Overview of the experimental constraints on
symmetry energy
Betty Tsang, NSCL/MSU
Nuclear Equation of State
Relationship between energy, temperature
pressure, density in nuclear matter
Nuclear Structure – What is the nature of the nuclear force that binds
protons and neutrons into stable nuclei and rare isotopes?
Nuclear Astrophysics – What is the nature of neutron stars and dense
nuclear matter?
E/A (,) = E/A (,0) + 2S()
 = (n- p)/ (n+ p) = (N-Z)/A
How does E/A depend on  and ?
EOS for Asymmetric Matter
30
E/A (MeV)
=1
Soft (=0, K=200 MeV)
asy-soft, =1/3 (Colonna)
asy-stiff, =1/3 (PAL)
=( - )/( + )
20
10
n
p
n
p
0
0
-10
-20
0
0.1
0.2
0.3
0.4
0.5
0.6
 (fm )
-3
E/A (,) = E/A (,0) + 2S()
 = (n- p)/ (n+ p) = (N-Z)/A
P  2
 E / A 
 s / a
Symmetry energy calculations with
effective interactions
constrained by Sn masses
This does not adequately constrain
the symmetry energy at higher
or lower densities
Brown, Phys. Rev. Lett. 85, 5296 (2001)
Symmetry energy at
Overview of the experimental constraints on
symmetry energy around saturation density
=1
0
E sym
L    0
 S o   B
3  0
L  3 0
 K sym
 
 18
Esym
 B

 B  0
  B  0

 0
3
0
Psym
2

  ...

