Symmetry Energy in Nuclei · 5/19/2009 · spectator remnant is to shift the isotopic...
Transcript of Symmetry Energy in Nuclei · 5/19/2009 · spectator remnant is to shift the isotopic...
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Neutron Number N
Pro
ton
Nu
mb
er Z
3/2AaAaB SV 3/1)1(
A
ZZaC
A
ZAasym
2)2(
Symmetry Energy in Nuclei
Inclusion of surface
terms in symmetry
2
23/2 )2()(
A
ZAAaAa SV
symsym
Hub
ble
ST
Crab Pulsar
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Physics of symmetry energy
• masses
• n-skin of heavy nuclei such as 208Pb
• Fission barriers
• .
• .
• Neutron star properties
Neutron Number N
Pro
ton
Nu
mb
er Z
3/2AaAaB SV 3/1)1(
A
ZZaC
A
ZAasym
2)2(
Hub
ble
ST
Crab Pulsar
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Density dependence of Symmetry Energy
Questions:
What have we learned about symmetry energy at low density?
What are the unique opportunities offered at high density?
F1
F3
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How to connect different symmetry energy representations
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Total Symmetry Energy
i=0.5
i=2.0
FSU GOLDAV18skm*NL3
Esy
m (
MeV
)
/0
Value of symmetry
energy at saturation
2
0
0
0
04
183
BsymBsym
KLaE
sym
B
symP
EL
B0
0
33
0
Expansion around 0:
Symmetry slope L &
curvature Ksym
Symmetry pressure Psym
40)( aEsym
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The National Superconducting
Cyclotron Laboratory
Michigan State University
Betty Tsang
Defining the neutron star crust: Giant
flares, superbursts and
x-ray bursts
May 17-21, 2009Santa Fe, NM
Learning about the EoS of Asymmetric Nuclear matter with Heavy Ion Collisiosn
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How to obtain the information about EoS?
Both astrophysical and laboratory observables can constrain the EoS, (,T,) or P(,T,) indirectly.
What are the experimental uncertainties of the constraints?
What are the model uncertainties of the constraints?
Experiments:
Accelerator: Projectile,
target, energy
Detectors: Information of
emitted particles – identity,
spatial info, energy, yields
construct observables
Models
Input: Projectile, target, energy.
Simulate the collisions with the
appropriate physics
Success depends on the
comparisons of observables.
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Transport models:
Describe dynamical evolution of the
collision process
•Self consistent mean field
•n-n collisions,
•Pauli exclusion
Uncertainties
Semi-classical
Approximations needed to make
computation feasible.
What are the uncertainties in theoretical predictions?
Symmetry energy included in the
nuclear EOS for infinite nuclear
matter at various densities from
the beginning of collision.
Low and Intermediate Energy
Heavy Ion Collisions Workshop
ECT*
5/11/2009 - 5/15/2009, Trento, Italy
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Reaction mechanism of fragment
productions depends on impact
parameters
Heavy Ion collision: 124Sn+124Sn, E/A=50 MeV
Neck fragments
b=7 fm
multifragmentationb=0 fm
S()=12.5(/o)2/3 +19(/o)
i
Charged fragments (Z=3-20) are formed at
subnormal density
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Strategies used to study the symmetry energy
with Heavy Ion collisions
• Vary the N/Z compositions of projectile and targets124Sn+124Sn, 124Sn+112Sn,
112Sn+124Sn, 112Sn+112Sn
• Measure N/Z compositions of emitted particles
n & p yields
isotopes yields – isospin diffusion
p+ & p- at high incident energy
Neutron Number N
Pro
ton
Nu
mb
er Z
3/2AaAaB SV 3/1)1(
A
ZZaC
A
ZAasym
2)2(
Isospin degree of freedom
Hub
ble
ST
Crab Pulsar
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Experimental Observables: n/p yield ratios
-100
-50
0
50
100
0 0.5 1 1.5 2
Li et al., PRL 78 (1997) 1644
Vasy
(M
eV
)
/ o
Neutron
Proton
F1=2u2/(1+u)
F2=u
F3=u
F1F2
F3
u =
stiff
soft
Uas
y(M
eV)
=0.3
•n and p potentials have opposite sign.
•n & p energy spectra depend on the symmetry energy softer density dependence emits more neutrons at low density.
•More n’s are emitted from the n-rich system and softer iso-EOS.
