Probing the Incompressibility of Neutron-Rich Matter from Heavy-Ion Reactions
Bao-An Li
Arkansas State University
Collaborators:Lie-Wen Chen, Shanghai Jiao Tong University
Che Ming Ko, Texas A&M University
Andrew W. Steiner, Los Alamos National Laboratory
1. Symmetry Energy and Incompressibility of Neutron-Rich Matter
•
Current status and major issues
•
Importance in astrophysics and nuclear physics
2. A Transport Model for Nuclear Reactions Induced by Neutron-Rich Nuclei
•
Some details of the IBUU04 model
•
Momentum dependence of the symmetry potential and neutron-proton effective mass splitting
in neutron-rich matter
3. Experimental Probes
Examples: isospin transport, n/p, π-/π+ ratio, and neutron-proton differential flow
4.
Summary
Equation of State of neutron-rich matter at T=0, density ρ and isospin
asymmetry   (  n   p ) /(  n   p ) is:
E (  ,  )  E (  ,0)  Esym (  ) 2   ( 4 ), K (  ,  )  K (  ,0)  Kasy (  ) 2
The key messages:
A great deal of information about the EOS of symmetric matter E(ρ,0) has been
obtained from studying heavy-ion reactions for almost 30 years.
The field has come to the point that further progress on determining the E(ρ,0)
and K∞ requires a better determination of the Esym(ρ) and Kasy.
The symmetry energy itself is very important for many interesting questions
in both nuclear physics and astrophysics
Momentum dependence of the symmetry potential and the corresponding
neutron-proton effective mass splitting in neutron-rich matter are critically
important for extracting novel properties of neutron-rich matter.
Nuclear reactions induced by neutron-rich nuclei provide a unique opportunity
to constrain the symmetry energy Esym(ρ) and the n-p effective mass splitting in
neutron-rich matter in a broad density range, -550 < Kasy < -450 MeV from MSU
isospin diffusion data
Esym (ρ) predicted by microscopic many-body theories
Symmetry energy (MeV)
Esym (  )  E pureneutron (  )  Enuclear (  )
BHF
Greens function
Variational
Density
A.E. L. Dieperink et al., Phys. Rev. C68 (2003) 064307
The major remaining uncertainty in determining the EOS of
symmetric matter is the poor knowledge about Esym(ρ)
Example 1: On extracting the incompressibility K∞ of symmetric nuclear matter at ρ0 from
giant monopole resonance, J. Piekarewicz, PRC 69, 041301 (2004);
G. Colo, N. Van Giai, J. Meyer, K. Bennaceur and P. Bonche, PRC 70, 024307 (2004).
K ( 0 )  9 02 (d 2 E / d  2 )0 , KA = m<r2>0E2ISGMR = K (1+cA-1/3) + Kasy δ2 + KCoul Z2 A-4/3
Kasy depends on the density dependence of the symmetry energy !
Kasy ( 0 )  9 02 (d 2 Esym / d  2 )0  180 (dEsym / d  ) 0
The latest conclusion:
data range K =230-240 MeV for
26 < Esym(ρ0) < 40 MeV
-566 ± 1350 < Kasy < 139 ±1617 MeV
Shlomo and Youngblood, PRC 47, 529 (1993)
Symmetry energy at ρ0
(Kasy ??)
The multifaceted influence of symmetry
energy in astrophysics and nuclear physics
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. (2005).
n/p π-/π+

Expanding fireball and
gamma-ray burst (GRB) from
the superdene neutron star
(magnetar) SGR 1806-20
on 12/27/2004. RAO/AUI/NSF
GRB and nucleosynthesis
in the expanding fireball
after an explosion of a
supermassive object
depends on the n/p ratio
In pre-supernova
explosion of massive

stars e  p  n   e
is easier with smaller
symmetry energy
t/3He K+/K0
Isospin
physics
isodiffusion
isotransport
isocorrelation
isofractionation
isoscaling
Promising Probes of the Esym(ρ) in Nuclear Reactions
(an incomplete list !)
At low 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
 n/p ratio of fast, pre-equilibrium nucleons
 Isospin fractionation and isoscaling in nuclear multifragmentation
 Isospin diffusion
 Neutron-proton differential flow
 Neutron-proton correlation functions at low relative momenta
 t/3He ratio
Towards high densities reachable at CSR/Lanzhou, GSI, RIA and RIKEN
 π – yields and π -/π + ratio
 Neutron-proton differential transverse flow
 n/p ratio at mid-rapidity
Detecting neutrons simultaneously with charged particles is critical !
Most probes in heavy-ion collisions are based on transport model studies
Hadronic transport equations for the reaction dynamics:
f b
p
b
b
Ub is the mean-field potential for

