Transcript Isospin Dependence of Fragmentation
Constraining the properties of dense matter William Lynch, Michigan State University
2. Example:T=0 with Skyrme 3. Present status
a) symmetric matter b) asymmetric matter and symmetry term.
4. Astrophysical relevance
B. Summary of first lecture
C. What observables are sensitive to the EOS and at what densities?
1. Binding energies
2. Radii of neutron and proton matter in nuclei
4. Particle flow and particle production: symmetric EOS
5. Particle flow and particle production: symmetry energy
D Summary
The EOS: Finite temperature:
The EOS describes how the pressure P depends on the temperature T, the density and the asymmetry =( n p )/ : P P ( , T , ) Here, designates the number density of nucleons ( nucleons/ fm 3 ) for nuclear matter at saturation density.
One can use =E/A, the average energy per particle in the system, to calculate P:
P
P (
, T , where f (
,
) T ,
F
V
T ,
)
F/A
2
f (
, T ,
)
T ,
(
, T ,
)
T
(
, T ,
).
(1) Here f is the helmholtz free energy per nucleon and is the entropy per nucleon. can be obtained from: 0 T V ,
T where c
V
T
V , 0 T
c
v
dT , T is the heat capacity per nucleon .
(2)
Homework 1: Obtain the approximate expression ) 2 T/(2 F ( )) for the fermionic nuclear system at density , assuming all temperature dependence in resides in the nucleon kinetic energies. Assume for simplicity that n = p . Zero temperature If one is at low enough temperature, one can ignore the dependence of the EOS on temperature and evaluate the EOS at T=0. For T=0,
P
P (
, 0 ,
)
2
f (
, 0 ,
)
T , 2
(
, 0 ,
)
, (3)
Phase transitions: Phase transitions can manifest themselves in the EOS if there are regions where dP/d | <0, making the matter mechanically unstable. Where this occurs, one must match the chemical potentials for the denser and more dilute phases by making a Maxwell construction, in which the area, A Vdp between the Maxwell construction line and the original 200 150 100
Demonstration of Maxwell construction
Bogota2 bog2_pt EOS is equal on the left and the right side. 50 The straight line shows an EOS (in red) 0 and its Maxwell construction in blue. (In most places, the red and blue coincide and -50 1 1.5
2 only the blue is visible. The red curve can only be seen in the 2.5
/ 0 unstable region.) Note, the system may follow the dashed curve for a while even in the mixed phase region if the expansion or compression is fast enough.
3 3.5
4
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Theoretical Approaches
• • • Variational and Bruekner model calculations with realistic two-body nucleon nucleon interactions: (see Akmal et al., PRC 58, 1804 (1998) and refs therein.) – Variational minimizes
– Incorporate three-body interactions.
• Some are "fundamental" • Others model relativistic effects.
Relativistic mean field calculations using relativistic effective interactions, (see Lalasissis et al., PRC 55, 540 (1997), Peter Ring lectures) – Well defined transformations under Lorentz boosts – Parameterization can be adjusted to incorporate new data.
Skyrme parameterizations: ( Vautherin and Brink, PRC
5
, 626 (1972).) – Requires transformation to local rest frame – Computationally straightforward -
Example: Skyrme interaction.
While it is possible to define effective two-body operators, etc., for a Skyrme iteraction, it is much easier to start with the corresponding mean field potential. For neutrons and protons, respectively, we have the following expressions for the respective
U
n
~
n p
a
b
c
1
~
n p and
U
p n
,
p
a
b
c
1
~
p
~
n where a is the coefficient for the attractive two-body interaction, b takes the short range repulsion and multi-nucleon diagrams into account, c describes the symmetry potential and
~
n p
, ~
.
When at this value of the mean field, increasing the neutron and proton densities by
d
n and
d ~
n , increases the potential energy per unit volume by:
d
n
V
n
U
n n
,
p
d ~
n and
d
p
V
p
U
p n
, ~
p
d ~
p . HW 2: Show the potential energy per unit volume is: n
V
n p
V
p
a
2
2
b
1
1
c
1
1
n p 2
. •Hint: use the expressions for the differential increases in potential energy per unit volume above and do a parametric integration over
from zero to one.
Dividing by the density, one obtains the average potential energy per nucleon: V
a
2
b
1
c
1
2
, where
n p .
