Transcript Probing the EOS of Neutron-Rich Matter with Heavy

Determining the Nuclear Symmetry Energy
of Neutron-Rich Matter and its Impacts on Astrophysics
Bao-An Li
Collaborators:
Wei-Zhou Jiang, Plamen Krastev, Aaron Worley, Texas A&M-Commerce
Lie-Wen Chen, Shanghai Jiao-Tung University
Che-Ming Ko, Texas A&M University, College Station
Andrew Steiner, Michigan State University
Gao-Chan Yong, Chinese Academy of Science, Lanzhou
Outline:
•
Theoretical predictions about density dependence of nuclear symmetry energy
•
How to constrain the symmetry energy with heavy-ion collisions
•
Astrophysical impacts of the partially constrained nuclear symmetry energy
Two examples:
(1) Mass-radius correlation of rapidly-rotating neutron stars
(2) The changing rate of the gravitational constant G due to the expansion of the Universe
•
Summary
What is the Equation of State in the extended isospin space
at zero temperature ?
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
density
0
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 
(1) Phenomenological models
23 RMF
models
Density
Symmetry energy (MeV)
(2) Microscopic model predictions
Brueckner HF
Greens function
Variational
many-body
theory
Density
A.E. L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier and V. Rodin, Phys.
Rev. C68 (2003) 064307
Interaction dependence
within the Bruckner Hartree-Fock Approach
With 3-body forces
Z.H. Li et al.,
PRC74, 047304 (2006)
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
A road map towards determining the nature
of neutron-rich nucleonic matter
?
The most important input to transport models for reactions involving neutron-rich nuclei
U n / p (  ,  , p)  Uisoscalar (  , p)  U sym (  , p) ,  for n/p
The most challenging unknown is the momentum-dependence of the symmetry potential
ρρ
Momentum and density dependence of the symmetry potential
?
δ
δ
?
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 Lane  (U n  U p ) / 2  V1   R  Ekin , Scientific, 1994
L. Ray, G.W. Hoffmann and W.R. Coker, Phys. Rep. 212,
V1 28  6MeV, R  0.1  0.2
(1992) 223.
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)
Symmetry energy and single nucleon potential used in the IBUU04 transport model
ρ
The x parameter is introduced to mimic
various predictions by different microscopic
Nuclear many-body theories using different
Effective interactions
soft
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).
Promising Probes of the Esym(ρ) in Nuclear Reactions
At sub-saturation densities
 Sizes of n-skins of unstable nuclei from total reaction cross sections
 Proton-nucleus elastic scattering in inverse kinematics
 Parity violating electron scattering studies of the n-skin in 208Pb at JLab
 n/p ratio of FAST, pre-equilibrium nucleons
 Isospin fractionation and isoscaling in nuclear multifragmentation
 Isospin diffusion/transport
 Neutron-proton differential flow
 Neutron-proton correlation functions at low relative momenta
 t/3He ratio
Towards high densities reachable at CSR/China, FAIR/GSI, RIA and RIKEN
 π -/π + ratio, K+/K0 ?
 Neutron-proton differential transverse flow
 n/p ratio of squeezed-out nucleons perpendicular to the reaction plane
 Nucleon elliptical flow at high transverse momenta
(1) Correlations of multi-observable are important
(2) Detecting neutrons simultaneously with charged particles is critical
Significant progress has been made recently in constraining the symmetry
energy at sub-saturation densities while NOTHING is known at higher densities
Neutron-skin in 208Pb and dEsym/dρ
B.A. Brown, S. Typel, C. Horowitz, J. Piekarewicz, R.J. Furnstahl, J.R. Stone, A. Dieperink et al.
C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001)
Rn-Rp (fm) for 208Pb
208Pb
B.A. Brown PRL85, 5296 (2000)
S. Typel and B.A. Brown,
PRC 64, 027302 (2001)
Pressure forces neutrons out against the surface tension from the symmetric core near ρ0
Eneutron  Enuclear  Esym ,
Rn  Rp  pnp  dEnuclear / d  |0  dEsym / d  |0  0  dEsym / d  |0
Extract the Esym(ρ) at subnormal densities from isospin diffusion/transport
A quantitative measure of isospin transport:
RXA B
2 X A B  X A  A  X B  B

X A A  X B  B
RXA A  1 and RXBB  1
RXA B  0 for complete isospin mixing
X is any isospin-sensitive observable,
F. Rami et al. (FOPI/GSI), PRL, 84 (2000) 1120.
Isospin transport/diffusion:
n   p   DI r
The degree of isospin transport/diffusion depends
on both the symmetry potential and the in-medium
neutron-proton scattering cross section.
For near-equilibrium systems, the mean-field contributes:

D  [ n   p  U n  U p ]  [4 Esym (  )  2U sym (  , p)]

m
I
L. Shi and P. Danielewicz,
PRC68, 064604 (2003)
During heavy-ion reactions, the collisional contribution to DI
is expected to be proportional to σnp
Isospin transport/diffusion experiments at the NSCL/MSU
M.B. Tsang et al., Phys. Rev. Lett. 92, 062701 (2004);
T.X. Liu et al., PRC 76, 034603 (2007)



X7=7Li/7Be
Transport model analysis of the NSCL/MSU data
L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. Lett 94, 32701 (2005);
Bao-An Li and Lie-Wen Chen, Phys. Rev. C72, 064611 (2005).
σ
σ



ρρ
Characterization of the symmetry energy
Slope :
L  3 0 (dEsym / d  )  0 ,
Asymmetry part of the isobaric incompressibility:
Kasy (  0 )  9  02 (d 2 Esym / d  2 ) 0  18 0 (dEsym / d  ) 0
MSU data range
Using in-medium NN xsections
(reduced wrt the free one)
PRL 99, 162503 (2007)
K  550  100 MeV
Consistent with conclusions based on
the isospin diffusion experiment:
32(  / 0 )0.7  Esym (  )  32(  / 0 )1.1
K asy ( 0 )  500  50 MeV,
L=88  25 MeV for   1.2 0
Constraining the dEsym/dρ with data from both isospin diffusion and n-skin in 208Pb
Andrew Steiner and Bao-An Li, PRC 72, 041601 (2005).
Isospin
fractionation
ρ
Neutron-rich
cloud
ρρ
Neutron-skin data: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994);
B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)
Comparing with calculations using 23 most popular RMF models widely
used in nuclear structure studies and astrophysics
L.W. Chen, C.M. Ko and B.A. Li, PRC 76, 054316 (2007)
Comparing with Hartree-Fock calculations using 21 most popular Skyrme
interactions widely used in nuclear structure studies and astrophysics
ρ
Predictions using most of the 21
widely used Skyrme interactions
are ruled out ! Only 5 survived !
Esym in the high density region
is still not constrained !
density
L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. C72, 064309 (2005).
Astrophysical impacts of the partially
constrained symmetry energy
•
Nuclear constraints on the moment of inertia of neutron stars
Aaron Worley, Plamen Krastev and Bao-An Li, The Astrophysical Journal (2008) in press .
•
Constraining properties of rapidly rotating neutron stars using data from
heavy-ion collisions
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
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).
•
Setting an upper limit on the gravitational waves from isolated, rotating
elliptical neutron stars with terrestrial nuclear laboratory data
Plamen Krastev, Bao-An Li and Aaron Worley, (2008) in preparation.
The proton fraction x at ß-equilibrium in proto-neutron stars is determined by
x
0.048[ Esym (  ) / Esym ( 0 )]3 (  / 0 )(1  2 x )3
The critical proton fraction for direct URCA process to happen is Xp=0.14 for npeμ
matter obtained from energy-momentum conservation on the proton Fermi surface
Slow cooling: modified URCA:
n  (n, p)  p  (n, p)  e   e
p  (n, p)  n  (n, p)  e   e
Consequence: long surface
thermal emission up to a few
million years
Faster cooling by 4 to 5 orders of
magnitude: direct URCA
n  p  e  e
p  n  e  e
PSR J0205+6449 in 3C58
was suggested as a candidate
Bao-An Li, Phys. Rev. Lett. 88, 192701 (2002).
The neutron star radius is primarily determined by
J.M. Lattime and M. Prakash, Phys. Rep. 333, 121 (2000).
dEsym (  )
d
near 0 20
The mass M and radius R are determined by solving the TOV eq.:
dP
G (e( r )  P( r ) / c 2 )(m( r )  4 r 3P ( r ) / c 2 )

,
dr
r 2 (1  2Gm( r ) / rc 2 )
r
m(r)=  4 r'2e( r ')dr ', e( r ) is the energy density at r,
0
the radius R is determined by P(R)=0.
The pressure P( ) for neutron stars (n,p,e) at  -equilibrium is
P( , )=Pnuclear  Pelectron   02 (
E
1
)   e e

4
dEnuclearmatter dEsym 2 1

 ) +  (1- ) Esym (  )
d
d
2
2 dE sym 2
 P0  Pasy   0

d
with  determined by chemical equilibrium: e  n   p  4 Esym (  ) and the charge neutrality:  e   p
  02 (
Constraining the radii of non-rotating neutron stars
Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006)
●
APR: K0=269 MeV.
The same incompressibility for symmetric nuclear
matter of K0=211 MeV for x=0, -1, and -2
.
Mass-radius correlation of non-rotating neutron stars and their EOS
Different EOS predicted
by various theories
Essentially, none of
them was ever tested
against reaction data
J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542.
Astronomers discover the fastest-spinning neutron-star
The latest report on the fastest spinning neutron star is XTE J1739-285 spinning at
1122 Hz, P. Kaaret et al., The Astrophysical Journal, V657, Issue 2, L97 (2007)
Science 311, 1901 (2006).
Rapidly rotating neutron stars
Plamen Krastev, Bao-An Li and Aaron Worley, The Astrophysical Journal, 676, 1170 (2008)
Assuming the observed frequency is the Kepler frequency
1
fk 
2
 GM
 3
 R
 eq
1/ 2