r f b   rU b  p f b  I bb
 I bm
t
Eb
baryons
f m
k
m
m
Mesons:

 r f m  I mm
 I bm
The phase space distribution
t
Em
Baryons:
functions, mean fields and
collisions integrals are all
isospin dependent
An example:
   N
f (r, p, t )
  N 
(gain)
(loss)
Simulate solutions of the coupled transport
equations using test-particles and Monte Carlo:
f ( r , p, t ) 
1
Nt
  (r  r ) ( p  p )
i
i
The evolution of f (r , p, t ) is followed
on a 6D lattice
i
Isospin splitting of nucleon mean field within the BHF approach
W. Zuo, L.G. Gao, B.A. Li, U. Lombardo and C.W. Shen, Phys. Rev. C (2005) in press.
The momentum dependence of the nucleon potential is a result of the non-locality
of nuclear effective interactions and the Pauli exclusion principle
m U 1
 [1 
] p
m
p p F
m*
Neutron-proton effective k-mass splitting
BHF vs. EBHF (with 3-bodt force)
U  e / 
e is the energy density
M*n > M*p
with
Without
3-body force
W. Zuo, L.G. Gao, B.A. Li, U. Lombardo and C.W. Shen, Phys. Rev. C (2005) in press.
Some Skyrme interactions gave mn* < mp* !
ρ
Symmetry energy and single nucleon potential
used in the IBUU04 transport code for reactions with radioactive beams
soft
HF using a modified Gogny force:
density
 '

 
B   1
2
U (  ,  , p, , x)  Au ( x)
 Al ( x)
 B( ) (1  x )  8 x
 '
0
0
0
  1 0

2C ,
0
2C , '
f (r , p ')
f ' (r , p ')
3
d
p
'

d
p
'
 1  ( p  p ')2 /  2 0  1  ( p  p ')2 /  2
3
1
2
 , '   , Al ( x)  121 
2 Bx
2 Bx
, Au ( x)  96 
, K  211MeV
 1
 1 0
B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).
Momentum and density dependence of the symmetry potential
U n / p  U isoscalar  U Lane 
δ
δ
Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions
provides only a constraint at ρ0:
U Lane  (U n  U p ) / 2  V1   R  Ekin ,
V1
28  6 MeV ,  R  0.1  0.2
for Ekin < 100 MeV
P.E. Hodgson, The Nucleon Optical Model,
World Scientific, 1994
G.W. Hoffmann et al., PRL, 29, 227 (1972).
G.R. Satchler, Isospin Dependence of Optical
Model Potentials, 1968
Although the uncertainities are large, NOT a
single experiment was analyzed assuming
the symmetry potential would INCREASE
with energy because then the errors will be
even larger
Neutron-proton effective k-mass splitting in neutron-rich matter
m*
m
 [1 
m U 1
] p
p p F
With the modified Gogny effective interaction
B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614 (2004); NPA 735, 563 (2004).
Probing the isospin-dependence of in-medium NN cross sections
How does the  np /  pp ratio change in neutron-rich medium?
(1) All published results are for symmetric matter
(2) Conflicting conclusions
(3) Need experimental constraints
NN cross section in free-space
G.Q. Li and R. Machleidt,
Phys. Rev. C48, 11702 and C49, 566 (1994).
Opposing conclusions 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)
Nucleon-nucleon cross sections and nuclear stopping power in neutron-rich matter
 medium /  free in neutron-rich matter
 
 medium /  free  


 NN 
*
NN
*
 NN
2
is the reduced mass of the
colliding pair NN in medium
Effects on the nuclear stopping power and
nucleon mean free-path in n-rich 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).
1. In-medium xsections are reduced
2. nn and pp xsections are splitted
due to the neutron-proton effective
mass slitting in neutron-rich matter
Isospin transport (diffusion) in heavy-ion collisions as a probe of Esym (ρ)
at subnormal densities
Particle Flux:
Isospin Flow:
The isospin diffusion coefficient DI depends
on both the symmetry energy and the
neutron-proton scattering cross section
Schematicaly for simplified situations:
DI 
1
 np
time
(
np