To this, one must add the average kinetic energy, which at zero temperature has the form: 3 5
f n
3 5
f
KE
p
where
f p
3 5
f
(
)
1 3
f
is the Fermi energy at density
. (
)
2
Putting the kinetic and potential terms together, one obtains an expression of the form:
E / A
(
, 0 , 0 )
S
2 where
(
, 0 , 0 , )
3 5
f
(
)
a
2
b
1 and S c (
0
)
1
1 3
f
(
)
c
1 .
Choosing
a
0 =-356 MeV,
b
0 =303 MeV, =1.17 and =18 MeV, provides a mean field with a bulk binding energy of 16 MeV and a reasonable value for the symmetry energy at normal density. The choice of gives a “soft” nuclear incompressibility constant of K nm =200 MeV.
The average energy for symmetric matter, with =0, and neutron matter, with =1, for this expression are shown in the left figure. In the right figure, we show results for the neutron matter energy for 19 different Skyrme interactions that were used in Hartree Fock calculations that reproduce the binding energy of Sn 20 10 0 -10 50 40 30 K nm =200 MeV Symmetric matter Neutron matter -20 0 0.05
0.1
0.15
/ 0 0.2
0.25
• Unlike symmetric matter, the potential energy of neutron matter is repulsive. 0.3
• The density dependence of symmetry energy is largely unconstrained.
O
Constraints on symmetric and asymmetric matter EOS
200 150 100 50
E/A (
,
) = E/A (
,0) +
2 S( )
= (
n
-
p
)/ (
n
+
p
) = (N-Z)/A
1
symmetric matter EOS
neutron matter RMF:DD RMF:NL3 Boguta Akmal experiment 100 10 1 1 1.5
2 2.5
3 / 0 3.5
Akmal av14uvII MS ( z =0, x =0) GWM:neutrons Fermi Gas Exp.+Asy_soft Exp.+Asy_stiff 4 4.5
5 • • • 0 1 2 3 / 0 4 5 Constraints come mainly from collective flow measurements.
Know pressure is zero at = 0 .
Results from variational calculations and Relativistic mean field theory with density dependent couplings lie within the allowed boundaries. • • Neutron matter EOS also includes the poorly constrained pressure from the symmetry energy. The uncertainty from the symmetry energy is larger than that from the symmetric matter EOS.
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Type II supernova: (collapse of 20 solar mass star)
• • Supernovae scenario: (Bethe Reference) – Nuclei H He C ...
Si Fe – Fe stable, Fe shell cools and the star collapses – Matter compresses to >4 s and then expands Relevant densities and matter properties – Compressed matter inside shock radius 0 < <10 0 , 0.4–0.9
• What densities are achieved? • What is the stored energy in the shock?
• What is the neutrino emission from the proto-neutron star?
– Clustered matter outside shock radius – mixed phase of nucleons and nuclear drops - nuclei: explosion?
< 0 , 0.3–0.5
• How much energy is dissipated in vaporizing the drops during the • What is the nature of the matter that interacts and traps the neutrinos?
• What are the seed nuclei that are present at the beginning of r-process which makes roughly half of the elements?
O
Neutron Stars
• • • • Neutron Star stability against gravitational collapse Stellar density profile Internal structure: occurrence of various phases.
– Stellar masses, radii and moments of inertia.
– Cooling rates of proto-neutron stars – Cooling rates for X-ray bursters.
Neutron Star Structure:
Pethick and Ravenhall, Ann. Rev. Nucl. Part. Sci. 45, 429 (1995)
O
Some examples Neutron star radii:
2 1.5
0.5
0
S
pot
const
.
F
(
u
);
u
/ 0 • • These equations of state differ only in their density dependent symmetry terms.
Clear sensitivity to the density dependence of the symmetry terms x
Cooling of proto-neutron stars:
10 7 Lattimer et al., Ap. J. 425 (1994) 802.
Standard cooling Direct URCA Isothermal t W 10 6 • • 10 5 1 10 100 1000 10 4 Age (y) 10 5 10 6 Neutrino signal from collapse.
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Feasibility of URCA processes for
proto-neutron star cooling if f p > 0.1. This occurs if S( ) is strongly density dependent. p+e n+ n p+e +
Summary of last lecture
• • The EOS describes the macroscopic response of nuclear matter and finite nuclei.
( ,0, ) = ( ,0,0) + 2 S( ) ; = ( n p )/ ( n + p ) = (N-Z)/A – It can be calculated by various techniques. Skyrme parameterizations are a relatively easy and flexible way to do so. .