Solving the Einstein equation in general relativity for
stationary axi-symmetric spacetime using the RNS code
written by Nikolaos Stergioulas and John L. Friedman,
Astrophysics J. 444, 306 (1995)
Testing the constancy of the “constant” G
P. Dirac, Nature 139, 323 (1937)
Suggested that the gravitational force might be weakening with the
continuous expansion of the Universe
Possibly many detectable astronomical consequences were suggested by
Chandrasekhar, Nature 139, 757 (1937); Kothar, Nature 142, 354 (1938)
Contrary to most of other physical constants, as the precision of measurements
increased, the disparity between measurements of G also increased. This promoted
the CODATA in 1998 to raise the uncertainty of G by about a factor of 12 from
0.013% to 0.15%
The CODATA is the Committee on Data for Science and Technology, http://www.codata.org/
The latest review: J. P. Uzan, Rev. Mod. Phys. 75: 403, 2003
Various Upper Bounds on
|G/G |
Terrestrial nuclear lab experiments + observations of old neutron stars:
5.4 10
12

 G/ G  22 1012 / yr
Plamen Krastev and Bao-An Li, PRC 76, 055804 (2007).
Only Modified URCA
allowed
Gravitochemical Heating Method – an outline
A change in G induces a variation in the internal composition of a neutron star,
causing dissipation and internal heating.
At the stationary temperature:
heating (relying on the changing rate of G)=cooling (relying on the symmetry energy)
P. Jofre, A. Reisenegger, and R. Fernadez, Phys. Rev. Lett. 97, 131102 (2006)

 ,eq
L
To obtain
G
G
~ G
D
G
8/7
L ,eq  4R2 (Ts )
Depending on the Esym(ρ)

s
T
from the Gravitochemical heating one needs to know:
(1) The surface temperature of a neutron star
(2) That the star is certainly older than the time-scale necessary to reach a
quasi-stationary state
(3) The density dependence of the nuclear symmetry energy
PSR J0437- 4715 – the closest millisecond pulsar is a good candidate
Surface temperature: By ultraviolet observations O. Kargaltsev et al., AJ 602, 327-335 (2004)
Mass: 1.58  0.18 M SOL
W. Van Straten et al., Nature 412, 158 (2001)
Stationary photon luminosity and surface temperature
Including hyperon-QGP
phase transition
Modified URCA only
Diract + Modified URCA
Surface temperature vs radius for PSRJ0437-4715

G/ G  22  1012 / yr

G/ G  5.4  1012 / yr
Summary and outlook
Significant progress has been made in determining the
density dependence of symmetry energy at sub-saturation
densities using heavy-ion reactions
• The partially constrained density dependence of symmetry
energy has already allowed us to put some constraints on
several neutron stras properties and the changing rate of G
• More challenges:
(1) Constraining the symmetry energy at supra-normal
densities with high energy radioactive beams
(2) probing the momentum and density dependence of the
isovector interaction
All 23 most popular RMF models give the WRONG momentum dependence of the Lane potential
Neutron-proton differential transverse flow:
x
n p
F
Bao-An Li, PRL 85, 4221 (2000).
1 iN ( y ) x
x
( y) 
(
p
(
neutron
)

p

i
i ( proton ))
N ( y ) i 1
Squeeze-out of neutrons perpendicular
to the reaction plane
Neutron-proton differential flow
in the reaction plane
Azimuthal angle
Ratio of charged pions
Another input to transport models:
nucleon-nucleon cross sections
Isospin-dependence of nucleon-nucleon
cross sections in symmetric matter
NN cross section in free-space
Experimental data
G.Q. Li and R. Machleidt,
Phys. Rev. C48, 11702 and C49, 566 (1994).
With other models:
1. Q. Li et al., PRC 62, 014606 (2000)
2. G. Giansiracusa et al., PRC 53, R1478 (1996)
3. H.-J. Schulze et al., PRC 55, 3006 (1997)
4. M. Kohno et al., PRC 57, 3495 (1998)
Isospin-dependence of nucleon-nucleon cross sections
in neutron-rich matter
 medium /  free in neutron-rich matter
The effective mass scaling model:
 medium /  free
*
NN
 