Symmetry energy calculations with
effective interactions
constrained by Sn masses
This does not adequately constrain
the symmetry energy at higher
or lower densities
Brown, Phys. Rev. Lett. 85, 5296 (2001)
Symmetry energy at
Overview of the experimental symmetry energy
constraints around nuclear matter density
Introduction
Experimental Observables:
•Heavy Ion Collisions
n/p ratios
Isospin diffusions
•Mass model (FRDM)
•Isobaric Analog States (IAS)
•Giant dipole Resonance (GDR)
208Pb
skin measurements
•PREX
•antiprotonic atom systematics
•208Pb(p,p)
•208Pb(p,p’) –dipole polarizability
•Dipole Pygmy Resonance (DPR)
Summary and Outlook
Strategies used to study the symmetry energy
with Heavy Ion collisions below E/A=100 MeV
 Vary the N/Z compositions of
Isospin degree of freedom
projectile and targets
124Sn+124Sn, 124Sn+112Sn,
Z ( Z  1)
•
B  a Aa A   a
112Sn+124Sn, 112Sn+112Sn
A
( A  2Z )
a
 Measure N/Z compositions of
A
emitted particles
• n & p yields
• isotopes yields: isospin
diffusion
 Simulate collisions with
transport theory
Crab Pulsar
• Find the symmetry energy
density dependence that
Neutron Number N
describes the data.
• Constrain the relevant input
transport variables.
B.A. Li et al., Phys. Rep. 464, 113 (2008)
S
C
1/ 3
2
sym
Hubble ST
Proton Number Z
2/3
V
E/A (,) = E/A (,0) + 2S()
 = (n- p)/ (n+ p) = (N-Z)/A
Two HIC observables: n/p ratios and isospin diffusion
Tsang et al., PRL 92 (2004) 062701
Li et al., PRL 78 (1997) 1644
100
Projectile
124Sn
stiff
50
(MeV)
soft
Neutron
Proton
V
asy
0
-50
Target
112Sn
-100
0
0.5
u=
1
/
1.5
o
Y(n)/Y(p); t/3He, p+/p-
2
Isospin Diffusion; low , Ebeam
Experimental Layout
Wall A
Wall B
Courtesy Mike Famiano
Neutron walls – neutrons
Forward Array – time start
Proton Veto scintillators
LASSA – charged particles
Miniball – impact parameter
E/A (,) = E/A (,0) + 2S()
 = (n- p)/ (n+ p) = (N-Z)/A
Two observables: n/p ratios and isospin diffusion
Li et al., PRL 78 (1997) 1644
100
gi
stiff
50
Esym=12.7(/o)2/3 + 19(/o)
(MeV)
soft
Neutron
Proton
V
asy
0
-50
-100
0
0.5
u=
Double Ratio
1
/
1.5
2
o
124Sn+124Sn;Y(n)/Y(p)
112Sn+112Sn;Y(n)/Y(p)
minimize
systematic errors
E/A (,) = E/A (,0) + 2S()
 = (n- p)/ (n+ p) = (N-Z)/A
Two observables: n/p ratios and isospin diffusion
• S()  12.3·(ρ/ρ0)2/3 +
17.6· (ρ/ρ0) γi
• IQMD calculations were
performed for gi=0.35-2.0.
• Momentum dependent
mean fields with mn*/mn
=mp*/mp =0.7 were used.
gi
Esym=12.7(/o)2/3 + 19(/o)
Zhang et.al.,Phys. Lett. B 664 (2008) 145
Double Ratio
124Sn+124Sn;Y(n)/Y(p)
112Sn+112Sn;Y(n)/Y(p)
minimize
systematic errors
E/A (,) = E/A (,0) + 2S()
 = (n- p)/ (n+ p) = (N-Z)/A
Two observables: n/p ratios and isospin diffusion
0.4gi 1.05
• S()  12.3·(ρ/ρ0)2/3 +
17.6· (ρ/ρ0) γi
• IQMD calculations were
performed for gi=0.35-2.0.
• Momentum dependent
mean fields with mn*/mn
=mp*/mp =0.7 were used.
Zhang et.al.,Phys. Lett. B 664 (2008) 145
Double Ratio
124Sn+124Sn;Y(n)/Y(p)
112Sn+112Sn;Y(n)/Y(p)
minimize
systematic errors
Isospin Diffusion
Projectile
124Sn
  ( AA   BB ) / 2
Ri ( AB )  2  AB
 AA   BB
Ri= 1 no diffusion; Ri= 0 equilibration
Extent of diffuseness reflects strength of SE
Target
112Sn
mixed 124Sn+112Sn
n-rich 124Sn+124Sn
p-rich 112Sn+112Sn
Projectile
Degree of isospin diffusion
0.4gi 1
124Sn
Target
112Sn
0.45gi 0.95
Consistent constraints from the 2
analysis of isospin diffusion data
S()=12.5(/o)2/3 +17.6 (/o)
gi
Symmetry energy constraints from HIC
S()=12.5(/o
)2/3 +17.6 (/
o)
gi
arXiv:1204.0466
0.4gi 1
E sym
L    0
 S o   B
3  0
 K sym
 
 18
  B  0

 0
2

  ...

L  3 0
Esym
 B

 B  0
Consistent with previous isospin diffusion constraints
Chen, Li et al., Phys. Rev. Lett. 94 (2005) 032701
3
0
Psym
Consistency in Symmetry Energy Constraints
Tsang et al, PRL 102,122701 (2009)
Giant Dipole Resonance
Collective oscillation of
neutrons against protons
Trippa et al., PRC 77, 061304(R) (2008)
Danielewicz & Lee, NPA818, 36 (2009)
Isobaric Analog States aa(A)
P. Moller et al, PRL 108,052501 (2012)
FRDM-2011a. s~0.57 MeV
Extrapolation from 208Pb radius to pressure in n-star
Typel & Brown, PRC 64, 027302 (2001)
P(0.1 fm (MeV/fm3)
Esym