– The high density behavior and the behavior at large isospin asymmetries of the EOS are not well constrained.
The behavior at large isospin asymmetries is described by the symmetry energy. – The symmetry energy has a profound influence on neutron star properties: stellar radii, maximum masses, cooling of proto-neutron stars, phases in the stellar interior, etc.
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Binding energies as probes of the EOS
B A,Z = a v [1-b 1 ((N-Z)/A)²]A - a s [1-b 2 ((N-Z)/A)²]A 2/3 - a c Z²/A 1/3 + δ A,Z A -1/2 + C d Z²/A, • • • • Fits of the liquid drop binding energy formula experimental masses can provide values for a Relationship to EOS v , a s
E
, a / c
A
, b 1 , b ( 2 , C , 0 d , and A , Z . ) ( , 0 , 0 )
S
2 ;
n
/
p
– a v = ( s ,0,0); a v b 1 =S( s ) – a s and a s b 2 provide information about the density dependence of ( s ,0,0) and S( s ) at subsaturation densities 1/2 s . (See Danielewicz, Nucl. Phys. A 727 (2003) 233.) – The various parameters are correlated. Coulomb and symmetry energy terms are strongly correlated. Shell effects make masses differ from LDM.
Measurement techniques: – Penning traps: =qB/m – Time of flight: TOF=distance/v B =mv/q – Transfer reactions: A(b,c)D Q=(m A +m b -m c -m D )c 2 Mass compilations exist: e.g. Audi et al,.,NPA 595, (1995) 409.
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Neutron and proton matter radii
• • A simple approximation to the density profile is a Fermi function (r)= 0 /(1+exp(r R)/a).
For stable nuclei, R p has been measured by electron scattering to about 0.02 fm accuracy. – (see G. Fricke et al., At. Data Nucl. Data Tables 60, 177 (1995).) 0.1
0.08
0.06
0.04
0.02
0 -2 0 n p (fm (fm -3 -3 ) ) 208
Pb
2 4 6 8 10 12 • • Neutron radii can be measured by hadronic scattering, which is more model dependent and less accurate ( R n 0.2 fm) because the interaction is mainly on the surface. a 0.5 – 0.6 fm for stable spherical nuclei, but near the neutron dripline, a n can be much larger.
– Strong interaction radius for 11 Li is about the same as that for 208 Pb.
Comparison of R
n
and R
p • • • • The asymmetry in the nuclear surface can be larger when S( ) is strongly density dependent because S( ) vanishes.more rapidly at low density when S( ) is stiff.
– Stiff symmetry energy larger neutron skins. (See Danielewicz lecture.) Measurements of 208 Pb using parity violating electron scattering are expected to provide strong constraints on
0.25
Brown, Phys. Rev. Lett. 85, 5296 (2001)
Shift in neutron radius for 208 Pb 0.2
0.15
0.1
0.05
0 -0.2 0 0.2 0.4 0.6 0.8
P sym (MeV/fm 3 ) 1 1.2 1.4
at =0.1 fm
O
Radii of Na isotopes
Suzuki, et al., PRL 75, 3241 (1995) •
R c
, int
R Na
, int ) 2 – Getting the actual neutron radius is model dependent.
• •
O
Proton radii are determined by measuring atomic transitions in Na, which has a 3s g.s. orbit. Neutron radii increase faster than R=r 0 A 1/3 , reflecting the thickness of neutron skin, e.g. RMF calculation.
Giant resonances
• • • Imagine a macroscopic, i.e. classical excitation of the matter in the nucleus.
– e.g. Isoscaler Giant Monopole (GMR) resonance GMR provides information about the curvature of ( ,0,0) about minimum.
Inelastic (1999).) ( , 0 , 0 ) 16
MeV
1 2 2 2 particle scattering e.g. 90 Zr( , )
s
90 2 Zr* can excite the GMR. (see Youngblood et al., PRL 92, 691 – Peak is strongest at 0 0 -5 -10 -15 -20 0 0.1
/ 0 0.2
0.3
Giant resonances 2
• HW 3: Assume that we can approximate a nucleus as having a sharp surface at radius R and ignore the surface, Coulomb and symmetry energy contributions to the nuclear energy. • – In the adiabatic approximation show that
PE
A
– Show that
KE
1 / 2
M
2 – Show that
E GMR
m K nm r
2 ; where K nm 9 2
s
2 2
s
s
R
0
R
3 , 0 , 0 In practice there are surface, Coulomb and symmetry energy corrections to the GMR energy. (see Harakeh and van der Woude, “Giant Resonances” Oxford Science...) – Leptodermous expansion:
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Giant Resonances 3
P N • • Isovector Giant Dipole Resonance: neutrons and protons oscillate against each other. The restoring force is the surface energy of the nucleus. Danielewicz has shown that E GDR depends on the surface symmetry energy but not on the volume symmetry energy. (Danielewicz,
NP A 727 (2003) 233.)