ImQMD
Y(n
)/Y
(p)
S()=12.5(/o)2/3 +17.6(/o)
i
-
80-380
420-620
n/p Experiment 124Sn+124Sn; 112Sn+112Sn; E/A=50 MeV
Scattering Chamber
Famiano et al
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n/p Double Ratios (central collisions)
124Sn+124Sn;Y(n)/Y(p)112Sn+112Sn;Y(n)/Y(p)
Double Ratiominimize systematic errors
Data : Famiano et al. PRL 97 (2006) 052701
Will repeat
experiment for
better accuracy
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Analysis of n/p ratios with ImQMD model
Esym=12.5(/o)2/3 + 17.6 (/o)
i
0.4i 1.05
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Analysis of n/p ratios with ImQMD model
Esym=12.5(/o)2/3 + 17.6 (/o)
i
0.4i 1.05
Data need better measurements but the trends and
magnitudes still give meaningful 2 analysis at 2 level
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Experimental Observable :
Isospin Diffusion
Isospin diffusion is measured with fragments emitted from the neck region.
Probe the symmetry energy at subsaturation densities in semi-peripheral collisions, e.g. 124Sn + 112Sn at b=6 fm
Isospin “diffuse” through low-density neck region
Symmetry energy drives system towards equilibrium.
• stiff EOS small diffusion; |Ri|>>0
• soft EOS fast equilibrium; Ri0
Projectile
Target
124Sn
112Sn
soft
stiff
BBAA
BBAAABi
xx
xxxR
2/)(2
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Isotope Distribution Experiment
MSU, IUCF, WU collaboration
Sn+Sn collisions involving 124Sn, 112Sn at E/A=50 MeV
Miniball + Miniwall
4 p multiplicity arrayZ identification, A
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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 :
)ZNexp(CZ,NY
Z,NY
1
2
Isotope distributions and isospin diffusions
1.0
0.33
-0.33
-1.0
Ri()
Isospin diffusionRi()
no diffusion
and are related to the
nucleon chemical potentials
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Analysis of isospin diffusion data with ImQMD model
BBAA
BBAAABi
xx
xxxR
2/)(2 iS()=12.5(/o)
2/3 + 17.6 (/o)
x(data)=x(QMD)=
EquilibriumRi=0
No diffusionRi =1; Ri =-1
0.4i 1
Impact parameter is not well determined in the experiment
b~5.8 – 7.2 fm
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Analysis of rapidity dependence of Ri with ImQMD model
BBAA
BBAAABi
xx
xxxR
2/)(2 iS=12.5(/o)
2/3 + 17.6 (/o)
x(data)=Y(7Li/7Be)x(QMD)=
EquilibriumRi=0
No diffusionRi =1; Ri =-1
0.4i 1
b~5.8 – 7.2 fm
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For the first time, we have a transport model that describes
np ratios and two isospin diffusion measurements
Consistent constraints from the 2 analysis of three observables
S()=12.5(/o)2/3 +17.6 (/o)
0.4i 10.4i 1.050.45i 0.95
0.4i 1i
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Expansion around 0:
slope L & curvature Ksym
...183
2
0
0
0
0
BsymBo
KLSS
S()=12.5(/o)2/3 +17.6 (/o)
i
IQM
D
S()~31.6(/o)
IBU
U04
LSymmetry pressure Psym
sym
B
symP
EL
B0
0
33
0
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IQM
D
IBU
U04
...183
2
0
0
0
0
BsymBo
KLSS
S=12.5(/o)2/3 + 17.6 (/o)
i S~31.6(/o)
Rnp=0.04 fm
LSymmetry pressure Psym
sym
B
symP
EL
B0
0
33
0
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S=12.5(/o)2/3 + Cs,p(/o)
i
No constraints on So
ImQMD
2 2 analysis
...183
2
0
0
0
0
BsymBo
KLSS sym
B
symP
EL
B0
0
33
0
Vary Cs,p and i
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Esym=12.5(/o)2/3 + Cs,p(/o)
i
...183
2
0
0
0
0
BsymBosym
KLSE sym
B
symP
EL
B0
0
33
0
Constraints from masses and
Pygmy Dipole Resonances
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Esym=12.5(/o)2/3 + Cs,p(/o)
i
...183
2
0
0
0
0
BsymBosym
KLSE sym
B
symP
EL
B0
0
33
0
Constraints from masses and
Pygmy Dipole Resonances
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Esym=12.5(/o)2/3 + Cs,p(/o)
i
...183
2
0
0
0
0
BsymBosym
KLSE sym
B
symP
EL
B0
0
33
0
Current constraints on
symmetry energy from HIC
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Constraints on the density dependence of symmetry energy
?
