Constraining the EOS of neutron-rich nuclear matter
and properties of neutron stars with central heavy-ion reactions
Bao-An Li
Outline:
•
& collaborators:
Wei-Zhou Jiang, Plamen G. Krastev, Richard Nobra, Will Newton,
De-Hua Wen and Aaron Worley, Texas A&M University-Commerce
Lie-Wen Chen and Hongru Ma, Shanghai Jiao-Tung University
Che-Ming Ko and Jun Xu, Texas A&M University, College Station
Andrew Steiner, Michigan State University
Zhigang Xiao and Ming Zhang, Tsinghua University, China
Gao-Chan Yong and Xunchao Zhang, Institute of Modern Physics, China
Champak B. Das, Subal Das Gupta and Charles Gale, McGill University
Indication on the symmetry energy at sub-saturation densities from the
NSCL/MSU isospin diffusion data
Astrophysical implications:
(1) Core-crust transition density of neutron stars
(2) Gravitational waves from elliptically deformed pulsars
•
Indication on the symmetry energy at supra-saturation densities from the
FOPI/GSI π-/π+ data
•
Summary
What is the Equation of State in the extended isospin space?
(EOS of neutron-rich matter)
symmetry energy
Isospin asymmetry δ
12
ρn : neutron density
ρp : proton density

   p 
E (  n ,  p )  E0 (  n   p )  Esym (  )  n
    

  
12
Nucleon density ρ=ρn+ρp
12
E (  n ,  p )
Energy per nucleon in symmetric matter
18
18
3
Energy per nucleon in asymmetric matter
Recent progress and new challenges in
isospin physics with heavy-ion reactions:
density
Bao-An Li, Lie-Wen Chen and Che Ming Ko
Physics Reports, 464, 113 (2008)
0
arXiv:0804.3580
Isospin asymmetry

ρ=ρn+ρp
The Esym (ρ) from model predictions using popular interactions
1 2 E
Esym (  ) 
 E (  ) pure neutron matter  E (  )symmetric nuclear matter
2
2 
Examples:
ρ
EOS of pure neutron matter
Alex Brown,
23PRL85,
RMF 5296 (2000).
models
APR
Density
-
The multifaceted influence of the isospin dependence of strong interaction
and symmetry energy in nuclear physics and astrophysics
J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542.
A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).
(Effective Field Theory)
n/p π-/π+

t/3He K+/K0
(QCD)
Isospin
physics
in
Terrestrial Labs
isodiffusion
isotransport
isocorrelation
isofractionation
isoscaling
Symmetry energy and single nucleon potential used in the IBUU04 transport model
ρ
The x parameter is introduced to mimic
various predictions on the symmetry energy
by different microscopic nuclear many-body
theories using different effective interactions
soft
Default: Gogny force
Density ρ/ρ0
Single nucleon potential within the HF approach using a modified Gogny force:
 '

 
B   1
2
U (  ,  , p, , x )  Au ( x )
 Al ( x )
 B( ) (1  x )  8 x
 '
0
0
0
  1  0
2C ,
2C , '
f ( r , p ')
f ' ( r , p ')
3
3

d
p
'