int
4 Esym
m
)
Isospin asymmetric force
L. Shi and P. Danielewicz, Phys. Rev. C68, 017601 (2003).
Extract the Esym(ρ) at subnormal densities from isospin transport
A quantitative measure of the isospin non-equilibrium and stopping using any
isospin tracer X, F. Rami (FOPI/GSI), PRL, 84, 1120 (2000).
RXA B 
2X
A B
A A
X
X
X A A  X B  B
B B
RXA A  1
RXA B  0 if complete isospin mixing
RXB  B  1
ρρ
MSU experiments: 124Sn+112Sn at
Ebeam/A=50 MeV Use X=7Li/7Be or δ
of the projectile residue.
M.B. Tsang et al.
PRL 92, 062701 (2004)
Comparing momentum-dependent IBUU04 calculations
using free NN xsections with data on isospin transport from NSCL/MSU
E/A=50 MeV and b=6 fm
MDI interaction
SBKD interaction
0.8
Experiments favors:
Esym()=32 (/ρ0 )1.1 for ρ<1.2ρ0
Kasy(ρ0)~-550 MeV
x=-2
(=1.60)
1-Ri
Strength of isospin transport
1.0
x=0
(=0.69)
L.W. Chen, C.M. Ko and B.A. Li,
PRL 94, 32701 (2005).
x=1
x=-1
(=1.05)
0.6
MSU data
0.4
-700
-600
124
Sn +
Next step:
1. Reduce the error bars of the data
and the calculations
2. Compare with results using other observables
3. Exam effects of in-medium cross sections
112
Sn
-500
-400
-300
Kasy (MeV)
Isobaric incompressibility of asymmetric nuclear matter
K (  ,  )  K0 (  )  Kasy (  ) 2
Kasy ( 0 )  9 02 (d 2 Esym / d  2 ) 0 180 (dEsym / d  ) 0
Correlation between isospin diffusion and n-skin in 208Pb
using the same EOS------- a more stringent constraint on
the symmetry energy
Softer symmetry energy
than E
 31.6(  / 
sym
1.1
)
0
IBUU04
corresponding to x=-1 is needed.
The way out: using reduced and isospin
dependent in-medium NN cross sections
would require a softer symmetry energy
to obtain the same value of isospin
diffusion at subnormal densities
MSU data
N-skin in 208Pb within HF
Andrew W. Steiner and Bao-An Li, nucl-th/0505051
Effects of the in-medium nucleon-nucleon cross sections
б
б
-550 < Kasy < -450 MeV
close to that extracted
from Osaka giant
resonsnces data by
Fujiwara et al.
in-medium xsection
0.7 <  < 1.1 in fitting
Esym=32(ρ/ρ0)
Strength of the symmetry potential with and
without momentum dependence
ρρ
The ultimate goal and major questions of studying heavy-ion
reactions induced by neutron-rich (stable and/or radioactive) nuclei
To understand the isospin dependence of the nuclear Equation of State, extract the
isospin dependence of thermal, mechanical and transport properties of asymmetric
nuclear matter playing important roles in nuclei, neutron stars and supernove.
Among the currently most interesting topics in isospin physics
1.
EOS of neutron-rich matter, especially the density dependence of symmetry energy
Esym(ρ) at abnormal densities.
2.
Momentum-dependence of the symmetry potential and the neutron-proton effective
mass splitting mn*-mp* in neutron-rich matter
3.
Isospin-dependence of in-medium nucleon-nucleon cross sections and the nuclear
stopping power in neutron-rich matter
4.
Explore the phase diagram (T-ρ-δ) of neutron-rich matter along the isospin asymmetry
δ axis (e.g, neutron distillation, n-Λ phase transition)
5.
Isospin mixing of vector mesons (ρ0-ω) and charge symmetry breaking in neutron-rich
matter
The most important question relevant to all of the above: What is the isospin dependence
of the in-medium nuclear effective interactions
Evolution of isospin diffusion
б
Predictions for reactions with high energy neutron-rich beams
at CSR/Lanzhou up to 500 MeV/A for 238U
at RIA/USA up to 400 MeV/A for 132Sn
at FAIR/GSI up to 2 GeV/A for 132Sn
Besides many other interesting physics, it allows the determination of nuclear
equation of state for neutron-rich matter at high densities where it is most uncertain
and most important for several key questions in astrophysics.
Examples:
• Isospin distillation/fractionation
• π - yields and π -/π + ratio
• Neutron-proton differential transverse flow
Formation of dense, asymmetric nuclear matter at CSR, RIA and GSI
Stiff Esym
Stiff Esym
n/p ratio of the
high density region
B.A. Li, G.C. Yong and W. Zuo, PRC 71, 014608 (2005)
Isospin fractionation (distillation): at isospin equilibrium Esym ( 1 ) 1  Esym ( 2 )  2
EOS requirement: E(  ,  )  E(  ,0)  Esym (  ) 2
low (high) density region is more neutron-rich with stiff (soft) symmetry energy
Isospin asymmetry of free nucleons
Symmetry enengy
ρ0
density
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 formed
preferentially with the soft symmetry energy
However, pion yields are also sensitive to the symmetric part of the EOS
E (  ,  )  E (  , 0)  Esym (  ) 2   ( 4 )
Pion ratio probe of symmetry energy
a) Δ(1232) resonance model
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