 NN 
*
NN
2
is the reduced effective mass of the
colliding nucleon pair NN
valid for   20
according to Dirac-Brueckner-Hatree-Fock calculations
F. Sammarruca and P. Krastev, nucl-th/0506081;
Phys. Rev. C73, 014001 (2005) .
Applications in symmetric nuclear matter:
J.W. Negele and K. Yazaki, PRL 47, 71 (1981)
V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992)
M. Kohno et al., PRC 57, 3495 (1998)
D. Persram and C. Gale, PRC65, 064611 (2002).
Application in neutron-rich matter:
nn and pp xsections are splitted due to
the neutron-proton effective mass slitting
Bao-An Li and Lie-Wen Chen, nucl-th/0508024,
Phys. Rev. C72, 064611 (2005).
at zero temperature
ρ
Near-threshold pion production with radioactive beams at RIA and GSI
stiff
soft
density
  yields are more sensitive to the symmetry energy Esym(ρ) since
they are mostly produced in the neutron-rich region
However, pion yields are also sensitive to the symmetric part of the EOS
E(  ,  )  E(  , 0)  Esym (  ) 2   ( 4 )
Isospin-dependence of nucleon-nucleon cross sections
in neutron-rich matter
 medium /  free in neutron-rich matter
The effective mass scaling model:
 medium /  free
*
NN
 



 NN 
*
NN
2
is the reduced effective mass of the
colliding nucleon pair NN
valid for   20 and relative momenta  240 MeV/c
according to Dirac-Brueckner-Hatree-Fock calculations
F. Sammarruca and P. Krastev, nucl-th/0506081
Phys. Rev. C (2005).
Applications in symmetric nuclear matter:
J.W. Negele and K. Yazaki, PRL 47, 71 (1981)
V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992)
M. Kohno et al., PRC 57, 3495 (1998)
D. Persram and C. Gale, PRC65, 064611 (2002).
Application in neutron-rich matter:
nn and pp xsections are splitted due to
the neutron-proton effective mass slitting
Bao-An Li and Lie-Wen Chen, nucl-th/0508024,
Phys. Rev. C (2005) in press.
at zero temperature
An input to transport models: nucleon-nucleon scattering cross sections
NN cross section in free-space
NN cross sections in nuclear medium
Experimental data
G.Q. Li and R. Machleidt,
Phys. Rev. C48, 11702 and C49, 566 (1994).
Formation of dense, asymmetric matter with high energy radioactive beams
at CSR/China, RIKEN/Japan, FAIR/Germany, RIA/USA, etc
Central density
Symmetry energy
n/p ratio of matter
with density ρ>ρ0
density
Stiff Esym
E( ,  )  ENuclear Matter ( )  Esym ( ) 2
B.A. Li, G.C. Yong and W. Zuo, PRC 71, 014608 (2005)
The n/p ratio of squeezed-out nucleons perpendicular to the reaction plane as a probe
of the high density symmetry energy, Yong, Li and Chen, Phys. Lett. B650, 344 (2007).
n
n/p
?
Tar.
Squeeze-out of participants
ø
Pro.
p
Azimuthal angle
around 900
Pion ratio probe of symmetry energy
GC
Coefficients
nn
a) Δ(1232) resonance model
i
n
f
i
r
s
t
c
h
a
n
c
e
N
N
s
c
a
t
t
e
r
p
i
n
g
s
:
(negelect rescattering and reabsorption)
 
5 N 2  NZ

 (
2

5Z
 NZ

n
p
(
N
Z
p
p
n
)


0
1
5
1
0
1
4
1

0


5
)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[( 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.
Time evolution of π-/π+ ratio in central reactions

(  )like 


1
2
       0  N *0


3
3

 
t 
1  2 *



     N
3
3
Radii of neutron stars: 10 – 20 Km ???
Challenges in Measuring the Radii of Neutron Stars
• Determine luminosity, temperature and deduce surface area.
Assume it is a black body L=4 R2T4, then the radius R can be inferred.
– T from X-ray spectrum, such as those from Chandra and X-MM Newton satellites
– Need distance to star in parallax and measured flux to get L.
– Deduce R from surface area (~30% corrections from curvature of space-time).
The radiation radius R∞ for an observer at infinity is related to the matter radius:
R  R (1  z )  R / 1  2GM / Rc 2
z is the gravitational red-shift that can be obtained from the obsorption lines
• Complications
– Spectrum peaks in UV and this is heavily absorbed by interstellar H.
– Not a black body: often black body fit to X-ray does not fit visible spectra.
– Model neutron star atmospheres (composition uncertain) to correct black body.
• Current status: available estimates give a wide range
“Although accurate masses of several neutron stars are available, a precise
measurement of the radius does not exist yet”……
Lattimer and Prakash, Science Vol. 304 (2004) 536
Incompressibility of npe-matter in neutron stars
at beta equilibrium:
Nuclear contribution Kµnucl
Electron contribution Kµβ
X=0
x
-2
-1
0
1