Probing reaction dynamics
with two-particle correlations
Zbigniew Chajęcki
National Superconducting Cyclotron Laboratory
Michigan State University
Outline
 p-p correlations (work with M. Kilburn, B. Lynch and collaborators)
 NSCL 03045 Experiment
 transport theory (BUU)
neutron and proton emission times and symmetry energy
(particle emission chronology)
 transport theory
 Summary
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Experimental correlation function
p1
x1
Experimental correlation function:
P( p1, p2 )
real event pairs
=
P( p1)P( p2 ) mixed event pairs
- single particle distribution
C( p1, p2 ) =
r
x2
P( p1 )
P( p1 , p2 ) - two particle distribution
p2
few fm
(p,p) correlation function
P(p1,p2)
P(p1)P(p2)
|q| = 0.5 |p1 - p2|
Z. Ch. - NuSYM 2011, June 17-20, 2011
|q| = 0.5 |p1 - p2|
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Femtoscopy
p1
Theoretical CF: Koonin-Pratt equation
C(q )= ò d 3r F q (r ) S (r )
2
x1
r
F q (r ) … 2-particle wave function
S (r ) … source function
x2
p2
(p,p) correlation function
few fm
S-wave
S-wave
interraction
interraction
S(r)
Coulomb
Coulomb
r1/2
0
S.E. Koonin,
PLB70 (1977) 43
S.Pratt et al.,
PRC42 (1990) 2646
r
uncorrelated
uncorrelated
|q| = 0.5 |p1 - p2|
Z. Ch. - NuSYM 2011, June 17-20, 2011
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NSCL experiments 05045: HiRA + 4 detector
= High Resolution Array
- 4π detector => impact parameter + reaction plane
- HiRA => light charge particle correlations
(angular coverage 20-60º in LAB,
-63 cm from target (= ball center))
Reaction systems:
40Ca
+ 40Ca @ 80 MeV/u
48Ca
+ 48Ca @ 80 MeV/u
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Momentum and rapidity dependence
C(q)
Measured correlation
functions depend on rapidity
and the transverse
momentum of the pair
Next step:
extract the sizes
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Fits to the data
Koonin-Pratt Equation
C ( q ) = 1 + 4p ò K ( q,r )S ( r ) r 2 dr
C(q)
Brown, Danielewicz, PLB398 (1997) 252
Danielewicz, Pratt, PLB618 (2005) 60
Two ways of characterizing the
size of the p-p source
1) S(r) - Gaussian shape
2) Imaged S(r) (Brown, Danielewicz)
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Fits to the data
Koonin-Pratt Equation
C ( q ) = 1 + 4p ò K ( q,r )S ( r ) r 2 dr
Brown, Danielewicz, PLB398 (1997) 252
Danielewicz, Pratt, PLB618 (2005) 60
C(q)
Two ways of characterizing the
size of the p-p source
1) S(r) - Gaussian shape
2) Imaged S(r) (Brown, Danielewicz)
Both methods give
consistent fits
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Fits to the data
Correlation function C(Q)
Source distribution : S(r) x103
r1/2
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Fit results
Small rapidity:
reflect the participant zone of
the reaction
Large rapidity:
reflect the expanding,
fragmenting and evaporating
projectile-like residues
Higher velocity protons are
more strongly correlated than
their lower velocity
counterparts, consistent with
emission from expanding and
cooling sources
Sensitivity to the initial size
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Modeling heavy-ion collisions : transport models
• BUU -
Danielewicz, Bertsch, NPA533 (1991) 712
B. A. Li et al., PRL 78 (1997) 1644
Boltzmann-Uehling-Uhlenbeck
• Simulates two nuclei colliding
• Parameter space
Micha Kilburn
NSCL/MSU
• not only about the symmetry energy
• also important to understand e.g. an effect of cross section
(free x-section, in-medium x-section), reduced mass
• Production of clusters: d,t, 3He (alphas)
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Comparing data to theory (pBUU)
.
BUU Pararameters
 No dependence on symmetry
energy
 Rostock in-medium reduction
 Producing clusters
BUU does reasonably well
Except at larger rapidities Spectator source
Where evaporation and
secondary decays are
important!