Au+Au
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Constraints on the density dependence of symmetry energy
Au+Au experiments in high energy are
not designed to measure symmetry energy
Need better experiments
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1. HI collision dynamics are complex but prove to be sensitive to density dependence of symmetry energy.
2. Need multiple observables to place various constraints on the input parameters of the transport models
3. Problems still remain, e.g.
• How to extract results to T=0;
• Control of input parameters in transport models.
Low and Intermediate Energy
Heavy Ion Collisions Workshop
ECT*
5/11/2009 - 5/15/2009, Trento, Italy
Lessons learned from LE measurements:
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International Collaboration
RIBF
FRIB
Outlook
MSU
GSI
FAIR
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Acknowledgements
NSCL Transport simulation group
Brent Barker, Abby Bickley, Dan Coupland, Krista Cruse, Pawel
Danielewicz, Micha Kilburn, Bill Lynch, Michelle Mosby, Scott Pratt,
Andrew Steiner, Josh Vredevooqd, Mike Youngs, YingXun Zhang
Experimenters
Michigan State University
T.X. Liu (thesis), W.G. Lynch, Z.Y. Sun, W.P.
Tan, G. Verde, A. Wagner, H.S. Xu
L.G. Sobotka, R.J. Charity (WU)
R. deSouza, V. E. Viola (IU)
M. Famiano: (Westen Michigan U)
Y.X. Zhang (ImQMD), P. Danielewicz, W.A. Friedman, A.
Ono, W.G. Lynch, L.J. Shi, Jenny Lee,
Trento workshop organizers and participants
Jorg Aichelin, Theo Gaitanos, Hermann Wolter
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Summary
The density dependence of the symmetry energy is of fundamental
importance to nuclear physics and neutron star physics.
Observables in HI collisions provide unique opportunities to probe
the symmetry energy over a range of density especially for dense
asymmetric matter
Need more guidance from theory regarding observables and
interpretation of experimental results.
The availability of intense fast rare isotope beams at a variety of
energies at RIKEN, FRIB & FAIR allows increased precisions in
probing the symmetry energy at a range of densities – international
co-ordination of a global program.
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Au+Au
Wed afternoon discussions
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Au+Au
stiffness
Inconsistencies in the
constraints on symmetry energy
at high and low density?
Wed afternoon discussions
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Wednesday afternoon discussions: Experiments in RIBF
?RIBF
Isospin diffusions
for fragments and
residues with RIBF,
~2010
Riken TPC &
to measure p+/p, t/3He, p/n (Nebula),
spectra ratios
RIBF
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E/A=200-300 MeV measure p+, p- spectra ratios, p,n, t/3He spectra ratios
and differential flow determine S(), m*, nn, pp, np at ~2o
E/A~50 MeV measure isospin diffusions for fragments and residues
determine S() at
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Measured Isotopic yieldsT.X Liu et al. PRC 69,014603
P T
Central collisions
Similar distributionsR21(N,Z)=Y2(N,Z)/ Y1(N,Z)
Xu et al, PRL, 85, 716 (2000)
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?MSUGSI
FRIB
FAIR
GSI (2011) : E/A~400 – 800 MeV measure p,n spectra ratios and differential
flow determine constraints S(, m*, nn, pp, np at 2.5o<
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MSU
FRIB
FAIR
MSU (2009-2012) : E/A
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?MSUGSI
FRIB
FAIR
GSI (2011) : E/A~400 – 800 MeV measure p,n spectra ratios and differential
flow determine constraints S(, m*, nn, pp, np at 2.5o<
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2020?
FRIB
FAIR
Outlook
Precision measurements in FRIB
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n detectors: NSCL/pre-FRIB; GSI; RIBF/Riken Pions/kaons & p, t, 3He detectors: TPC
NSCL Dual purpose AT-TPC: proposal to be submitted to DOE
RIBF TPC: SUMARAI magnet funded, TPC – Japan-US collaboration: proposal to be submitted to DOE.
