Constraining the properties of dense matter

Constraining the properties of dense matterConstraining the properties of dense matter

A. What is the EOS 1. Theoretical approaches 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 3. Giant resonances 4. Particle flow and particle production: symmetric EOS 5. Particle flow and particle production: symmetry energy

D Summary

A. What is the EOS 1. Theoretical approaches 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 3. Giant resonances 4. Particle flow and particle production: symmetric EOS 5. Particle flow and particle production: symmetry energy

D Summary

William Lynch, Michigan State University

T h e E O S : F i n i t e t e m p e r a t u r e : T h e E O S d e s c r i b e s h o w t h e p r e s s u r e P d e p e n d s o n t h e t e m p e r a t u r e T , t h e d e n s i t y a n d t h e a s y m m e t r y = ( n - p ) / :

),T,(PP H e r e , d e s i g n a t e s t h e n u m b e r d e n s i t y o f n u c l e o n s ( n u c l e o n s / f m 3 ) f o r n u c l e a r m a t t e r a t s a t u r a t i o n d e n s i t y .

O n e c a n u s e = E / A , t h e a v e r a g e e n e r g y p e r p a r t i c l e i n t h e s y s t e m , t o c a l c u l a t e P :

).,T,(T),T,(F/A ),T,(f where

),T,(f

V

F),T,(PP

,T

2

,T

( 1 )

H e r e f i s t h e h e l m h o l t z f r e e e n e r g y p e r n u c l e o n a n d i s t h e e n t r o p y p e r n u c l e o n . c a n b e o b t a i n e d f r o m :

.nucleonper capacity heat theisT

c where

,T

dTc

T

d

,VV

T

0

vT

0

,V

( 2 )

H o m e w o r k 1 : O b t a i n t h e a p p r o x i m a t e e x p r e s s i o n ) 2 T / ( 2

F ( ) ) f o r t h e f e r m i o n i c n u c l e a r s y s t e m a t d e n s i t y , a s s u m i n g a l l t e m p e r a t u r e d e p e n d e n c e i n r e s i d e s i n t h e n u c l e o n k i n e t i c e n e r g i e s . A s s u m e f o r s i m p l i c i t y t h a t n =

p . Z e r o t e m p e r a t u r e I f o n e i s a t l o w e n o u g h t e m p e r a t u r e , o n e c a n i g n o r e t h e d e p e n d e n c e o f t h e E O S o n t e m p e r a t u r e a n d e v a l u a t e t h e E O S a t T = 0 . F o r T = 0 ,

,

2

,T

2 ),0,(),0,(f),0,(PP ( 3 )

P h a s e t r a n s i t io n s : P h a s e t r a n s i t io n s c a n m a n ife s t th e m s e lv e s in th e E O S i f th e re a r e r e g io n s w h e re d P /d |< 0 , m a k in g th e m a t te r m e c h a n ic a l l y u n s ta b le . W h e re th is o c c u r s , o n e m u s t m a tc h th e c h e m ic a l p o te n t ia ls fo r th e d e n s e r a n d m o re d i lu te p h a s e s b y m a k in g a M a x w e l l c o n s t ru c t io n ,

in w h ic h th e a r e a , VdpA b e tw e e n th e

M a x w e l l c o n s t ru c t io n lin e a n d th e o r ig in a l E O S is e q u a l o n th e le f t a n d th e r ig h t s id e . T h e s t r a ig h t l in e s h o w s a n E O S ( in r e d ) a n d i t s M a x w e l l c o n s t ru c t io n in b lu e . ( In m o s t p la c e s , th e r e d a n d b lu e c o in c id e a n d o n ly th e b lu e is v is ib le . T h e r e d c u rv e c a n o n ly b e s e e n in th e u n s ta b le r e g io n .) N o te , th e s y s te m m a y fo l lo w th e d a s h e d c u rv e fo r a w h i le e v e n in th e m ix e d p h a s e r e g io n i f th e e x p a n s io n o r c o m p re s s io n is fa s t e n o u g h .

