Constraining the EoS and Symmetry Energy from HI collisions

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Constraining the EoS and Symmetry Energy from HI
collisions
William Lynch, Yingxun Zhang, Dan Coupland, Pawel
Danielewicz, Micheal Famiano, Zhuxia Li, Betty
Tsang

• Statement of the problem
• Demonstration symmetric matter EOS
• Laboratory constraints on the symmetry energy
  from nuclear collisions
• Summary and outlook
•  

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EOS symmetric matter and neutron matter
Brown, Phys. Rev. Lett. 85, 5296 (2001)
Neutron matter EOS

• E/A (?,?) E/A (?,0) d2?S(?)
• d (?n- ?p)/ (?n ?p) (N-Z)/A

• The density dependence of symmetry energy is
  largely unconstrained.
• What is stiff or soft is density dependent

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Need for probes sensitive to higher
densities(Why the monopole is not sufficient and
Knm is somewhat irrelevant.)

• If the EoS is expanded in a Taylor series about
  ?0, the incompressibility, Knm provides the term
  proportional to (?-?0)2. Higher order terms
  influence the EoS at sub-saturation and
  supra-saturation densities.
• The solid black, dashed brown and dashed blue
  EoSs all have Knm300 MeV.

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Constraining the EOS at high densities by nuclear
collisions
AuAu collisions E/A 1 GeV)
pressure contours
density contours

• 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. .

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Constraints from collective flow on EOS at ??gt2
?0.
E/A (?, ?) E/A (?,0) ?2?S(?) ? (?n-
?p)/ (?n ?p) (N-Z)/A?1
Danielewicz et al., Science 298,1592 (2002).
Danielewicz et al., Science 298,1592 (2002).

• Note analysis required additional constraints on
  m and ??NN.
• Flow confirms the softening of the EOS at high
  density.
• Constraints from kaon production are consistent
  with the flow constraints and bridge gap to GMR
  constraints.

• The symmetry energy dominates the uncertainty in
  the n-matter EOS.
• Both laboratory and astronomical constraints on
  the density dependence of the symmetry energy at
  supra-saturation density are urgently needed.

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Probes of the symmetry energy
E/A(?,?) E/A(?,0) d2?S(?) d (?n- ?p)/
(?n ?p) (N-Z)/A

• To maximize sensitivity, reduce systematic
  errors
• Vary isospin of detected particle
• Vary isospin asymmetry ?(N-Z)/A of reaction.
• Low densities (?lt?0)
• Neutron/proton spectra and flows
• Isospin diffusion
• High densities (??2?0) ?
• Neutron/proton spectra and flows
• ?? vs. ?- production

?lt?0
?lt?0
symmetry energy
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Why choose to measure Isospin Diffusion, n/p
flows and pion production?

• Supra-saturation and sub-saturation densities are
  only achieved momentarily
• Theoretical description must follow the reaction
  dynamics self-consistently from contact to
  detection.
• Theoretical tool transport theory
• The most accurately predicted observables are
  those that can be calculated from
  i.e. flows and other average properties of the
  events that are not sensitive to fluctuations.
• Isospin diffusion and n/p ratios
• Depends on quantities that can be more accurately
  calculated in BUU or QMD transport theory.
• May be less sensitive to uncertainties in (1) the
  production mechanism for complex fragments and
  (2) secondary decay.

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Measurement of n/p spectral ratios At E/A 50
MeV, it probes the pressure due to asymmetry
term at ???0.

• Expulsion of neutrons from bound neutron-rich
  system by symmetry energy. At E/A50 MeV, ???0
  is the relevant domain.
• Has been probed by direct measurements of n vs.
  proton emission rates in central SnSn
  collisions.

• Double ratio removes the sensitivity to neutron
  efficiency and energy calibration.

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Isospin diffusion in peripheral collisions, also
probes symmetry energy at ?lt?0.

• Collide projectiles and targets of differing
  isospin asymmetry
• Probe the asymmetry ?(N-Z)/(NZ) of the
  projectile spectator during the collision.
• The use of the isospin transport ratio Ri(?)
  isolates the diffusion effects
• Useful limits for Ri for 124Sn112Sn collisions
• Ri 1 no diffusion
• Ri ?0 Isospin equilibrium

mixed 124Sn112Sn n-rich 124Sn124Sn p-rich
112Sn112Sn
Systems
measure asymmetry after collision
Example
proton-rich target
P
?
N
neutron-rich projectile
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What influences isospin diffusion?

• Isospin diffusion equation
• Naive expectations
• D? increases with S(?)
• D? decreases with ?np
• We tested this by performing extensive BUU and
  QMD calculations with S(?) for the form
• S(?) 12.5(?/?0)2/3 Sint (?/?0) ?i
• Results
• Diffusion sensitive to S(0.4?0)
• Diffusion increases with Sint and decreases with
  ?i
• Diffusion increases with ?np
• Diffusion decreases when mean fields are momentum
  dependent and neck fragments emerge.
• Diffusion decreases with cluster production.