5
Z

NZ


0
1
5
1
0
1
4
1

nn
i

n
p
(
N
Z
p
p
n
)
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):
see, e.g., Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701.
Time evolution of π-/π+ ratio in central reactions at RIA and GSI
From the overlapping
n-skins of the colliding nuclei

(  )like 


1
2
       0  N *0


3
3

 
t 
1  2 *



     N
3
3


Differential  /  ratios
Transverse flow as a probe
of the nuclear EOS:
px
px
y
y
Neutron-proton differential flow as a
probe of the symmetry energy:
x
n p
F
1 iN ( y) x
( y) 
pi  i

N ( y ) i 1
 i  1
for n and p
symmetry potential is generally
repulsive for neutrons and
attractive for protons
Bao-An Li, PRL 85, 4221 (2000).
G.C. Yong, B.A. Li, W. Zuo, High energy physics
and nuclear physics (2005).
Summary
• The EOS of n-rich matter, especially the Esym(ρ) is very
important for many interesting questions in both
astrophysics and nuclear physics
• Transport models are invaluable tools for studying the
EOS of neutron-rich matter
• Isospin transport experiments at intermediate energies
allowed us to extract Esym()=32 (/ρ0 )1.1 for ρ<1.2ρ0,
and Kasy(ρ0)~-550 MeV
• High energy radioactive beams at RIA and FAIR/GSI will
allow us to study the EOS of n-rich matter up to 3ρ0
using several sensitive probes of the symmetry energy
Esym().
Extract the Esym(ρ) from isospin transport
A quantitative measure of the isospin non-equilibrium and stopping power
In A+B using any isospin tracer X, F. Rami (FOPI), PRL, 84, 1120 (2000).
RXA B 
2X
A B
A A
X
X
X A A  X B  B
RXA A  1
RXA B  0
RXB  B  1
If complete isospin mixing/ equilibrium
1.5
MSU experiments: R=0.42-0.52 in
124Sn+112Sn at E
beam/A=50 MeV
mid-central collisions.
With 112Sn+112Sn and 124Sn+124Sn
as references. Use X=7Li/7Be or δ
of the projectile residue, etc.
124
Sn +
1.0
Ri
M.B. Tsang et al.
PRL 92, 062701 (2004)
B B
x=-1
MDI
SBKD
/0 (MDI)
112
Sn
/0
/0 (SBKD)
0.5
MSU data
0.0
SBKD: momentum-independent
Soft Bertsch-Kruse-Das Gupta EOS
MDI: Momentum-Dependent Interaction
0
50
t (fm/c)
Momentum-independent
100
150
Momentum-dependent
All having the same Esym (ρ)=32 (ρ/ρ0)1.1
Coulomb effects on π-/π+ ratio is well known
The n/p ratio of pre-equilibrium nucleons
is one of the most sensitive observables to the Esym(ρ) as
expected from the statistical consideration and predicted by
several dynamical models
Experimental situation: (several examples)
• Unusually high n/p ratio of pre-equilibrium nucleons beyond the Coulomb
effect was observed by Dieter Hilscher et al. using the Berlin neutron-ball
in both heavy-ion and pion induced reactions around 1987
• Similar phenomenon was observed in heavy-ion experiments at MSU
using the Rochester neutron-ball by Udo Schröder et al. around 1997
• More recent experiments dedicated to the study of n/p ratio of preequilibrium nucleons are being analyzed by the MSU-UW collaboration
Generally, a high n/p ratio of pre-equilibrium nucleons emitted from
subnormal densities requires a soft symmetry energy. However, no model
comparison has been made, thus no indication on the Esym(ρ) has been
obtained yet.
The n/p ratio of pre-equilibrium nucleons as a probe of Esym(ρ)
Symmetry energy
Symmetry potential
Statistically one expects: n / p  exp(  n   p ) / KT
n
p
n   p  (Vasy
 Vasy
)  VCoul  KT {ln
Dynamical simulations
B.A. Li, C.M. Ko and. Z. Ren,
PRL 78 (1997) 1644
B.A. Li, PRL 85 (2000) 4221
(N/Z)free/(N/Z)bound
at 100 fm/c
stiff
m
m
n
m 1
1
3