-5 0

0

5 0

1 0 0

1 5 0

2 0 0

1 1 .5 2 2 .5 3 3 .5 4

D e m o n st r a t io n o f M a x w e ll c o n s t r u c t io n

B o g o ta 2b o g 2 _ p t

P

/0

O

Theoretical ApproachesTheoretical 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 <H> with elaborate grounds state wavefunction that includes nucleon-nucleon correlations.

– 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

• 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 <H> with elaborate grounds state wavefunction that includes nucleon-nucleon correlations.

– 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

Example: Skyrme interaction.Example: Skyrme interaction.

•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.

D iv id in g b y th e d e n s i ty , o n e o b ta in s th e a v e ra g e p o te n t ia l e n e rg y p e r n u c le o n :

pn2 where,1

c1

b2

aV .

T o th is , o n e m u s t a d d th e a v e ra g e k in e t ic e n e rg y , w h ic h a t z e ro te m p e ra tu re h a s th e fo r m :

2

ff

ppfnnf)(

3

1)(

5

353

53

KE

w h e re f i s th e F e rm i e n e rg y a t d e n s i ty .

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 isotopes but differ in their symmetry energies.

Brow

n, Phys. R

ev. Lett. 85, 5296 (2001)

• The density dependence of symmetry energy is largely unconstrained.

• The density dependence of symmetry energy is largely unconstrained.

• Unlike symmetric matter, the potential energy of neutron matter is repulsive.

• Unlike symmetric matter, the potential energy of neutron matter is repulsive.

O

-20

-10

0

10

20

30

40

50

0 0.05 0.1 0.15 0.2 0.25 0.3

Calculations with simple skyrme.

Knm

=200 MeV Symmetric matter

Neutron matter

(M

eV

)

/0

• 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.

• 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.

Constraints on symmetric and asymmetric matter EOSConstraints on symmetric and asymmetric matter EOS

E/A (, ) = E/A (,0) + 2S() = (n- p)/ (n+ p) = (N-Z)/A1

• 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.

• 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.

0

50

100

150

200

1 2 3 4 5

symmetric matter EOS

RMF:DDRMF:NL3BogutaAkmalexperiment

P (M

eV/f

m3 )

/0

1

10

100

1 1.5 2 2.5 3 3.5 4 4.5 5

neutron matter

Akmalav14uvIIMS (=0,=0)GWM:neutronsFermi GasExp.+Asy_softExp.+Asy_stiff

P (

MeV

/fm

3 )

/0

O

Danielew

icz et al., Scie

nce

298,1

59

2 (2

002).

Danielew

icz et al., Scie

nce

298,1

59

2 (2

002).

Type II supernova: (collapse of 20 solar mass star)Type II supernova: (collapse of 20 solar mass star)

• Supernovae scenario: (Bethe Reference)– Nuclei HHeC...SiFe– Fe stable, Fe shell cools and the star collapses– Matter compresses to >4s and then expands

• Relevant densities and matter properties– Compressed matter inside shock radius 0<<100, 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: <0, 0.3–0.5

• How much energy is dissipated in vaporizing the drops during the explosion?

• 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?

• Supernovae scenario: (Bethe Reference)– Nuclei HHeC...SiFe– Fe stable, Fe shell cools and the star collapses– Matter compresses to >4s and then expands

• Relevant densities and matter properties– Compressed matter inside shock radius 0<<100, 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: <0, 0.3–0.5

• How much energy is dissipated in vaporizing the drops during the explosion?

• 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 StarsNeutron Stars

• Neutron Star stability against gravitational collapse

• Stellar density profile

• Internal structure: occurrence of various phases.

• Observational consequences: – Stellar masses, radii and

moments of inertia.

– Cooling rates of proto-neutron stars

– Cooling rates for X-ray bursters.

• Neutron Star stability against gravitational collapse

• Stellar density profile

• Internal structure: occurrence of various phases.

• Observational consequences: – 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)

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Some examplesSome examples

• These equations of state differ only in their density dependent symmetry terms.

