  )=R(  ) Statistical theory: –Final isoscaling parameters are often similar to those of the primary distribution –Both depend linearly on   R(  )=R(  ) Dynamical theories: –Final isoscaling parameters are often smaller than those of primary distribution –Both depend linearly on   R(  )=R(  ) –Doesn't matter which one is correct. Primary: Before decay of excited fragments, Final: after decay of excited fragments final R

39. Test of linearity using central collisions Data analyzed in well-mixed region at 70  cm  110 . Linearity is demonstrated for ,  and ln(Y( 7 Li)/Y( 7 Be))  -  Liu et al., (2006) R

40. Densities of  2  0 can be achieved at E/A  400 MeV. –Provides information about direct URCA cooling in proto-neutron stars, stability and phase transitions of dense neutron star interior. Densities of  2  0 can be achieved at E/A  400 MeV. –Provides information about direct URCA cooling in proto-neutron stars, stability and phase transitions of dense neutron star interior. Densities of  2  0 can be achieved at E/A  400 MeV. –Provides information about direct URCA cooling in proto-neutron stars, stability and phase transitions of dense neutron star interior. S(  ) influences diffusion of neutrons from dense overlap region at b=0. –Diffusion is reduced, neutron-rich dense region is formed for soft S(  ). Densities of  2  0 can be achieved at E/A  400 MeV. –Provides information about direct URCA cooling in proto-neutron stars, stability and phase transitions of dense neutron star interior. S(  ) influences diffusion of neutrons from dense overlap region at b=0. –Diffusion is reduced, neutron-rich dense region is formed for soft S(  ). Asymmetry term studies at  2  0 Yong et al., Phys. Rev. C 73, 034603 (2006) R

41. First observable: pion production The enhanced neutron abundance at high density for the soft asymmetry term (x=0) leads to a stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1). –In delta resonance model, Y(  - )/Y(  + )  (  n, /  p ) 2  - /  + means Y(  - )/Y(  + ) Coulomb interaction has a strong effect on the pion spectra: –Coulomb repels  + and attracts  -. The enhanced neutron abundance at high density for the soft asymmetry term (x=0) leads to a stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1). –In delta resonance model, Y(  - )/Y(  + )  (  n, /  p ) 2  - /  + means Y(  - )/Y(  + ) Coulomb interaction has a strong effect on the pion spectra: –Coulomb repels  + and attracts  -. The density dependence of the asymmetry term changes ratio by about 15% for neutron rich system. –How does one reduce sensitivity to systematic errors? Yong et al., Phys. Rev. C 73, 034603 (2006) soft stiff

42. Double ratio: pion production Double ratio involves comparison between neutron rich 132 Sn+ 124 Sn and neutron deficient 112 Sn+ 112 Sn reactions. Double ratio maximizes sensitivity to asymmetry term. –Largely removes sensitivity to difference between  - and  + acceptances. Double ratio involves comparison between neutron rich 132 Sn+ 124 Sn and neutron deficient 112 Sn+ 112 Sn reactions. Double ratio maximizes sensitivity to asymmetry term. –Largely removes sensitivity to difference between  - and  + acceptances. Yong et al., Phys. Rev. C 73, 034603 (2006) soft stiff R

43. Independent observable: n/p spectra Neutrons are repelled and protons are attracted by the asymmetry term (in neutron rich matter). The Coulomb interaction has somewhat the opposite effect. Sensitivity can be maximized by constructing a double ratio: Removes sensitivity to calibration and efficiency problems Neutrons are repelled and protons are attracted by the asymmetry term (in neutron rich matter). The Coulomb interaction has somewhat the opposite effect. Sensitivity can be maximized by constructing a double ratio: Removes sensitivity to calibration and efficiency problems Li et al., arXiv:nucl-th/0510016 (2005) soft stiff

44. Alternate observable: n-p differential transverse flow Transverse directed flow is usually obtained by plotting the mean transverse momentum vs. the rapidity y. The neutron-proton differential flow is defined here to be: Sensitivity to acceptance effects might be minimized by constructing the difference: Transverse directed flow is usually obtained by plotting the mean transverse momentum vs. the rapidity y. The neutron-proton differential flow is defined here to be: Sensitivity to acceptance effects might be minimized by constructing the difference: Li et al., arXiv:nucl-th/0504069 (2005)

45. Constraints on momentum dependence of mean fields and in-medium cross sections We need calculations of the corresponding double ratios. –Not clear that we have a good way to distinguish momentum and density dependencies. We need calculations of the corresponding double ratios. –Not clear that we have a good way to distinguish momentum and density dependencies. Important to control the number of n-p collisions, p-p and n-n collisions –compare 37 Ca+ 112 Sn to 37 Ca+ 124 Sn –compare 52 Ca+ 112 Sn to 52 Ca+ 124 Sn. 40 Ca+ 100 Zn E/A=200 MeV Li et al., Phys. Rev. C 71, 054603 (2005) Li et al., Phys. Rev. C 69, 011603 ( R ) ( 2004 ) R