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Probes of the symmetric matter EOS
• • Nuclear collisions are the only way to make variations in nuclear density under “experimentally controlled” conditions and obtain information about the EOS.
Theoretical tool: transport theory: – Example Boltzmann-Uehling-Uhlenbeck eq. (Bertsch Phys. Rep. 160, 189 (1988).):
f
1
t
v
r
p
f
1 4 3
d
3
k
2
d
d
d
nn v
12
f
3
f
4 1
f
1 1
f
2
f
1
f
2 1
f
3 1
f
4 – Describes the time evolution of the Wigner transform of the one-body density matrix: (quantum analogue to classical phase space distribution) – classically, f= ( the number of nucleons/d 3 rd 3 p at ) . – Semiclassical: “time dependent Thomas-Fermi theory” – Each nucleon is represented by ~1000 test particles that propogate classically under the influence of the mean field U and subject to collisions due to the residual interaction. The mean field is self consistent, at each time step, one: • propogates nucleons, etc. subject to the mean field and collisions, and • recalculates the mean field potential according to the new positions.
Constraining the EOS at high densities by laboratory collisions
Au+Au collisions E/A = 1 GeV)
pressure contours density contours
• Two observable consequences of the high pressures that are formed: – Nucleons deflected sideways in the reaction plane.
– Nucleons are “squeezed out” above and below the reaction plane. .
Procedure to study high pressures
• • • • • • • • Measure collisions Simulate collisions with BUU or other transport theory Identify observables that are sensitive to EOS (see Danielewicz et al., Science 298,1592 (2002). for flow observables) Directed transverse flow (in-plane) – “Elliptical flow” out of plane, e.g. “squeeze-out” – Kaon production. ( Schmah, PRC C
71
, 064907 (2005)) Analyze data and model calculations to measured and calculated observable assuming some specific forms of the mean field potentials for neutrons and protons. At some energies, produced particles, like pions, etc. must be calculated as well.
Find the mean field(s) that describes the data. If more than one mean field describes the data, resolve the ambiguity with additional data. Constrain the effective masses and in-medium cross sections by additional data. Use the mean field potentials to apply the EOS information to other contexts like neutron stars, etc.
projectile target
Directed transverse flow
p x y Partlan, PRL
75
, 2100 (1995).
Au+Au collisions EOS TPC data E beam /A • • • Event has “elliptical” shape in momentum space. The long axis lies in the reaction plane, perpendicular to the total angular momentum.
Analysis procedure: – Find the reaction plane – Determine
in this plane – note: y 1 2 ln
E
E
p z c p z c
v
||
c
non relativist ically • y/y beam (in C.M) The data display the “s” shape characteristic of directed transverse flow.
– The TPC has in-efficiencies at y/y beam < -0.2.