• Clear sensitivity to the density dependence of the symmetry terms

• These equations of state differ only in their density dependent symmetry terms.

• Clear sensitivity to the density dependence of the symmetry terms

• Neutrino signal from collapse.• Feasibility of URCA processes for

proto-neutron star cooling if fp > 0.1. This occurs if S() is strongly density dependent. p+e- n+ n p+e-+

• Neutrino signal from collapse.• Feasibility of URCA processes for

proto-neutron star cooling if fp > 0.1. This occurs if S() is strongly density dependent. p+e- n+ n p+e-+

0/ );(. uuFconstS pot

Neutron star radii: Cooling of proto-neutron stars:

O

0

0.5

1.5

2

Lattim

er , Ap. J., 550, 426 (2001).

105

106

107

1 10 100 1000 104 105 106

Lattimer et al., Ap. J. 425 (1994) 802.

Standard coolingDirect URCAIsothermaltW

TS

oo (

deg)

Age (y)

x

Summary of last lectureSummary of last lecture

• The EOS describes the macroscopic response of nuclear matter and finite nuclei.

– 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.


– It can be calculated by various techniques. Skyrme parameterizations are a relatively easy and flexible way to do so. .




(,0,) = (,0,0) + 2S() ; = (n- p)/ (n+ p) = (N-Z)/A(,0,) = (,0,0) + 2S() ; = (n- p)/ (n+ p) = (N-Z)/A

O

Binding energies as probes of the EOSBinding energies as probes of the EOS

• Fits of the liquid drop binding energy formula experimental masses can provide values for av, as, ac, b1, b2, Cd and A,Z.

• Relationship to EOS

– av = (s,0,0); avb1=S(s)

– as and asb2 provide information about the density dependence of (s,0,0) and S(s) at subsaturation densities 1/2s . (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=(mA+mb-mc-mD)c2

• Mass compilations exist: e.g. Audi et al,.,NPA 595, (1995) 409.

• Fits of the liquid drop binding energy formula experimental masses can provide values for av, as, ac, b1, b2, Cd and A,Z.

• Relationship to EOS

– av = (s,0,0); avb1=S(s)

– as and asb2 provide information about the density dependence of (s,0,0) and S(s) at subsaturation densities 1/2s . (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=(mA+mb-mc-mD)c2

• Mass compilations exist: e.g. Audi et al,.,NPA 595, (1995) 409.

BA,Z = av[1-b1((N-Z)/A)²]A - as[1-b2((N-Z)/A)²]A2/3 - ac Z²/A1/3 + δA,ZA-1/2 + CdZ²/A, BA,Z = av[1-b1((N-Z)/A)²]A - as[1-b2((N-Z)/A)²]A2/3 - ac Z²/A1/3 + δA,ZA-1/2 + CdZ²/A,

O

/;)0,0,(),0,(/ 2pnSAE

Neutron and proton matter radiiNeutron 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, Rp 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).)

• A simple approximation to the density profile is a Fermi function (r)=0/(1+exp(r-R)/a).

• For stable nuclei, Rp 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).)

• Neutron radii can be measured by hadronic scattering, which is more model dependent and less accurate (Rn 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, an can be much larger.

– Strong interaction radius for 11Li is about the same as that for 208Pb.

• Neutron radii can be measured by hadronic scattering, which is more model dependent and less accurate (Rn 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, an can be much larger.

– Strong interaction radius for 11Li is about the same as that for 208Pb.

0

0.02

0.04

0.06

0.08

0.1

-2 0 2 4 6 8 10 12

n (fm-3)

p (fm-3)

n (fm

-3)

R (fm)

208Pb

(r)

(fm

-3)

r (fm)

Comparison of Rn and RpComparison of Rn and Rp

• 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 208Pb using parity violating electron scattering are expected to provide strong constraints on <rn

2>1/2- <rp

2>1/2 and on S() for < s. Uncertainties are of order 0.06 fm. (see Horowitz et al., 63, 025501(2001).)