d
p
'
0 
1  ( p  p ') 2 /  2
0 
1  ( p  p ') 2 /  2
 , '  
1
2 Bx
2 Bx
, Al ( x )  121 
, Au ( x )  96 
, K  211MeV
2
 1
 1 0
The momentum dependence of the nucleon potential is a result of the non-locality
of nuclear effective interactions and the Pauli exclusion principle
C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003).
B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).
Momentum dependence of the isoscalar potential
Compared with variational many-body theory
Momentum and density dependence of the symmetry (isovector) potential
Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions
provides only a constraint at ρ0:
P.E. Hodgson, The Nucleon Optical Model, World
U n / p  U isoscalar  U Lane 
Scientific, 1994
U Lane  (U n  U p ) / 2  V1   R  Ekin , G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972).
V1
28  6MeV, R  0.1  0.2
for E kin  100 MeV
G.R. Satchler, Isospin Dependence of Optical Model
Potentials, in Isospin in Nuclear Physics,
D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)
Constraints from both isospin diffusion and n-skin in 208Pb
Isospin diffusion data:
Transport model calculations
B.A. Li and L.W. Chen, PRC72, 064611 (05)
ρ
M.B. Tsang et al., PRL. 92, 062701 (2004);
T.X. Liu et al., PRC 76, 034603 (2007)
ρρ
J.R. Stone
implication
PREX?
Hartree-Fock calculations
A. Steiner and B.A. Li, PRC72, 041601 (05)
Neutron-skin from nuclear scattering: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994);
B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)
A conservative conclusion about the density dependence of symmetry energy
at sub-saturation densities based on 4 kinds of independent experiments:
(1) isospin diffusion experiment,
(2) n-skin in 208 Pb from hadronic probes
(3) isoscaling in heavy-ion reactions
(4) Isospin dependence of giant monopole resonance
31.6(  / 0 )0.69  Esym (  )  31.6(  /  0 )1.05
K asy ( 0 )  500  50 MeV,
L=86  25 MeV
L.W. Chen, C.M. Ko and B.A. Li,
Phys. Rev. Lett 94, 32701 (2005);
Characterization of the symmetry energy
Slope :
L  3 0 (dEsym / d  )   0 ,
Asymmetry part of the isobaric incompressibility:
K asy (  0 )  9  02 (d 2 Esym / d  2 ) 0  18 0 ( dEsym / d  ) 0
• Rotational glitches: small
changes in period from
sudden unpinning of
superfluid vortices.
Neutron Star Crust
– Evidence
for solid crust.
crust
– 1.4% of Vela moment of
Kazuhiro Oyamatsu, Kei Iida
Phys. glitches.
Rev. C75 (2007) 015801
inertia
– Needs to know the density
and pressure at the
transition to calculate the Can one extract transition density
fractional moment of
from heavy-ion collisions?
inertia of the curst
Chuck Horowitz at WCI3, Texas, 2005
Yes, the symmetry energy constrained by
the isospin diffusion experiments at the
NSCL is in the same density range of the
inner crust
Onset of instability in the uniform n+p+e matter
Dynamical approach
K0
Thermodynamic approach
If one uses the parabolic approximation (PA)
Then the stability condition is:
Stability condition:
>0
Similarly one can use the RPA
What we found about the core-crust transition density
Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, arXiv:0807.4477
It is NOT accurate enough to know the symmetry energy,
one almost has to know the exact EOS of n-rich matter
Why?
Because it is the determinant of the curvature matrix
that determines the stability condition
Example:
Thermodynamical method
The quartic term is also important
for direct URCA, Andrew Steiner,
arXiv:nucl-th/0607040
Constraint on the core-crust transition density
Transition pressure
pasta
Need to reduce the error bars
with more precise data and calculations!
Kazuhiro Oyamatsu, Kei Iida
Phys. Rev. C75 (2007) 015801
Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, arXiv:0807.4477
Partially constrained EOS for astrophysical studies
Plamen Krastev, Bao-An Li and Aaron Worley,
Phys. Lett. B668, 1 (2008).
Danielewicz, Lacey and Lynch,
Science 298, 1592 (2002))
Astrophysical impacts of the partially
constrained symmetry energy
•
Nuclear constraints on the moment of inertia of neutron starsarXiv:0801.1653
Aaron Worley, Plamen Krastev and Bao-An Li, The Astrophysical Journal 685, 390 (2008).
•
Constraining properties of rapidly rotating neutron stars using data from
heavy-ion collisions arXiv:0709.3621
Plamen Krastev, Bao-An Li and Aaron Worley, The Astrophysical Journal, 676, 1170 (2008)
•
Constraining time variation of the gravitational constant G with terrestrial
nuclear laboratory data arXiv:nucl-th/0702080
Plamen Krastev and Bao-An Li, Phys. Rev. C76, 055804 (2007).
•
Constraining the radii of neutron stars with terrestrial nuclear laboratory data
Bao-An Li and Andrew Steiner, Phys. Lett. B642, 436 (2006). arXiv:nucl-th/0511064
•
Nuclear limit on gravitational waves from elliptically deformed pulsars
Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008). arXiv:0805.1973
•
Locating the inner edge of neutron star crust using nuclear laboratory data,
Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma arXiv:0807.4477
What are Gravitational Waves?
• Gravitational Waves = “Ripples in space-time”
Traveling GW
Gravity
J.B. Hartle
Lx
Lx[1 + h(t)]
Amplitude parameterized by (tiny) dimensionless strain h:
h(t) = DL/L
proper separation
between two masses
The expected signal has the form (P. Jaranowski, Phys. Rev. D58, 063001 (1998) ):
 1  cos2  
h  t   F  t;  h 0 
 cos (t )  F  t;  h 0 cos sin (t )
2


