Symmetry Energy Effects in Heavy Ion Collisions

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Symmetry Energy Effects in Heavy Ion Collisions
Che-Ming Ko Texas AM University

• Nuclear symmetry energy
• Isospin-dependent transport model (IBUU)
• Nucleon emission source
• Two-nucleon correlation functions
• Light clusters production

Collaborators Lie-Wen Chen, Vincenzo Greco,
Bao-An Li (Arkansas State
University)
PRL 90, 162701 (2003) nucl-th/0302068 0305036
0306032
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Nuclear symmetry energy
EOS of asymmetric nuclear matter
Symmetry energy
Symmetry energy coefficient
theoretical values -50 to 200 MeV
Slope
theoretical values -700 to 466 MeV
Curvature
experimental values
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Symmetry energy
Pressure
Potential
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Isospin-dependent transport model (IBUU)
NN Cross sections
Isospin-independent potential
In-medium cross sections from Dirac- Brueckner
approach based on Bonn potential (Li, PRC 48,
1702 (93))
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• using free cross
• sections, soft
• symmetry potential
• 10,000 events
• with 100 test
• particles for a
• physical nucleon

Formation of ring structure in transverse plane
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Quadrupole moment of momentum distribution
Maximum density
More sensitive to incompressibility of
isospin-independent part of EOS than stiffness
of symmetry energy
Thermal equilibrium after about 40 fm/c
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Nucleon emission times

• Longer emission duration for lower momentum
  nucleons
• Earlier emissions for stiffer symmetry energy
• Larger separation in neutron and proton emission
  times for softer
• symmetry energy
• High momentum nucleons emitted earlier than low
  momentum ones

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Size of nucleon emission source

• Broader emission source size distribution for
  lower momentum nucleons
• Source size larger in transverse direction than
  in longitudinal direction
• Larger emission source size for softer symmetry
  energy

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Momentum distributions of emitted nucleons

• Peak momentum lower for stiffer symmetry energy
• Symmetry energy effect larger for low momentum
  protons than neutrons

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Effects of isocalar and Coulomb potentials, and
NN cross sections

• Nucleon emission rates insensitive to
  incompressibility of isospin-
• independent part of EOS
• In-medium cross sections enhance slightly
  nucleon emission rate
• at later stage of collisions
• Coulomb potential shortens slightly proton
  emission time

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Two-nucleon correlation functions
emission function, i.e., probability for
emitting a nucleon with momentum p from the
space-time point x(r,t)
relative wave function of two nucleons
Correlation After Burner including final-state
nuclear and Coulomb
interactions (Scott Pratt, NPA 566,
103 (1994))
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Two-nucleon correlation functions in
central collisions of 52Ca48Ca at 80 AMeV

• Correlations of low momentum or very energetic
  pairs insensitive to
• symmetry energy
• Symmetry energy effects on high momentum pairs
  about 20-30 with
• stiffer one giving stronger correlations

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Time evolution of two-nucleon correlation
functions
Effects of isoscalar potential and NN cross
sections
Symmetry energy effect appears at 50 fm/c when
density is slightly below normal density
Effects less than 10 for both isoscalar
potential and NN cross sections
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Impact parameter and incident energy dependence
Strength of correlation functions increases with
incident energy, and symmetry energy effect
remain similar
Symmetry energy effect decreases with impact
parameter with stiffer one reduced more
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Two nucleon correlation functions in central
collisions of 132Sn124Sn at 80 AMeV

• Symmetry energy effect
• on correlation functions of
• high momentum pairs of
• pp and pn are reduced by
• 2 compared to 52Ca48Ca
• collisions due to larger
• Coulomb effect
• Effect on nn correlation
• function similar to 52Ca48Ca
• collisions, i.e., 20

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Light clusters production
Coalescence model
nucleon phase-space distribution function
Wigner phase-space distribution function for
clusters
G statistical factor 3/8 for deuteron, 1/3 for
triton and 3He
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Wigner phase-space distribution function for
deuteron
Hulthen wave function
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Wigner phase-space distribution function for
triton and 3He
Gaussian wave function
Jacobi coordinates
b1.61 fm for triton and 1.74 for 3He ? correct
radii
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Light clusters production from collisions of
symmetric nuclei

• Deuteron energy spectra
• reproduced
• Low energy tritons slightly
• underestimated
• Inverse slope parameter of
• 3He underestimated probably
• due to neglect of

• larger binding effect
• stronger Coulomb effect

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Yields and energy spectra of light clusters

• Symmetry energy effects are about 51, 73, and
  100 on deuteron,
• triton and 3He yields with stiffer one
  producing more
• Symmetry energy effects stronger on lower energy
  light clusters
• Effects of isoscalar potential and NN cross
  sections small

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Isobaric yield ratio of t/3He

• Stiffer symmetry energy gives smaller t/3He
  ratio
• With increasing kinetic energy, t/3He ratio
  increases for stiff symmetry
• energy but slightly decreases for soft
  symmetry energy

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Impact parameter and incident energy dependence
Symmetry energy effects decrease with increasing
incident energy but only slightly with
increasing impact parameter
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Light clusters production in central
collisions of 132Sn124Sn at 80 AMeV
Symmetry energy effects similar to 52Ca48Ca
collisions 53, 74 and 120 for d, t, and 3He
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Emission times of light clusters
Average emission time earlier for heavier ones
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Two-deuteron correlation functions
Include final-state repulsive nuclear s-wave and
Coulomb interactions
Anticorrelations of pairs of high total momentum
are affected by symmetry energy with stiffer one
giving a larger strength, about 20 at q40 MeV/c
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Two-triton or two-3He correlation functions
Include only final-state repulsive Coulomb
interaction
Stiffer symmetry energy gives stronger
anticorrelation but the effect is smaller than in
two-nucleon and two-deuteron correlation functions
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Summary
Density dependence of nuclear symmetry energy
affects dynamics of heavy ion collisions induced
by neutron-rich nuclei at intermediate energies
with stiffer density dependence giving

• Earlier emissions of nucleons and light clusters
• Stronger two-nucleon correlation functions
• Larger light clusters production
• Larger ratio of high energy tritons and 3He
• Larger light clusters correlation functions

Light clusters production and two-particle
correlation functions are useful probes of
nuclear symmetry energy
Effects of momentum-dependence? in progress