• The upper figure shows how the predicted neutron skins depend on Psym=2dS()/d

• Analyses of <rn2>1/2- <rp

2>1/2 for Na isotopes have placed some constraints on S() for < s, (see Danielewicz, NPA 727, 203 (2003).

• 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 208Pb using parity violating electron scattering are expected to provide strong constraints on <rn

2>1/2- <rp

2>1/2 and on S() for < s. Uncertainties are of order 0.06 fm. (see Horowitz et al., 63, 025501(2001).)

• The upper figure shows how the predicted neutron skins depend on Psym=2dS()/d

• Analyses of <rn2>1/2- <rp

2>1/2 for Na isotopes have placed some constraints on S() for < s, (see Danielewicz, NPA 727, 203 (2003).

Brown, Phys. Rev. Lett. 85, 5296 (2001)

0

0.05

0.1

0.15

0.2

0.25

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Shift in neutron radius for 208Pb

<rn

2>1/2-<rp

2>1/2

<r n

2 >1/

2 - <

r p

2 >1/

2

Psym

(MeV/fm3) at =0.1 fm

O

stiffer

softer

Radii of Na isotopesRadii of Na isotopes

• The relationship between cross-section and Na interaction radius is:

– Getting the actual neutron radius is model dependent.

• The relationship between cross-section and Na interaction radius is:

– Getting the actual neutron radius is model dependent.

• Proton radii are determined by measuring atomic transitions in Na, which has a 3s g.s. orbit.

• Neutron radii increase faster than R=r0A1/3, reflecting the thickness of neutron skin, e.g. RMF calculation.

• Proton radii are determined by measuring atomic transitions in Na, which has a 3s g.s. orbit.

• Neutron radii increase faster than R=r0A1/3, reflecting the thickness of neutron skin, e.g. RMF calculation.

xexpNN cintincidentdtransmitte

2int,int,int )( Nac RR

<rp2>1/2 ~ 0.1 fm

O

Suzuki, et al., PRL 75, 3241 (1995)

Giant resonancesGiant 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 particle scattering e.g. 90Zr(, )90Zr* can excite the GMR. (see Youngblood et al., PRL 92, 691 (1999).)– Peak is strongest at 0

• 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 particle scattering e.g. 90Zr(, )90Zr* can excite the GMR. (see Youngblood et al., PRL 92, 691 (1999).)– Peak is strongest at 0

-20

-15

-10

-5

0

0 0.1 0.2 0.3

(M

eV)

/0

22

2

2

116)0,0,( sMeV

Giant resonances 2Giant 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

– Show that

– Show that

• 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:

• 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

– Show that

– Show that

• 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:

0,0,

3

0

R

RAPE s

25/32/1 RMKE

s

snmGMR

rm

KE

2

22

nm2

9K where;

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Giant Resonances 3Giant Resonances 3

• 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 EGDR depends on the surface symmetry energy but not on the volume symmetry energy. (Danielewicz, NP A 727 (2003) 233.)

• 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 EGDR depends on the surface symmetry energy but not on the volume symmetry energy. (Danielewicz, NP A 727 (2003) 233.)

NP

O

Probes of the symmetric matter EOSProbes 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).):

– 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/d3rd3p 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.

• 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).):

– 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/d3rd3p 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.

),,( tprf

p and

r

Constraining the EOS at high densities by laboratory collisions

Constraining the EOS at high densities by laboratory collisions

• 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. .

• 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. .

pressure contours

density contours

Au+Au collisions E/A = 1 GeV)Au+Au collisions E/A = 1 GeV)

Procedure to study high pressuresProcedure 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.

• 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.