F+ and Fx : plus and cross polarization, bounded between -1 and 1
h0 – amplitude of the gravitational wave signal,  – polarization angle of signal
 – inclination angle of source with respect to line of sight, (t)- phase of pulsar
Why do we need to study • Test General Relativity:
Gravitational Waves?
– Quadrupolar radiation? Travels at speed of light?
Michael Landry
LIGO Hanford Observatory
and California Institute of
Technology
– Unique probe of strong-field gravity
• Gain different view of Universe:
– Sources cannot be obscured by dust / stellar envelopes
– Detectable sources are some of the most interesting,
least understood in the Universe
– Opens up entirely new non-electromagnetic spectrum
Gravitational Wave
Interferometer
Projects
LISA
Michelson-Morley IFO
GEO
LIGO
TAMA
VIRGO
LIGO, GEO, TAMA; VIRGO taking
data; LISA is a ESA-NASA project
18
ACIGA
Gravitational Waves
Possible sources of Gravitational Waves:
Compact binary inspiral: “chirps”
Examples
Orbital decay of the
Hulse-Taylor binary
neutron star system
(Nobel prize in 1993)
is the best evidence
so far.
Elliptically deformed pulsars:
“periodic”
Non-radial oscillations of neutron stars
Supernovae / GRBs: “bursts”
Gravitational waves from elliptically deformed pulsars
Solving linearized Einstein’s field equation of General Relativity, the leading contribution
to the GW is the mass quadrupole moment
Frequency of the pulsar
Distance to the observer
Breaking stain of crust
Mass quadrupole moment
EOS
B. Abbott et al., PRL 94, 181103 (2005)
B.J. Owen, PRL 95, 211101 (2005)
Estimate of gravitational waves from spinning-down of pulsars
Assumption: spinning-down is completely due to the GW radiation
“Standard fiducial value”
•
•
•
Solid black lines: LIGO and
GEO science requirement,
for T=1 year
Circles: upper limits on
gravitational waves from
known EM pulsars,
obtained from measured
spindown
Only known, isolated
targets shown here
The LIGO Scientific Collaboration,
Phys. Rev. D 76, 042001 (2007)
GEO
LIGO
Testing the standard fudicial value of the moment of inertia
Aaron Worley, Plamen Krastev and Bao-An Li,
The Astrophysical Journal 685, 390 (2008).
The ellipticity of pulsars

I xx  I yy
I zz

EOS
Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008).
Constraining the strength of gravitational waves
Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008).
Compare with the latest upper limits
from LIGO+GEO observations
Phys. Rev. D 76, 042001 (2007)
It is probably the most uncertain factor
B.J. Owen, PRL 95, 211101 (05)
Pion ratio probe of symmetry energy
at supra-normal densities
GC
Coefficients2
a) Δ(1232) resonance model
in first chance NN scatterings:
(negelect rescattering and reabsorption)

