Motivation Current status Outlook This work was supported in part by: The Robert A. Welch Foundation: Grant Number A-1266 and, The Department of Energy: Grant Number DE-FG03-93ER40773 D.V. Shetty, G. A. Souliotis, D. Rowland, A. Keksis, E. Bell, M. Jandel, M. Veselsky, R. Laforest, E. Ramakrishnan, H. Johnston, L. Trache, F. Gimeno-Nogues, A. Ruangma Cyclotron Institute, Texas A&M University, College Station, Texas S.J. Yennello Using Isoscaling to understand the symmetry energy of the nuclear equation of state.

A heavy nucleus (like 208 Pb) is 18 orders of magnitude smaller and 55 orders of magnitude lighter than a neutron star ! Atomic nuclei & Neutron star ( two vastly different systems ) Yet bounded by a common entity, the nuclear Equation Of State (EOS) !

H. Huber et al, Phys. Rev. C 50 (1994) 1287(R) Softer EOS leads to a smaller star mass and radius for a given central density EOS in Astrophysics and Stellar Properties Size & Structure of Neutron Star depends on EOS EOS influence R,M relationship maximum mass. cooling rate. core structure

Studying Nuclear Equation of State (EOS) Using Heavy Ions  High temperature & density can be achieved in intermediate energy heavy ion collision. ( At relativistic energies : T ~ MeV,  (10 – 20)  o )  Direct excess to supernova core or neutron star impossible  Coupled with the possibility of neutron rich beams, very asymmetric nuclear matter (N/Z > 1) can be probed.  The largely unconstrained density dependence of the asymmetry term in the EOS is sensitive to many observables in heavy ion collisions

Saturation point : a single (equilibrium) point in the EOS of nuclear matter at T = 0  o ~ 0.17 fm -3 ; E/A ~ -16 MeV ; K ~ 220 MeV ( compressibility from giant monopole resonance studies ) Equation of state for symmetric (N = Z) nuclear matter Symmetric (N = Z )matter EOS

Density dependence of symmetry energy Equation of State of Asymmetric (N/Z > 1) Nuclear Matter B.A. Brown, Phys. Rev. Lett. 85 (2000) 5296 Largely unconstrained below and above saturation density

Host of nuclear EOS employed in astrophysical modeling of neutron star & supernova explosion Still leaves a wide range of possibilities ! Some are excluded by causality & some by known masses of existing neutron stars F. Weber, IoP publishing, Bristol (1999)

 E sym (    = (  n -  p )/(  n +  p ) Not well constrained Danielewicz, Lacy, Lynch, Science 298,1592 (2002) Pressure (MeV/fm 3 ) What is known about the EOS of symmetric matter?

Binding Energy Effects of Stiff and Soft Asymmetry Terms  (n/fm 3 ) B.E. (MeV/A)  =0.3  =0 Stiff Soft We really don’t know what the nuclear asymmetry term is. Recall:  0 ~0.15n/fm 3.

Symmetry Energy in Nuclear Matter  =0 Lattimer & Prakash (2000) Opposite signs for neutrons and protons. “Asy-soft” potential has a higher absolute magnitude for protons. “Asy-stiff” potential has equal magnitudes for protons and neutrons. S(  )~  2 S(  )~   2

EOS of asymmetric nuclear matter Symmetry energy Symmetry energy coefficient Pressure Compressibility Nuclear EOS

Inclusion of surface terms in symmetry Neutron Number N Proton Number Z Constraints from Nuclei

Multifragmentation reaction (Probing the low density dependence)

Observables sensitive to the asymmetry term in the EOS ? Moderate density (  < 1.5  o ) : Fragment isotope distribution, isotopic & isobaric yield ratios Isospin distillation/fractionation, relative n & p densities Isospin diffusion Nuclear stopping & N/Z equilibration Pre-equilibrium emission Particle - particle correlation Light cluster production High density (  > 1.5  o ) : Collective flow Subthreshold particle production

An inhomogeneous distribution of the neutrons and protons within the system is predicted, resulting in a dilute neutron rich (N/Z > 1) gas (light clusters) and a dense and symmetric (N/Z ~ 1) liquid (heavy fragments) Isospin distillation in asymmetric (N/Z > 1) nuclear matter ( H. Muller & B. Serot. Phys. Rev. C 52, 2072 (1995) )

Studying isospin distillation Measure the yields of the light clusters (gas phase) Determine the n & p densities Compare them from one reactions to another with different isospin (N/Z)

Relative n density Relative p density Relative n & p density from isotope & isotone yields free neutron & proton densities

Relative n & p densities : 58 Ni + 58 Ni (reaction 1, N/Z = 1.07)) 58 Ni + 58 Fe (reaction 2, N/Z = 1.15 ) 58 Fe + 58 Fe (reaction 2, N/Z = 1.23) 58 Ni + 58 Ni (reaction 1, N/Z = 1.07) Relative n & p density & 30, 40, 47 MeV/nucl.