AT-TPC: FRIB
~ 1.2 M
TPC ~ 0.78 M
Travel: 0.37 M
Detectors needed:
-
Density region sampled
depends on collision
observable & beam
energy
• >0 examples:
– Pion energy spectra
– Pion production ratios
– Isotopic spectra
– Isotopic flow
– With NSCL beams, densities
up to 1.7 0 are accessible
– Beams: 50-150 MeV,
50,000pps
106Sn-126Sn, 37Ca-49Ca
Riken Samurai TPC
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Density region sampled
depends on collision
observable & beam
energy
• >0 examples:
– Pion energy spectra
– Pion production ratios
– Isotopic spectra
– Isotopic flow
– With NSCL beams, densities
up to 1.7 0 are accessible
– Beams: 50-150 MeV,
50,000pps
106Sn-126Sn, 37Ca-49Ca
Riken Samurai TPC
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Experimental Observable :
Isospin Diffusion
Probe the symmetry energy at
subsaturation densities in
peripheral collisions, e.g. 124Sn + 112Sn
Isospin “diffuse” through
low-density neck region
Projectile
Target
124Sn
112Sn
BBAA
BBAAABi
xx
xxxR
2/)(2
x(calc)=
soft
stiff
Symmetry energy drives system towards equilibrium.
•stiff EOS small diffusion; |Ri|>>0
•soft EOS fast equilibrium; Ri0
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Isospin Diffusion
from isoscaling from Y(7Li)/Y(7Be)
Projectile
Target
124Sn
112Sn
No diffusion
Complete mixing
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Au+Au
stiffness
Inconsistencies in the constraints on
symmety energy at high and low density?
Wed afternoon discussions
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n-star HI collisions
/0~ 0.1 - 10 ~ 0.1-5
ye~0.1 ~0.38-0.5
T(MeV) ~1 ~ 4-50
Extrapolate information from
limited asymmetry and
temperature to neutron stars!
Laboratory experiments to study properties of neutron stars
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Laboratory experiments to study properties of neutron stars
208Pb
extrapolation from 208Pb radius to n-star radius
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Isotope Distribution Experiment
MSU, IUCF, WU collaboration
Sn+Sn collisions involving 124Sn, 112Sn at E/A=50 MeV
Miniball + Miniwall
4 p multiplicity arrayZ identification, A
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N/Z ratios from bound fragments (Z=3-8)
complementary to n/p ratios
EC.M.
P T
Esym=12.7(/o)2/3 + 19(/o)
i
Effects are
small
Hot fragments
produced in
calculations.
-
P T
Sequential
decay effects
are important
Data consistent
with soft EOS
N/Z ratios from bound fragments (Z=3-8)
complementary to n/p ratios
Esym=12.7(/o)2/3 + 19(/o)
i
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Experimental Observables to probe the symmetry energy
E/A(,) = E/A(,0) + 2S() ; = (n- p)/ (n+ p) = (N-Z)/A
• Collision systems: 124Sn+124Sn, 124Sn+112Sn, 112Sn+124Sn, 112Sn+112Sn
• E/A=50 MeV
• Low densities (
-
asy-stiff
asy-soft
Exploring Bulk properties of Nuclear Matter with Heavy Ion Collisions
High density/energy
• differential flow
• n/p, LIF ratios
• pions ratios
• kaon ratios
• neutron stars
Low density/energy
• fragments, ratios
• isospin diffusion
• isoscaling
• migration/fractionat.
• collective excitations
• surface phenomena
• phase transitions
QF Li, Di Toro
Di Toro, Lukasic
DiToro, Reisdorf, QF Li
Prassa, QF Li
BA Li, Kubis
Tsang, HW
BA Li, HW
Tsang
Di Toro
Aumann, Ducoin
Lehaut
Danielewicz
Hermann Wolter
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Cluster effects
Cluster effects
are important
for low energy
nucleons but
cannot explain
the large
discrepancy
between data
and IBUU04
calculations
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Constraints from Isospin Diffusion Data
124Sn+112Sn data
Data:
•Collisions of
Sn+Sn isotopes
at E/A=50 MeV
•b/bmax>0.8
~7.2 fm
from
multiplicity
gates.