Micha Kilburn, NSCL/MSU
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Averaged emission time
of particles in transport theory
Z. Ch. - NuSYM 2011, June 17-20, 2011
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L-W Chen et al., PRL90 (2003) 162701
Emission of p’s and n’s: Sensitivity to
SymEn
52Ca
Stiff
EoS
Stiff EoS (γ=2)
48Ca
Soft
EoS
Soft
Soft EoS (γ=0.5)
p’s and n’s emitted
at similar time
p’s emitted
after n’s
faster emission times
later emission times
Z. Ch. - NuSYM 2011, June 17-20, 2011
Stiff
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n-p correlation function
p1
Theoretical CF: Koonin-Pratt equation
C(q )= ò d 3r F q (r ) S (r )
2
x1
r
S.E. Koonin,
PLB70 (1977) 43
S.Pratt et al.,
PRC42 (1990) 2646
F q (r ) … 2-particle wave function
S (r ) … source function
x2
p2
few fm
(n,p) correlation function
S(x)
0
x
q = 0.5(p1 - p2)
Z. Ch. - NuSYM 2011, June 17-20, 2011
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L-W Chen et al., PRL90 (2003) 162701
Emission of p’s and n’s: Sensitivity to
SymEn
52Ca
Stiff
EoS
48Ca
Soft
EoS
Soft EoS (γ=0.5)
Stiff EoS (γ=2)
p’s and n’s emitted at similar time
p’s emitted after n’s
faster emission times
later emission times
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Possible emission configurations (stiff sym. pot.)
Catching up
Catching up
p
n
p
n
qx<0
Moving away
p
n
n
qx<0
q=pp -pn =(qx, qy=0, qz=0); r =(x, y=0,z=0)
S(x)
p
qx>0
Moving away
(n,p) correlation function
qx<0
qx>0
qx>0
C(q )= ò d r F q (r ) S (r )
3
0
2
x
Z. Ch. - NuSYM 2011, June 17-20, 2011
q = 0.5(pp - pn)
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L-W Chen et al., PRL90 (2003) 162701
Emission of p’s and n’s: Sensitivity to
SymEn
52Ca
Stiff
EoS
48Ca
Soft
EoS
Soft EoS (γ=0.5)
Stiff EoS (γ=2)
p’s and n’s emitted at similar time
p’s emitted after n’s
faster emission times
later emission times
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Sensitivity to particle emission (soft sym. pot.)
Moving away
p
Catching up
p
n
n
qx<0
Experimentally, we measure
the2
3
C(q )distribution!
= d r F q (r ) S (r )
CF, not the source
ò
S(x)
C( p1, p2 ) =
qx>0
(n,p) correlation function
qx<0
qx>0
P( p1, p2 )
real event pairs
=
P( p1)P( p2 ) mixed event pairs
x
P( p1 ) - single particle distribution
P( p1 , p2 ) -0two particle distribution
q=pp -pn =(qx, qy=0, qz=0); r =(x, y=0,z=0)
Z. Ch. - NuSYM 2011, June 17-20, 2011
qx = 0.5(px,p - px,n)
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Relating asymmetry in the CF to space-time asymmetry
(n,p) correlation function
qx<0
qx>0
C ( qx ) = ò dx Fq ( x ) S ( x ( t ))
2
S(x)
Soft
Stiff
EoS
EoS
<x>
x
0
qx = 0.5(px,p - px,n)
Classically, average separation
b/t protons and neutrons
æ
x < 0 if ç
ç
è
x
p
< x
t p > tn
(
) (
x ( t ) = x p - xn - V t p - t n
)
x = x p - xn - V t p - tn
n Not expected if n,p emitted from the same source
(no n-p differential flow)
Protons emitted later
Voloshin et al., PRL 79:4766-4769,1997
Z. Ch. - NuSYM 2011, June 17-20, 2011 Lednicky et al., PLB 373:30-34,1996
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L-W Chen et al., PRL90 (2003) 162701
IBUU: more calculations
Figure obtained from calculations with
momentum-independent potential
Stiff
AsyEoS
Calculations with momentum
-dependent nuclear potential
Soft
AsyEoS
L-W Chen et al., PRC69 (2004) 054606
Z. Ch. - NuSYM 2011, June 17-20, 2011
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IBUU: averaged emission time
Momentum
dependent
(isoscalar)
Momentum independent
52Ca+48Ca
Momentum
dependent
(isoscalar & isovector)
@ 80 MeVA
Z. Ch. - NuSYM 2011, June 17-20, 2011
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IBUU vs pBUU: Averaged emission time
IBUU
pBUU
Z. Ch. - NuSYM 2011, June 17-20, 2011
52Ca+48Ca
@ 80 MeVA
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pBUU: Averaged emission time
Danielewicz, Bertsch, NPA533 (1991) 712
momentum dependent
WITHOUT CLUSTERS
WITH CLUSTERS
No effect of symmetry energy on averaged emission time of particles
Clusters affect the space-time picture of the HIC (t-3He correlations could show
possible sensitivity to the relative emission time analogously to n-p correlations)
Z. Ch. - NuSYM 2011, June 17-20, 2011
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Summary
 Two particle correlations provide a unique probe to study
the space-time extend of the source
 add constrains on the in-medium cross-section
 importance of the clusters, symmetry energy
 validate theoretical models
 The average relative emission time of n’s and p’s
potentially sensitive to the symmetry energy and can be
“measured” with two particle correlations
 Transport models
 Predictions are model dependent
 Collaboration between theorists and experimentalists beneficial
for both sides
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