If the clusters in reaction 2 are more neutron-rich than in reaction 1 Relative n density, Relative p density, > 1 < 1

Ratio of isotopic yields 58 Ni, 58 Fe + 58 Ni, 58 Fe 30 MeV/nuc R 21 (N,Z) = C e  N +  Z

      Relative neutron and proton densities at 30 MeV/nuc Isotope ratios Isotone ratios

neutron and proton densities at 30, 40, 47 MeV/nuc       Isotope ratios Isotone ratios

      Experimental SMM model comparison Excitation energy (MeV/A) Isotone ratios Isotope ratios

30 MeV40 MeV47 MeV   Fe + 58 Fe / 58 Ni + 58 Ni Tsang PRC64, (2002) Temperature dependence of the scaling parameter 

Symmetry energy and the fragment yield distribution in Multifragmentation reaction Symmetry energy of the primary fragments are significantly lower D. V. Shetty et al, (2004) Primary fragments Secondary fragments

EOS and dynamical simulation of fragment production (AMD model calculations) A. Ono et al, Phys. Rev. C 68 (2003) (R)

D. V. Shetty et al, Phys. Rev. C 70 (2004) (R) Symmetry energy and the scaling parameter  Csym ~ 18 – 20 MeV ;  ~ 0.08 fm -3

Formation of hot neutron rich nuclei in supernova explosion During supernova II type explosion the thermodynamical conditions of stellar matter between the protoneutron star & the shock front correspond to nuclear liquid- gas coexistence region. Neutron rich hot nuclei can be produced in this region which can influence the dynamics of the explosion contribute to the synthesis of heavy elements A slight decrease in the symmetry energy co-efficient can shift the mass distribution to higher masses A. Botvina et al, Phys. Lett. B 584 (2004) 233

Deep InelasticTransfer mechanism can produce neutron-rich heavy residues Neutron richness AA

FAUST FAUST has solid angle coverage of 90% from 2.31º to 33.63º, 71% from 1.6º to 2.3º, and 25% from 33.6º to 44.9º mg/cm mg/cm mg/cm2 Mylar F. Gimeno-Nogues, et al. NIM A 399 (1997) 94

Neutron richness Production of quasi projectiles over a wide range in N/Z from 33Mev/nuc 20 Na, 20 Ne, 20 F + Au reactions Rowland, Phys. Rev. C 67, (2003). Beam proj DIT DIT+evap exp 20 Na Ne F

Botvina Isoscaling R 21 = C exp(  ’A +  ’(N-Z) 30 MeV/nuc 50 MeV/nuc  ’ = 0.237±0.038  ’ = 0.186±0.017  ’ =  (  n +  p )/2T  ’ =  (  n -  p )/2T 28 Si + 112,124 Sn

Botvina Isoscaling R 21 = C exp(  ’A +  ’(N-Z) 28 Si + 112,124 Sn ; 50 MeV/nuc E * (MeV) ’’

Al quasiprojectiles

 ’T  ’ =  (  n -  p )/2T

MARS Recoil Separator and Setup PPAC2 Stop T Silicon Telescope: ΔE 1, X,Y (Strips) ΔE 2, E residual PPAC1 Start T X,Y Production Target D1D2 D3 Wien Filter Q1 Q2 Q3 Q4 Q5 Dispersive Image Final Achromatic Image Rotatable Arm Reaction Angle: 0-12 o (selectable) MARS Acceptances: Anglular: 9 msr Momentum: 4 % Beam angle set to: 0 o ( o ) for Kr+Ni (  gr = 3.5 o ) 4 o ( o ) for Kr+Sn (  gr = 6.5 o )

BigSol at TAMU BigSol at TAMU

The Superconducting Solenoid Rare Isotope Line at TAMU: Schematic diagram of the setup for heavy-residue studies from DIC:

Ge Data using targets:  124 Sn  112 Sn  64 Ni EPAX2 Target N/Z Valley of stability 124 Sn 1.48 n-rich 118 Sn 112 Sn 1.24 n-poor 64 Ni 1.29 n-rich 60 Ni DIT mechanism can produce neutron-rich heavy residues Souliotis, Phys. Rev. Lett. 91, (2003) Mass Distribution of Germanium from 86Kr(25MeV/u) + 124Sn,112Sn, 64Ni

Comparison: Data/EPAX : 86 Kr (25 MeV/u) + 64 Ni, 124 Sn For near-projectile fragments and above the projectile N/Z, the cross sections are larger. DIC between massive n-rich nuclei appear to be advantageous for very high N/Z RIB production

Scaling of Yield Ratios : 86 Kr+ 124 Sn, 112 Sn (data inside  gr =6.2 ο ) R 21 (N,Z) = Y 2 /Y 1 R 21 = C exp ( α N ) 86 Kr+ 64 Ni, 58 Ni (data outside  gr =3.5 o )

Isoscaling Parameter α : * 86 Kr+ 64 Ni, 58 Ni R 21 = C exp ( α N ) 86 Kr+ 124 Sn, 112 Sn α =0.43 α =0.27 * G.A. Souliotis et al., Phys. Rev. C 68, α = 4 C sym /T ( (Z/A) 1 2 – (Z/A) 2 2 ) Quasi-projectiles 1: n-poor 2:nrich

Heavy Residue Isoscaling and N/Z equilibration R 21 ~ exp ( α N ) N/Z 86 Kr+ 124 Sn => [ 210 Rn] Kr+ 112 Sn => [ 198 Rn] 1.30 Δ(N/Z) = 0.14 N/Z equilibrated quasi-projectile α = 4 C sym /T ( (Z/A) 1 2 – (Z/A) 2 2 ) * α = 8 C sym /T (Z/A) 3 ave Δ(N/Z) 86 Kr+ 124 Sn, 112 Sn Evolution towards N/Z equilibration of quasi-projectile** ** G.A. Souliotis et al., Phys. Rev. C 68, For each Z: get Δ(N/Z) from α and T ( from ε* ) :

Isoscaling data from residues of 64 Ni (25MeV/nucleon) R 21 (N,Z) = Y 2 /Y 1 R 21 = C exp ( α N ) BigSol data

BigSol Data: Isoscaling parameter and Velocity vs Z: 64 Ni+ 124,112 Sn  64 Ni + 64,58 Ni  64 Ni+ 232 Th, 208 Pb υ min E*/A ~2.9 MeV

86 Kr, 64 Ni, 136 Xe data: Isocaling parameter  vs Δ(Z/A) 2 : Quasi-projectiles : E/A ~20-25 MeV N/Z equilibrated, no effect of pre-eq. 86 Kr+ 124,112 Sn  *  2.0 MeV/u 86 Kr+ 64,58 Ni  *  2.4 MeV/u 64 Ni+ 124,112 Sn 64 Ni+ 64,58 Ni 64 Ni+ 232 Th, 208 Pb  *  2.9 MeV/u α = 4C sym /T ( (Z/A) 1 2 – (Z/A) 2 2 ) c 136 Xe+ 124,112 Sn 136 Xe+ 64,58 Ni 136 Xe+ 232 Th,Au  *  2.5 MeV/u

Data : 86 Kr+ 124,112 Sn 86 Kr+ 64,58 Ni 64 Ni+ Ni,Sn,Th-Pb 136 Xe+Ni,Sn,Th-Au Calculation: Fermi Gas (K=13) Mononucleus expansion model (L. Sobotka, J. Toke) C sym = c T / 4 Variation w.r.t excitation energy:

lProjectile residue distributions from peripheral to mid-peripheral collisions show enhanced production of neutron –rich nuclei lHeavy-Residue Isoscaling can be used as probe of N/Z Equilibration, E sym (  ) lMultifragmentation reactions may be able to distinguish between various functional form of the density dependence of the symmetry energy lSymmetry energy can be significantly below saturation density and could be interesting in the studies relating to the elemental abundances in core collapse supernova explosion lEffect of Secondary Decay on isoscaling parameters needs to be understood lComparisons with Reaction Models: Extension of studies using higher  N/Z beams and heavy targets Summary and Outlook