•y/yb>0.7
•Results
obtained with
clusters
IBUU04:
Collisions of
Sn+Sn isotopes
at E/A=50 MeV
b=6 fm
Projectile-like
residue
determined from
density &
y/yb>0.5
No clusters
x=0
x=1(soft)
x=-1
x=-2
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Isospin diffusion in the projectile-like region
Basic ideas:
• Peripheral reactions
• Asymmetric collisions 124Sn+112Sn, 112Sn+124Sn
-- diffusion
• Symmetric Collisions 124Sn+124Sn, 112Sn+112Sn
-- no diffusion
• Relative change between
target and projectile is the
diffusion effect
Projectile
Target
-
Density region sampled
depends on collision
observable & beam
energy
• >0 examples:
– Pion energy spectra
– Pion production ratios
– Isotopic spectra
– Isotopic flow
– With NSCL beams, densities
up to 1.7 0 are accessible
– Beams: 50-150 MeV,
50,000pps
106Sn-126Sn, 37Ca-49Ca
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• The density dependence of
symmetry energy is largely
unconstrained.
• Pressure, i.e. EOS is rather
uncertain even at 0.
Isospin Dependence of the
Nuclear Equation of State
E/A (,) = E/A (,0) + 2S()
= (n- p)/ (n+ p) = (N-Z)/A
-20
-10
0
10
20
30
0 0.1 0.2 0.3 0.4 0.5 0.6
EOS for Asymmetric Matter
Soft (=0, K=200 MeV)asy-soft, =1/3 (Colonna)asy-stiff, =1/3 (PAL)
E/A
(M
eV)
(fm-3
)
=(n-
p)/(
n+
p)
PAL: Prakash et al., PRL 61, (1988) 2518.
Colonna et al., Phys. Rev. C57, (1998) 1410.
Brown, Phys. Rev. Lett. 85, 5296 (2001)
a/s
2 A/EP
Neutron matter EOS
-
dailygalaxy.com
•
-
Transport description of heavy ion collisions:
,fIfUfm
p
t
fcollp
2-body hard collisions
)1)(1()1)(1(
)()2(
1
432432
43213412
124322
3
ffffffff
ppppd
dvdpdpdp
p
loss term gain term
11 1
1
2
3 4
non-relativistic: BUU
effective mass
Kinetic momentum
Field tensor
Relativistic BUU
),( fIcoll
Vlasov eq.; mean field
EOS
Simulation with Test Particles:
me
an
px(y
0)
[Me
V]
-
Analysis of rapidity dependence of Ri with ImQMD model
Esym=12.5(/o)2/3 + 17.6 (/o)
i
New analysis on rapidity dependence of isospin diffusion ratios –
not possible with BUU type of simulations due to lack of fragments.
0.45i 0.95
-
Extracting Density dependence of Symmetry Energy in Heavy Ion Reactions
F1
F3
HIC provides a range of density determined
from incident energy and impact parameter
Micha Kilburn
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The National Superconducting
Cyclotron Laboratory
Michigan State University
Betty Tsang
RIBF Mini-workshop on
Nuclear Collisions and
Nuclear Matter
Dec 16-17, 2008Riken
Learning about the Density dependence of Symmetry Energy in Heavy Ion Reactions
?RIBF
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n/p Double Ratios (central collisions)
•Effect is much larger
than IBUU04
predictions
inconsistent with
conclusions from
isospin diffusion data.
124Sn+124Sn;Y(n)/Y(p)112Sn+112Sn;Y(n)/Y(p)
Double Ratiominimize systematic errors
Double
Rat
io
Center of mass EnergyFamiano et al. RPL 97 (2006) 052701
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What are the densities created in Heavy Ion Collisions
Highest density reached by
central collisions depends
on incident beam energy.
Types of particles formed depend
on emission times and density.
Pions, n, p fragments
E/A=1600 MeV
200 MeV50 MeV
-
• Experiment: measure collective flow (emission patterns) of particles emitted
in Au+Au collisions from (E/A~1-8 GeV).
• Transport model (BUU) relates the measurements to pressure and density.
pressure
contours
density
contours
Danielewicz, Lacey, Lynch, Science 298,1592 (2002)
-
E/A(,) = E/A(,0) + 2S() ; = (n- p)/ (n+ p) = (N-Z)/A
Equation of State of Nuclear Matter
1
10
100
1 1.5 2 2.5 3 3.5 4 4.5 5
symmetric matter
RMF:NL3AkmalFermi gasFlow Exp.