d
p x
/
A
/ determined at –0.2 Danielewicz et al., Science 298,1592 (2002). p x y / A • The curves labeled by K nm represent calculations with parameterized Skyrme mean fields – They are adjusted to find the pressure that replicates the observed transverse flow. • • The boundaries represent the range of pressures obtained for the mean fields that reproduce the data. They also reflect the uncertainties from the effective masses in in medium cross sections. • • • ( ,0, ) = ( ,0,0) + 2 S( ) ; = ( n p )/ ( n + p ) = (N-Z)/A Common features of some of these studies – Vary isospin of detected particle • Sign in U asy is opposite for n vs. p. • Shape is influenced by “stiffness”. – Vary isospin asymmetry of reaction. Low densities ( < 0 ): – Isospin diffusion – Neutron/proton spectra and flows – Neutron, proton radii, E1 collective modes. High densities ( 2 0 ) : – Neutron/proton spectra and flows – + vs. production 100 =0.3 50 0 -50 -100 0 F 1 =2u 2 /(1+u) F 2 =u F 3 = u 0.5 u = Li et al., PRL 78 (1997) 1644 stiff 1 / o F 1 1.5 F 2 F 3 soft Neutron Proton 2 • • • – n v n p v p n D p / D the isospin diffusion coef. Two effect contribute to diffusion – Random walk – Potential (EOS) driven flows D governs the relative flow of neutrons and protons – – D D decreases with np increases with S int ( ) softer stiffer •R is the ratio between the diffusion coefficient with a symmetry potential and without a symmetry potential. • • • • Vary isospin driving forces by changing the isospin of projectile and target. Probe the asymmetry =(N-Z)/(N+Z) of the projectile spectator after the collision. The asymmetry of the spectator can change due to diffusion, but it also can changed due to pre-equilibrium emission. The use of the isospin transport ratio R i ( ) isolates the diffusion effects: R i ( ) 2 ( both _ neut . rich both _ neut . rich both _ prot . rich ) / both _ prot . rich 2 • Useful limits for R i collisions: for 124 Sn+ 112 Sn – – R i R i =±1: 0: no diffusion Isospin equilibrium mixed 124 Sn+ 112 Sn n-rich 124 Sn+ 124 Sn p-rich 112 Sn+ 112 Sn P neutron-rich No isospin diffusion 0.0 Complete mixing R i N proton-rich • • The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre equilibrium emission. The use of the isospin transport ratio R effects: i ( ) isolates the diffusion R i Neutron rich Neutron rich Pr oton rich Pr oton rich Stronger density dependence Weaker density dependence Tsang et al., PRL92(2004) • The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay Liu et al., (2006) • The the shift can be compactly described by the isoscaling parameters and obtained by taking ratios of the isotopic distributions: 2 1 Tsang et. al.,PRL 92, 062701 (2004 ) i R i Neutron rich Neutron rich Pr oton rich Pr oton rich 2 1 • Statistical theory provides: n p T 2 C sym T 2 C sym 1 1 / / T T , and n p where : 2 1 T 4 C sym / T , • Consider the isoscaling ratio R i (X), where X = or R i ( X ) 2 X ( X Neutron rich X Pr oton rich X Neutron rich X Pr oton ) / 2 • • 2 By direct substitution: R i R i 2 – true for known production models – linear dependences confirmed • The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay Liu et al., (2006) • The the shift can be compactly described by the isoscaling parameters and obtained by taking ratios of the isotopic distributions: 2 1 Tsang et. al.,PRL 92, 062701 (2004 ) no diffusion 1.0 0.33 R i ( ) -0.33 -1.0 124 Sn+ 112 Sn data A B C M.B. Tsang et. al., PRL 92, 062701 (2004) L.W. Chen, C.M. Ko, and B.A. Li, PRL 94, 032701 (2005) C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001) B.A. Li and A.W. Steiner, nucl-th/0511064 Approximate representation of the various asymmetry terms used in BUU calcuations: E sym ( ) ~ 32( / 0 ) [( n p ) /( n + p )] 2 ~5 for cases A, B, C) •Interpretation requires assumptions about isospin dependence of in-medium cross sections and effective masses • • • The EOS describes the macroscopic response of nuclear matter and finite nuclei. ( ,0, ) = ( ,0,0) + 2 S( ) ; = ( n p )/ ( n + p ) = (N-Z)/A – It can be calculated by various techniques. Skyrme parameterizations are relatively easy. – The high density behavior and the behavior at large isospin asymmetries of the EOS are not well constrained. The behavior at large isospin asymmetries is described by the symmetry energy. – It influences many nuclear physics quantities: binding energies, neutron skin thicknesses, isovector giant resonances, isospin diffusion, etc. Measurements of these quantities can constrain the symmetry energy. – The symmetry energy has a profound influence on neutron star properties: stellar radii, maximum masses, cooling of proto-neutron stars, phases in the stellar interior, etc. Constraints on the symmetry energy and on the EOS will be improved by planned experiments. Some of the best ideas have not yet been discovered. • • Primary: Before decay of excited fragments, Final: after decay of excited fragments Statistical theory: – Final isoscaling parameters are often similar to those of the primary distribution – Both depend linearly on R( )=R( ) • • Dynamical theories: – Final isoscaling parameters are often smaller than those of primary distribution – Both depend linearly on R( )=R( ) – Doesn't matter which one is correct. 