Steiner et al., Phys. Rep. 411, 325 (2005)
0
=L/30
An experimental overview of the symmetry
energy around nuclear matter density
Introduction
Experimental Observables:
•Heavy Ion Collisions
n/p ratios
Isospin diffusions
•Mass model (FRDM)
•Isobaric Analog States (IAS)
•Giant dipole Resonance (GDR)
208Pb
skin measurements
•PREX
•antiprotonic atom systematics
•208Pb(p,p)
•208Pb(p,p’) –dipole polarizability
•Dipole Pygmy Resonance (DPR)
Summary and Outlook
Laboratory experiments to measure the 208Pb neutron skin
Rnp = 0.21±0.06 fm
L=67±12.1 MeV
So=33±1.1 MeV
Pb(p,p)
18
Zenihiro et al., PRC82, 044611 (2010)
Pb skin thickness from dipole polarizability
E
Pb(p,p’) Ebeam=295 MeV
n
p
p++
p+
p
p+
p  O E
polarizability
Tamii et al., PRL107, 062502 (2011)
Reinhard & Nazarewicz ,PRC81, 051303 (R) (2010)
J.Piekarewicz et al., arXiv:1201.3807
Rnp = 0.156+0.025-0.021 fm
Theoretical uncertainties
underpredicted ?
Summary of Pb skin thickness and symmetry energy constraints
arXiv:1204.0466
Diverse experiments but consistent results
Importance of 3n force in the EoS of pure n-matter
See references in
arXiv:1204.0466
Challenge I: constraints from neutron star observations
arXiv:1204.0466
Steiner PRL108, 081102 (2012)
Challenge II: Constraints at very low density
Esym(n)/Esym(n0)
J.B. Natowitz et al, PRL 104 (2010) 202501
Symmetry energy at <0.05o, T<10 MeV is
dominated by correlations and cluster formation
Challenge III: Constraints at supra-normal densities
with Heavy Ion Collisions
Isospin Observables:
Neutron/proton and t/3He and light
isotopes energy spectra
• flow
– px vs. y (v1)
– Elliptic flow (v2)
– Disappearance of flow
(balance energy)
• π+/π- spectra
• π+/π- flow
RIBF, FRIB, KoRIA
International Collaboration
ASY-EOS experiment @GSI, May 2011
Au+Au @ 400 AMeV
96Zr+96Zr @ 400 AMeV
96Ru+96Ru @ 400 AMeV
~ 5x107 Events for each system
Krakow array
Beam Line
Chimera
TofWall
MicroBall
target
Shadow Bar
Russotto & Lemmon
Land
(not splitted
SAMURAI-TPC will be installed inside the SAMURAI
dipole magnet in RIKEN
Electronics and pad plane
AT/TPC @ MSU
Field Cage
Voltage
step down
Outer
enclosure
TPC chamber
arXiv:1204.0466
Acknowledgement:
NSCL/HiRA group
Nuclear Symmetry Energy (NuSym) collaboration
http://groups.nscl.msu.edu/hira/sep.htm
Determination of the Equation of State of Asymmetric Nuclear Matter
MSU: B. Tsang & W. Lynch, G. Westfall, P. Danielewicz, E. Brown, A. Steiner
Texas A&M University : Sherry Yennello, Alan McIntosh
Western Michigan University : Michael Famiano
RIKEN, JP: TadaAki Isobe, Atsushi Taketani, Hiroshi Sakurai
Kyoto University: Tetsuya Murakami
Tohoku University: Akira Ono
GSI, Germany: Wolfgang Trautmann , Yvonne Leifels
Daresbury Laboratory, UK: Roy Lemmon
INFN LNS, Italy: Giuseppe Verde, Paulo Russotto
GANIL, France: Abdou Chbihi
CIAE, PU, CAS, China: Yingxun Zhang, Zhuxia Li, Fei Lu, Y.G. Ma, W. Tian
Korea University, Korea: Byungsik Hong
A Time projection chamber is being built in the US to
measure p+/p- & light charge particles in RIKEN
Summary
Importance of 3n force in the EoS
of pure n-matter at high density.
Consistent Symmetry Energy constraints for 0.3</o<1
using different experimental techniques and theories.
arXiv:1204.0466
Extracting the 208Pb skin thickness from HIC
arXiv:1204.0466
E sym
L    0
 S o   B
3  0
 K sym
 
 18
  B  0

 0
2

  ...