bm (  T ) m (    )}
n
p
p
m
2
m
t/3He ratio as a probe of the symmetry energy
Correlation between n/p and t/3He ratios in advanced coalescence model
• It is expected to be linear ONLY within the simplified momentum-space coalescence model
• In more advanced coalescence models where the overlap between the product of the neutron and
proton phase space distribution functions at freeze-out and the Wigner functions of light clusters
are used, this correlation is NOT NECESSARILY linear
• Can we use this correlation itself as a probe of the symmetry energy?
X=1
X=-2
Jorge Piekarewicz et al., RMF with FSU-Gold parameter set (NL3 with
two additional couplings). It reproduce the GMR in 90Zr and 208Pb,
and the isovector giant dipole resonance of 208Pb. The symmetry energy
is significantly softer than what is extracted from the isospin diffusion.
Isospin mixing of vector mesons (ρ0-ω) and charge
symmetry breaking (CSB) in neutron-rich matter
proton
ρ0
neutron
ω
+
ρ0
Anti-proton
ω
Anti-neutron
Vertex (ρpp)= - Vertex (ρnn), Vertex (ωpp)= Vertex (ωnn)
the two contributions almost cancel out in symmetric matter, the small remains due
to Mn > Mp is enough to explain the CSB in vacuum and the Nolen-Schiffer effect
in mirror nuclei.
In neutron-rich matter due to both N > Z and the neutron-proton effective mass
splitting Mn* - Mp*, there is a large (ρ0-ω) mixing. The consequences of the mixing
are: (1) Separation of ρ0 and ω increases from about 12 MeV in vacuum to about
150 MeV in dense neutron-rich matter; (2) additional reduction of the
ρ mass besides that due to the chiral symmetry restoration which happens also in
symmetric matter
Probing the Neutron-Proton Effective Mass Splitting
in Neutron-Rich Matter at CSR?
Bao-An Li
Arkansas State University, USA
and
Institute of Modern Physics, Chinese Academy of Science
Collaborators:Lie-Wen Chen, Shanghai Jiao Tong University
Che Ming Ko, Texas A&M University
Pawel Danielewicz and Bill Lynch, Michigan State University
Gaochan Yong and Wei Zuo, IMP, Lanzhou, Chinese Academy of Science
Andrew W. Steiner, Los Alamos National Laboratory
Champak B. Das, Charles Gale and Subal Das Gupta, McGill University
1. The ultimate goal and major issues of studying heavy-ion reactions induced by
neutron-rich (stable and/or radioactive) nuclei
2. Determining the momentum dependence of the symmetry potential and neutron-proton
effective mass splitting in neutron-rich matter
3. Determining the symmetry energy of neutron-rich matter at abnormal densities
Isospin- asymmetric nuclear matter
with Three Body Force
with
3-body force
without
Zuo Wei and U. Lombardo
(2) Present constraints on the EOS of symmetric and pure neutron matter for
2ρ0< ρ < 5ρ0 using flow data from BEVALAC, SIS/GSI and AGS
P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002)
neutron matter
P (MeV/fm
3)
100
10
1
•
Use constrained mean fields to predict
the EOS for symmetric matter
– Width of pressure domain reflects
uncertainties in comparison and of
assumed momentum dependence.
Akmal
av14uvII
NL3
DD
Fermi Gas
Exp.+Asy_soft
Exp.+Asy_stiff
1
•
1.5
2
2.5
3
/
3.5
4
4.5
0
Corresponding EOS for neutron
matter E
pureneutron (  )  E symmetric (  )  E sym (  )
- Two regions correspond to two
different Esym(ρ), the width is
completely due to the
uncertainty in Esym(ρ).
5
Simulation as the Third Branch of Science
(experiment/observation, theory, simulation)
1. Applications of transport models
Transport models have been widely used in astrophyics, plasma physics,
semiconductor and nanostructures, particle and nuclear physics.
2. Development of transport models
Transport codes often implement extra physical assumptions and dynamical
mechanisms which go beyond the equations used to motivate their designs. These
algorithms often undergo evolutions with time as we make progresses in our R&D