5 N 2  NZ

 (
2

5Z
 NZ

nn
pp
np(pn)
N
Z


0
5
1
1
1
4
0


5
0
1
)2
R. Stock, Phys. Rep. 135 (1986) 259.
b) Thermal model:
(G.F. Bertsch, Nature 283 (1980) 281; A. Bonasera and G.F. Bertsch, PLB195 (1987) 521)

 exp[2( n   p ) / kT ]


m
n
m 1 1 3 m m
n   p  (V  V )  VCoul  kT {ln  
bm (  T ) (    )}
n
p
p m m
2
n
asy
p
asy
H.R. Jaqaman, A.Z. Mekjian and L. Zamick, PRC (1983) 2782.
c) Transport models (more realistic approach):
Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701, and several papers by others
2
Isospin asymmetry reached in heavy-ion reactions E (  ,  )  E (  , 0)  Esym (  )
48
124
48
124
197
197
E/A=800 MeV,
b=0, t=10 fm/c
t=10 fm/c
t=10 fm/c
Correlation between the N/Z and the π-/ π+
Another advantage: the π-/ π+ is NOT sensitive
to the incompressibility of symmetric matter,
but the high density behavior of the symmetry
energy (K0=211 MeV is used in the results
shown here)
(distance from the center of the reaction system)
Formation of dense, asymmetric nuclear matter
E (  ,  )  E (  , 0)  Esym (  ) 2
Symmetry energy
Central density
density
π-/ π+ probe of dense matter
Stiff Esym
n/p ratio at supra-normal densities
π-/π+ ratio as a probe of symmetry energy at supra-normal densities
W. Reisdorf et al. for the FOPI/GSI collaboration , NPA781 (2007) 459
IQMD: Isospin-Dependent Quantum Molecular Dynamics
C. Hartnack, Rajeev K. Puri, J. Aichelin, J. Konopka,
S.A. Bass, H. Stoecker, W. Greiner
Eur. Phys. J. A1 (1998) 151-169
corresponding to Esym (  ) 
100 
3

 (22 / 3  1) EF0 ( ) 2 / 3
8 0
5
0
Need a symmetry energy softer than the above
to make the pion production region more neutron-rich!
E(  ,  )  E(  ,0)  Esym (  ) 2
low (high) density region is more neutron-rich
with stiff (soft) symmetry energy
N/Z dependence of pion production and effects of the symmetry energy
Softer symmetry energy
Stiff symmetry energy
Zhigang Xiao, Bao-An Li, Lie-Wen Chen,
Gao-Chan Yong, Ming Zhang
arXiv:0808.0186
Excitation function
Central density
?
MSU-TPC?
Astrophysical implications
For pure nucleonic matter
K0=211 MeV is used, higher incompressibility
for symmetric matter will lead to higher
masses systematically
The softest symmetry energy
that the TOV is still stable is
x=0.93 giving M_max=0.1
solar mass and R=>40 km
Can the symmetry energy becomes negative at high densities?
Yes, due to the isospin-dependence of the nuclear tensor force
The short-range repulsion in n-p pair is stronger than that in pp and nn pairs
At high densities, the energy of pure neutron matter can be lower than symmetric matter leading to negative symmetry energy
Example: proton fraction with 10 interactions leading to negative symmetry energy
Negative symmetry energy  Isospin separation instability
because of the E sym 2 term,
for symmetric matter,
it is energetically more favoriable to write  =0=1-1,
i.e., pure neutron matter + pure proton matter
x  0.048[ Esym (  ) / Esym (  0 )]3 (  /  0 )(1  2 x )3
 n  e  

Summary
• Based on the NSCL/MSU data, the symmetry energy at sub-saturation
densities is constrained to
31.6(  /  0 ) 0.69  Esym (  )  31.6(  /  0 )1.05
K asy (  0 )  500  50 MeV,
L=86  25 MeV
• The FOPI/GSI pion data indicates a symmetry energy at supra-saturation densities
softer than the APR prediction