L  3 0
Esym
 B

 B  0
3
0
Psym
Typel & Brown, PRC 64, 027302 (2001)
Steiner et al., Phys. Rep. 411, 325 (2005)
Laboratory experiments to measure the 208Pb neutron skin
208Pb
Existence of a neutron skin
3%  1% measurement
PREXII approved (2013-2014)
Phys Rev Let. 108, 112502 (2012)
Symmetry Energy in Nuclei
B  aV A  a S A
2/3
2
Z ( Z  1)
(
A

2
Z
)
   aC
 a sym
1/ 3
A
A
Inclusion of surface
terms in symmetry
Crab Pulsar
Neutron Number N
Hubble ST
Proton Number Z
2
(
A

2
Z
)
(aVsym A  a Ssym A2 / 3 )
A2
Fitting of Empirical Binding Energies
B  aV A  a S A
2/3
Z ( Z  1)
( A  2Z ) 2
   aC
 Csym
1/ 3
A
A
Csym=22.4 MeV
Souza et al., PRC 78, 014605 (2008)
(aVsym A  a Ssym A2 / 3 )
a2 / b
(a  1/ 3 )
A
Ambiquities in the volume and surface terms of asym
Finite nuclei  infinite nuclear matter
Symmetry coefficient from Isobaric Analog States aa(A)
E   aV A  aS A
2/3
Z2
( N  Z )2
 aC 1/ 3  aa
 Emic
A
A
Charge invariance: invariance of nuclear interactions under rotations in n-p space
Corrections for microscopic
effects + deformation
So = 31.5 MeV; L = 75.6 – 107.1 MeV
So = 33.5 MeV; L = 80.4 – 113.9 MeV
Danielewicz & Lee, NPA818, 36 (2009)
Finite Range Droplet Model (FRDM) – Peter Moller
Sophisticated mass model by adding improvements
FRDM-2011a includes symmetry energy in the fit
So = 32.5 ± 0.5 MeV;
L = 70 ± 15 MeV
P. Moller et al, PRL 108,052501 (2012)
Correlation analysis of Pb skin thickness
Reinhard & Nazarewicz ,PRC81, 051303 (R) (2010)
Skin thickness from Pygmy Dipole Resonance
Collective oscillation of neutron skin against the Core  Rnp
Rnp (208Pb)= 0.20±0.02 fm
L=65.1±15.5 MeV
So=32.3±1.3 MeV
Klimkiewicz et al., PRC 76, 051603(R) (2007)
Wieland et al., PRL102, 092502 (2009)
Carbonne et al., PRC 81, 041301 (2010)
Systematics of anti-protonic atoms
A. Trzcinska et al, PRL 87, 082501 (2001); M. Centelles et al, PRL
102, 122502 (2009); B.A. Brown et al., PRC 76, 034305 (2007); B.
Klos et al., PRC76, 014311 (2007)
Data from 26
nuclei: 40Ca to 238U
Strong correlation
between Rnp and
asymmetry .
Rnp
Antiproton probe
the extreme tail of
nuclei.
NZ/A
Rnp = 0.16±(0.02)stat±(0.04)syst±(0.05)theo
Large experimental and theoretical uncertainties
Need better quality data & theory
Constraining the EoS using Heavy Ion collisions
E/A (,) = E/A (,0) + 2S();
 = (n- p)/ (n+ p) = (N-Z)/A
Au+Au collisions E/A = 1 GeV)
pressure
contours
density
contours
Two observable due to the high pressures formed in the overlap region:
– Nucleons are “squeezed out” above and below the reaction plane.
– Nucleons deflected sideways in the reaction plane.
Determination of symmetric matter EOS
from nucleus-nucleus collisions
Danielewicz et al., Science 298,1592 (2002).
symmetric matter
P (MeV/fm3)
Sideward Flow
in plane
100
RMF:NL3
Fermi gas
Boguta
Akmal
K=210 MeV
K=300 MeV
experiment
10
out of plane
1
The curves represent calculations with
parameterized Skyrme mean fields
They are adjusted to find the pressure
that replicates the observed flow.
1
1.5
2
2.5
3
/
3.5
4
4.5
0
The boundaries represent the range of
pressures obtained for the mean
fields that reproduce the data.
They also reflect the uncertainties from
the input parameters in the model.
5
NSCL experiments 05049 & 09042
Density dependence of the symmetry energy
with emitted neutrons and protons
& Investigation of transport model parameters
(May & October, 2009 )
GSI S394, May 2011
Travel
(k) $
0
year
density
n, p,t,3He
Beam
Energy
50-140
2009
<o
GSI
n, p, t, 3He
400
25
2010/2011
2.5 o
MSU
iso-diffusion
50
0
2011
<o
RIKEN
MSU
RIKEN
iso-diffusion
p+,pn,p,t, 3He,p+,p-
50
140
200-300
25
0
85
GSI
n, p, t, 3He
800
25
2014
2-3o
FRIB
n, p,t,3He, p+,p-
200
0
2018-
2-2.5 o
FAIR
K+/K-
800-1000
?
2018
3o
facility
Probe
MSU
2012
<o
2014-2015 1-1.5 o
2013-2014
2o
NSCL experiment 07038
Precision Measurements of
Isospin Diffusion, June 2011)
arXiv:1204.0466
Definition of Symmetry Energy
B  aV A  a S A
2/3
2
Z ( Z  1)
(
A