P (
MeV
/fm
-3)
/0
Danielewicz, Lacey, Lynch, Science 298,1592 (2002)
-
1
10
100
1 1.5 2 2.5 3 3.5 4 4.5 5
symmetric matter
RMF:NL3AkmalFermi gasFlow ExperimentKaons ExperimentFSU Au
P (
MeV
/fm
-3)
/0
Danielewicz, Lacey, Lynch, Science 298,1592 (2002)
•Newer calculation and
experiment are consistent with
the constraints.
E/A(,) = E/A(,0) + 2S() ; = (n- p)/ (n+ p) = (N-Z)/A
Equation of State of Nuclear Matter
??
•Transport model includes
constraints in momentum
dependence of the mean field
and NN cross-sections
-
EOS: symmetric matter and neutron matter
E/A (,) = E/A (,0) + 2S()
= (n- p)/ (n+ p) = (N-Z)/A
Brown, Phys. Rev. Lett. 85, 5296 (2001)
Neutron matter EOS
-20
0
20
40
60
80
100
120
0 1 2 3 4
symmetric matter
NL3Bog1:e/aBog2:e/aK=300, m*/m=70Akmal_corr.
E/A
( M
eV)
/0
•The density dependence
of symmetry energy is
largely unconstrained.
-
N-detection – neutron wall
-
p-detection: Scattering Chamber
3 particle telescopes
(p, d, t, 3He, …)
WU MicroBall
(b determination)
~6in
2.0ˆ b
# of charged particles
central
-
n/p Double Ratios (central collisions)
124Sn+124Sn;Y(n)/Y(p)112Sn+112Sn;Y(n)/Y(p)
Double Ratiominimize systematic errors
Double
Rat
io
Center of mass EnergyFamiano et al. RPL 97 (2006) 052701
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Nuclear Collisions simulations with Transport Models
– Nuclear EOS included from beginning of collisions
BUU models:
Semiclassical solution of one-
body distribution function.
Pros
Derivable, approximations
better understood.
Cons
Mean field no fluctuations
BUU does not predict cluster
formation
QMD:
Molecular dynamics with
Pauli blocking.
Pros
Predicts cluster production
Cons
Cluster properties (masses,
level densities) approximate
Need sequential decay codes
to de-excite the hot fragments
Code used: ImQMD
At high incident energies, cluster production is weak
the two models yield the same results.
Clusters are important in low energy collisions.
-
y/ybeam
QMDisoscaling
R7
R7Ri()=Ri()
-
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? What is the origin of elements heavier than iron in the
cosmos? What are the nuclear reactions that drive stars and stellar
explosions?
Tests of Fundamental Symmetries – Why is there now more matter
than antimatter in the universe?
Examples of possible research areas in FRIB
Learning about the Equation of State of Asymmetric
Nuclear Matter with Heavy Ion Collisions
-
Nuclear Equation of State
E/A (,) = E/A (,0) + 2S()
= (n- p)/ (n+ p) = (N-Z)/A
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? What is the origin of elements heavier than iron in the
cosmos? What are the nuclear reactions that drive stars and stellar
explosions?
Tests of Fundamental Symmetries – Why is there now more matter
than antimatter in the universe?
Examples of possible research areas in FRIB
-
xAB, yAB experimental or
theoretical observable for AB
yAB= a xAB+b
Ri(xAB )= Ri(yAB )
Rami et al., PRL, 84, 1120 (2000)
BBAA
BBAAABi
xx
xxxR
2/)(2
Experimental Observable : Isospin Diffusion --Isospin Transport Ratio
No isospin diffusion between
symmetric systems124
124112
112
Isospin diffusion occurs only
in asymmetric systems A+B124
112
Non-isospin diffusion effects
same for A in A+B & A+A ;same for B in B+A & B+B
Non-isospin transport effects are “cancelled”??
Ri = 1
Ri = -1
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0.4i 1
0.4i 1.050.45i 0.95
Trautmann et al
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Betty Tsang
Learning about Symmetry Energy at Low Density with Heavy Ion Reactions
Defining the neutron star crust: Giant
flares, superbursts and
x-ray bursts
May 17-21, 2009Santa Fe, NM
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Constraints from Isospin Diffusion Dataof calculation!
M.B. Tsang et. al., arXive:0811.3107 L.W. Chen, … B.A. Li, PRL 94, 032701 (2005)
iQMD:S()=12.5(/o)2/3 + 19.6 (/o) IBUU04 : S()~31.6(/o)
stiffness
Are these constraints consistent? What are the model uncertainties?
How to connect different representations of the symmetry energy
stiffness
0.69 1.05i ~0.7