efforts and also as our needs for including new processes arise. They may involve
many phenomenological parameters which are not all well experimentally constrained
yet because of the lack of the relevant experimental data, and some of them
are exactly what we want to learn.
3. Challenges of transport models for reactions involving radioactive beams
• Develop practically implementable quantum transport theories ( the de Broglie
wavelenght may be comparable to the nucleon mean free path in energetic central
reactions, and the uncertainty principle imposes a strong correlation between
delocalization of nucleons and their momentum distribution)
• Include more structure information in the initial state especially for peripheral reactions.
• Use consistently all inputs (initial state, mean field and бNN) from the same interactions
Initialization procedures:
(1) r-space: distribute n and p according to
predictions by structure models**, e.g., RMF
(2) p-space: using local Thomas-Fermi
(3) check stability
** Trade-off: drawbacks/advantages
• The initial state may not be the ground
state corresponding to the effective
interactions used in the subsequent
reactions. This is, however, not a serious
problem for high energy central reactions.
RMF (TM1)
• None of the existing transport models can
generate the initial nucleon density profiles
BETTER than dedicated structure models
for radioactive nuclei, and there is no data
to constrain the neutron density profiles.
• If one insists on using the same effective
interactions in generating the initial state
and carrying out the subsequent reactions,
it is then hard to tell whether differences in
final observables obtained from using
different interactions, e.g., symmetry
energies, are from the different initial states
or from the reaction dynamics.
A compromise has to be made
Sensitivity of the Various Structure Measurements
..
Iso Tanihata
Radii
Density
Skin
A=58
0 .1 5
rrms 0.10
A=68 A=72
(c)
RIA
GSI
2%
0 .0 5
Sk in Th ick n ess
0
Radii of Ni isotopes
0.4
R rms n - R rms p
Radii rrms [fm]
5.6
RMF
5.4
Accessible at GSI
5.2
5.0
SHF
( SIII)
Accessible at RIA
4.6
0.2
SIII
0.0
-0.2
4.8
20
A=58
20
30
30
40
50
N
A=68 A=72
40
50％
RMF
50
N
60
60
Equation of State of Neutron-Rich Matter:
E (  ,  )  E (  , 0)  Esym (  ) 2   ( 4 )
Isospin asymmetry
  (  n   p ) /(  n   p )
RMF
(TM1)
SHF (SIII)
(a)
0
10
neutron matter
0
neutron matter
saturation lines
N/Z
Binding energy per nucleon [MeV]
Two typical equation-of-states
0
0.2
0.2
0.8
0.8
0.4
-10
0.4
0.6
0.6
stable nuclei
-20
0
0.05
0.1
1.0
0.15
nucleon density [fm -3]
1.0
stable nuclei
0
0.05
0.1
0.15
nucleon density [fm -3]
0.2
K. Oyamatsu, I. Tanihata, Y. Sugahara, K. Sumiyoshi and H. Toki, NPA 634 (1998) 3.
Esym(ρ) from Hartree-Fock approach using different effective interactions
B. Cochet, K. Bennaceur, P. Bonche, T. Duguet and J. Meyer, Nucl. Phys. A731, 34 (2004).
J. Stone, J.C. Miller, R. Koncewicz, P.D. Stevenson and M.R. Strayer, PRC 68, 034324 (2003).
Bao-An Li, PRL 88, 192701 (2002) (where paramaterizations of Easym and Ebsym are given)
New Skyrme
interactions
Amplication around
normal density
Ebsym
J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542.
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 Xc=1/9
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
Critical points of the
direct URCA process
APR
Akmal et al.
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
A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).
Probing the momentum-dependence of the symmetry
potential and n-p effective mass splitting
X=0
X=1
Bao-An Li et al., NPA735, 563 (2004)
J.Rizzo, M. Di Toro et al., (2005)
Sn124+Sn124 @50 AMeV, b=2 fm
m*n>m*p
m*n<m*p
Asy-soft
t=60 fm/c
t=80 fm/c
t=100 fm/c
Asy-stiff
Asy-soft
Asy-stiff