2
Z
)
   aC
 a sym
1/ 3
A
A
E/A (,) = E/A (,0) + 2S();
 = (n- p)/ (n+ p) = (N-Z)/A
EOS for Asymmetric Matter
30
Soft (=0, K=200 MeV)
asy-soft, =1/3 (Colonna)
asy-stiff, =1/3 (PAL)
=( - )/( + )
E/A (MeV)
20
10
n
p
n
p
0
-10
-20
0
0.1
0.2
0.3
0.4
 (fm )
-3
0.5
0.6
Challenges: Constraints on the density dependence of
symmetry energy at supra normal density
Xiao et al., PRL102, 062502 (2009)
Russotto et al., PL B697 (2011) 471
Fourpi experiments are not optimized to measure symmetry energy
Need better experiments (ASYEOS & SAMURAI/TPC collaborations)
How to obtain the information about EoS
using heavy ion collisions?
Experiments:
Models
Accelerator: Projectile,
target, energy
Input: Projectile, target, energy.
Detectors: Information of
emitted particles – identity,
spatial info, energy, yields
construct observables
Simulate the collisions with the
appropriate physics
Success depends on the
comparisons of observables.
Theory must predict how reaction evolves from initial
contact to final observables
Transport models (BUU, QMD, AMD)
Describe dynamical evolution of the collision process
Summary of Pb skin thickness and symmetry energy constraints
Hebeler et al., PRL 105, 161102 (2010)
arXiv:1204.0466
Laboratory experiments to measure the 208Pb neutron skin
PREX (Parity Radius experiment)
measures Rnp of 208Pb
Why 208Pb?
It has excess of neutrons
Rnp=<rn2>1/2 -<rp2>1/2
It is doubly-magic
Its first excited state is 2.6 MeV
208Pb
Expected uncertainty:
APV (Parity-violating Asymmetry )~3%
Rn ~1%
Ran in Jefferson Lab in Spring 2010.
Phys Rev Let. 108, 112502 (2012)
Density dependence of Symmetry Energy
E/A (,) = E/A (,0) + 2S();
 = (n- p)/ (n+ p) = (N-Z)/A
neutron matter
??
symmetric matter
P (MeV/fm
Danielewicz, Lacey, Lynch, Science 298,1592 (2002)
3)
100
Akmal
av14uvII
NL3
DD
Fermi Gas
Exp.+Asy_soft
Exp.+Asy_stiff
10
-3
P (MeV/fm )
100
1
RMF:NL3
Akmal
Fermi gas
Flow Experiment
Kaons Experiment
FSU Au
10
1
1
1.5
2
2.5
3
/
3.5
0
4
4.5
5
1
1.5
2
2.5
3
/
3.5
0
4
4.5
5
symmetry energy constraints from IAS
Tsang et al, PRL 102,122701 (2009)
E sym
L    0
 S o   B
3  0
 K sym
 
 18
  B  0

 0
2

  ...