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Sensitivity to symmetry energy
Stronger density dependence

• 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 Ri(?)
  isolates the diffusion effects

Weaker density dependence
Tsang et al., PRL92, 062701 (2004)
Lijun Shi, thesis
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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

• This can be described by the isoscaling
  parameters ? and ?

Liu et al.PRC 76, 034603 (2007).
Tsang et. al.,PRL 92, 062701 (2004)
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Determining ?Ri(?)

• In statistical theory, certain observables depend
  linearly on ?
• Calculations confirm this
• Experiments confirm this

• Consider the ratio Ri(X), where X ??, X7 or
  some other observable
• If X depends linearly on ?2
• Then by direct substitution

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Probing the asymmetry of the Spectators

• This can be described by the isoscaling
  parameters ? and ?

• The main effect of changing the asymmetry of the
  projectile spectator remnant is to shift the
  isotopic distributions of the products of its
  decay

Liu et al.PRC 76, 034603 (2007).
Tsang et. al.,PRL 92, 062701 (2004)
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Quantitative values

• Reactions
• 124Sn112Sn diffusion
• 124Sn124Sn neutron-rich limit
• 112Sn112Sn proton-rich limit
• Exchanging the target and projectile allowed the
  full rapidity dependence to be measured.
• Gates were set on the values for Ri(?) near beam
  rapidity.
• Ri(?) ? 0.47?0.05 for 124Sn112Sn
• Ri(?) ? -0.44 ?0.05 for 112Sn124Sn
• Obtained similar values for Ri(ln(Y(7Li)/
  Y(7Be))
• Allows exploration of dependence on rapidity

Liu et al., (2006)
Liu et al.PRC 76, 034603 (2007).
? v??/vbeam
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Comparison to QMD calculations

• IQMD calculations were performed for ?i0.35-2.0,
  Sint17.6 MeV.
• Momentum dependent mean fields with mn/mn
  mp/mp 0.7 were used. Symmetry energies S(?) ?
  12.3(?/?0)2/3 17.6 (?/?0) ?i

• Experiment samples a range of impact parameters
• b?5.8-7.2 fm.
• larger b, smaller ?i
• smaller b, larger ?i

mirror nuclei
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Diffusion is sensitive to S(0.4?), which
corresponds to a contour in the (S0, L) plane.
Tsang et al., PRL 102, 122701 (2009).
ImQMD fits for S030.1 MeV
fits to IAS masses
ImQMD CONSTRAINTS
fits to
ImQMD fits for variable S0

• Expansion around r0
  
• Symmetry slope L curvature Ksym

• Symmetry pressure Psym

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Why probe higher densities?Example EoS ?
neutron star radius
?L

• The neutron star radius is not strongly
  correlated with the symmetry pressure at
  saturation density.
• This portends difficulties in uniquely
  constraining neutron star radii for constraints
  at subsaturation density.

• The correlation between the pressure at twice
  saturation density and the neutron star radius is
  much stronger.
• additional measurements at supra-saturation
  density will lead to stronger constraints.

• Would be advisable to have multiple probes that
  can sample different densities

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High density probe pion production
Li et al., arXivnucl-th/0312026 (2003).

• Larger values for ?n/ ? p at high density for the
  soft asymmetry term (x0) causes stronger
  emission of negative pions for the soft asymmetry
  term (x0) than for the stiff one (x-1).
• ?- /?? means Y(??-)/Y(??)
• In delta resonance model, Y(??-)/Y(??)?(?n,/?p)2
• In equilibrium,
• ?(?)-?(?-)2( ?p-?n)
• The density dependence of the asymmetry term
  changes ratio by about 10 for neutron rich
  system.

soft
soft
stiff
stiff
t (fm/c)

• This can be explored with stable or rare isotope
  beams at the MSU/FRIB and RIKEN/RIBF.
• Sensitivity to S(?) occurs primarily near
  threshold in AA

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Preliminary results puzzling
Riken
Future determination of the EoS of neutron-rich
matter
?
FRIB
S(?) MeV
GSI
MSU
Isospin diffusion, n-p flow
Xiao, et al., arXiv0808.0186 (2008) Reisdorf, et
al., NPA 781 (2007) 459.
Pion production
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Double ratio pion production

• Double ratio involves comparison between neutron
  rich 132Sn124Sn and neutron deficient
  112Sn112Sn 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
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Summary and Outlook

• Heavy ion collisions provide unique possibilities
  to probe the EOS of dense asymmetric matter.
• A number of promising observables to probe the
  density dependence of the symmetry energy in HI
  collisions have been identified.
• Isospin diffusion, isotope ratios, and n/p
  spectral ratios provide some constraints at ???0,
  .
• ?? vs. ?- production, neutron/proton spectra
  and flows may provide constraints at ??2?0 and
  above.
• The availability of fast stable and rare isotope
  beams at a variety of energies will allow
  constraints on the symmetry energy at a range of
  densities.
• Experimental programs are being developed to do
  such measurements at MSU/FRIB, RIKEN/RIBF and
  GSI/FAIR