0.4 0.3 Statistical Multifragmentation Model Sn+Sn collisions 0.2 1 AMD - Gogny: Ca+Ca collisions 0.8 0.6 primary secondary final 0.1 0.4 Primary_stiff Final_stiff 0 0.2 -0.1 0.1 0.12 0.14 0.16 0.18 0.2 0 0 0.05 0.1 0.15 0.2 • Data analyzed in well-mixed region at 70 cm 110 . • Linearity is demonstrated for , and ln(Y( 7 Li)/Y( 7 Be)) Liu et al., (2006) 0 • • Densities of 2 0 0 can be achieved at E/A 400 MeV. – Provides information about direct URCA cooling in proto-neutron stars, • S( ) influences diffusion of neutrons from dense overlap region at b=0. – Diffusion is reduced, neutron-rich dense region is formed for soft S( ). Yong et al., Phys. Rev. C 73 , 034603 (2006) • • • The enhanced neutron abundance at high density for the soft asymmetry term (x=0) leads to a stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1). – In delta resonance model, Y( )/Y( + ) ( n, / p ) 2 / + means Y( )/Y( + ) Coulomb interaction has a strong effect on the pion spectra: – Coulomb repels + attracts . and Yong et al., Phys. Rev. C 73 , 034603 (2006) soft stiff • The density dependence of the asymmetry term changes ratio by about 15% for neutron rich system. – How does one reduce sensitivity to systematic errors? • • Double ratio involves comparison between neutron rich 132 Sn+ 124 Sn and neutron deficient 112 Sn+ 112 Sn R reactions. / 132 Sn Y Y 132 124 112 112 124 / / Y Y Sn 112 Sn 132 124 112 112 112 Sn Double ratio maximizes sensitivity to asymmetry term. – Largely removes sensitivity to difference between and + acceptances. Yong et al., Phys. Rev. C 73 , 034603 (2006) soft stiff • • Neutrons are repelled and protons are attracted by the asymmetry term (in neutron rich matter). • The Coulomb interaction has somewhat the opposite effect. • Sensitivity can be maximized by R constructing a double ratio: n / Y Y p 132 Sn 124 132 124 112 112 / / Sn Y Y 112 Sn 132 124 112 112 112 Sn Removes sensitivity to calibration and efficiency problems stiff soft • • • Transverse directed flow is usually obtained by plotting the mean transverse momentum vs. the rapidity y. The neutron-proton differential flow is defined here to be: F n x p N 1 N ( y ) w i i p x i w i 1 (-1); neut. (prot.) Sensitivity to acceptance effects might be minimized by constructing the difference: D x n p F x n F n x p p 132 Sn 112 Sn 124 Sn 112 Sn F Li et al., arXiv:nucl-th/0504069 (2005) Li et al., Phys. Rev. C 69 , 011603 ( R ) ( 2004 ) 0.6 0.5 0.4 =1 np = pp = < > =1 free cross sections =2 np = pp = < > =2 free cross sections 0.3 • We need calculations of the corresponding double ratios. – Not clear that we have a good way to distinguish momentum and density dependencies. 0.2 0.1 40 Ca+ 100 Zn E/A=200 MeV • 0 -1 -0.5 0 0.5 (y/y beam ) cm (normal kinematics) Important to control the number of n-p collisions, p-p and n-n collisions – compare 37 Ca+ 112 Sn to 37 Ca+ 124 Sn – compare 52 Ca+ 112 Sn to 52 Ca+ 124 Sn. 1Determination of symmetric matter EOS from nucleus-nucleus collisions
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Probes of the symmetry energy
Constraining the density dependence of the symmetry energy Observable: Isospin diffusion in peripheral collisions
Probe: Isospin diffusion in peripheral collisions
1.0
-1.0
Sensitivity to symmetry energy
(
)
2
(
)
/ 2
Lijun Shi, thesis
Probing the asymmetry of the Spectators
Y
Y
N , Z
C exp(
N
Z )
Determining R
(
)
(
)
2
(
)
/ 2 Y
Y
N N , , Z Z
C exp(
N
Z )
X
a
b
Probing the asymmetry of the Spectators
Y
Y
N , Z
C exp(
N
Z )
Constraints from Isospin Diffusion Data
O
Final Summary
Influence of production mechanism on isoscaling parameters
Test of linearity using central collisions
Asymmetry term studies at
2
First observable: pion production
Double ratio: pion production
Independent observable: n/p spectra
Alternate observable: n-p differential transverse flow
Constraints on momentum dependence of mean fields and in-medium cross sections