Equation of State (EoS)
Ideal gas: PV=nRT
EOS for Asymmetric Matter
30
Soft (=0, K=200 MeV)
asy-soft, =1/3 (Colonna)
asy-stiff, =1/3 (PAL)
=( - )/( + )
E/A (MeV)
20
10
n
p
n
p
0
-10
-20
0
0.1
0.2
0.3
0.4
 (fm )
-3
0.5
0.6
Definition of Symmetry Energy
B  aV A  a S A
2/3
2
Z ( Z  1)
(
A

2
Z
)
   aC
 a sym
1/ 3
A
A
FRDM
IAS
E/A (,) = E/A (,0) + 2S();
 = (n- p)/ (n+ p) = (N-Z)/A
Reactions
B.A. Brown, PRL 85 (2000) 5296
nuclear
structure
U.S. effort at NSCL
Active Target Time Projection Chamber
(MRI)
•
•
Two alternate modes of operation
• Fixed Target Mode with target wheel inside chamber:
– 4p tracking of charged particles allows full event characterization
– Scientific Program » Constrain Symmetry Energy at >0
• Active Target Mode:
– Chamber gas acts as both detector and thick target (H2, D2, 3He, Ne, etc.)
while retaining high resolution and efficiency
– Scientific Program » Transfer & Resonance measurements, Astrophysically
relevant cross sections, Fusion, Fission Barriers, Giant Resonances
U.S. collaboration on AT-TPC, U.S. – French GET collaboration on electronics.
Symmetry Energy from phenomenological interactions
S()=12.5(/o)2/3 +C (/o)
gi
arXiv:1204.0466
B.A. Brown, PRL 85 (2000) 5296
E sym
L    0
 S o   B
3  0
L  3 0
 K sym
 
 18
Esym
 B

 B  0
  B  0

 0
3
0
Psym
2

  ...

Acknowledgement:
NSCL/HiRA group
Symmetry Energy from Giant dipole Resonsnce
Tsang et al, PRL 102,122701 (2009)
Giant Dipole Resonance
Collective oscillation of
neutrons against protons
Trippa et al., PRC 77, 061304(R) (2008)
Danielewicz & Lee, NPA818, 36 (2009)
Isobaric Analog States aa(A)
P. Moller et al, PRL 108,052501 (2012)
FRDM-2011a. s~0.57 MeV
Mini-summary of symmetry energy constraints from
FRDM and IAS
E sym
L    0
 S o   B
3  0
 K sym
 
 18
  B  0

 0
2

  ...

L  3 0
Esym
 B

 B  0
3
0
Psym
An experimental overview of the symmetry
energy around nuclear matter density
Introduction
Experimental Observables:
•Heavy Ion Collisions
n/p ratios
Isospin diffusions
•Giant dipole Resonance (GDR)
•Mass model (FRDM)
•Isobaric Analog States (IAS)
208Pb
skin measurements
•PREX
•antiprotonic atom systematics
•208Pb(p,p)
•208Pb(p,p’) –dipole polarizability
•Dipole Pygmy Resonance (DPR)
Summary and Outlook
symmetry energy constraints from nuclear masses
Danielewicz & Lee, NPA818, 36 (2009)
Isobaric Analog States aa(A)
relates the asymmetry term to
isospin and energy differences of
two IAS
P. Moller et al, PRL 108,052501 (2012)
Sophisticated mass model by
adding improvements
FRDM-2011a includes symmetry
energy in the fit. s~0.57 MeV
E sym
L    0
 S o   B
3  0
 K sym
 
 18
Tsang et al, PRL 102,122701 (2009)
  B  0

 0
2

  ...