Directed transverse flowDirected transverse flow

• 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 <px(y)> in this plane

– note:

• 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 <px(y)> in this plane

– note:

• The data display the “s” shape characteristic of directed transverse flow.– The TPC has in-efficiencies at

y/ybeam< -0.2.– Slope is

determined at –0.2<y/ybeam<0.3

• The data display the “s” shape characteristic of directed transverse flow.– The TPC has in-efficiencies at

y/ybeam< -0.2.– Slope is

determined at –0.2<y/ybeam<0.3

y

px

projectile

target

icallyrelativist-non

ln2

1 ||

c

v

cpE

cpE

z

z

y

Ebeam/A

Au+Au collisions

EOS TPC data

dyApd x //

y/ybeam (in C.M)

Partlan, PRL 75, 2100 (1995).

Determination of symmetric matter EOS from nucleus-nucleus collisions

Determination of symmetric matter EOS from nucleus-nucleus collisions

• The curves labeled by Knm represent calculations with parameterized Skyrme mean fields– They are adjusted to find the

pressure that replicates the observed transverse flow.

• The curves labeled by Knm represent calculations with parameterized Skyrme mean fields– They are adjusted to find the

pressure that replicates the observed transverse flow.

y

Apx

/

• 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.

• 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.

Danielewicz et al., Science 298,1592 (2002).

O

Probes of the symmetry energyProbes of the symmetry energy

• Common features of some of these studies

– Vary isospin of detected particle

• Sign in Uasy 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 (20) :

– Neutron/proton spectra and flows + vs. - production

• Common features of some of these studies

– Vary isospin of detected particle

• Sign in Uasy 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 (20) :

– Neutron/proton spectra and flows + vs. - production


-100

-50

0

50

100

0 0.5 1 1.5 2

Li et al., PRL 78 (1997) 1644

Vas

y (M

eV)

/o

NeutronProton

F1=2u2/(1+u)

F2=u

F3=u

F1F2

F3

u =

stiff

soft

Uas

y (M

eV)

=0.3

Constraining the density dependence of the symmetry energyObservable: Isospin diffusion in peripheral collisions

Constraining the density dependence of the symmetry energyObservable: Isospin diffusion in peripheral collisions

• In a reference frame where the matter is stationary:

– 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 decreases with np

– D increases with Sint()

• In a reference frame where the matter is stationary:

– 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 decreases with np

– D increases with Sint()

Dvv ppnn

/pn Shi et al, C

68, 064604 (2003)

softer

stiffer

•R is the ratio between the diffusion coefficient with a symmetry potential and without a symmetry potential.

•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 Ri() isolates the diffusion effects:

• Useful limits for Ri for 124Sn+112Sn collisions:

– Ri =±1: no diffusion

– Ri 0: Isospin equilibrium

• 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.


• Useful limits for Ri for 124Sn+112Sn collisions:

– Ri =±1: no diffusion

– Ri 0: Isospin equilibrium

Probe: Isospin diffusion in peripheral collisionsProbe: Isospin diffusion in peripheral collisions

rich.prot_bothrich.neut_both

rich.prot_bothrich.neut_bothi

2/)(2)(R

P

N

mixed 124Sn+112Sn

n-rich 124Sn+124Sn

p-rich 112Sn+112Sn

mixed 124Sn+112Sn

n-rich 124Sn+124Sn

p-rich 112Sn+112Snneutron-rich

No isospin diffusion

1.0

-1.0

0.0Complete mixing

proton-rich

Ri

Sensitivity to symmetry energySensitivity to symmetry energy

Tsang et al., PRL92(2004)

Stronger density dependence

Weaker density dependence

Lijun Shi, thesis

richotonrichNeutron

richotonrichNeutroniR

Pr

Pr 2/)(2)(

• The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre-equilibrium emission.


• The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre-equilibrium emission.


Probing the asymmetry of the SpectatorsProbing the asymmetry of the Spectators

• The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay


• The the shift can be compactly described by the isoscaling parameters and obtained by taking ratios of the isotopic distributions:

• The the shift can be compactly described by the isoscaling parameters and obtained by taking ratios of the isotopic distributions: )ZNexp(C

Z,NY

Z,NY

1

2 Liu et al., (2006)

Tsang et. al.,PRL 92, 062701 (2004)

Determining Ri()Determining Ri()

• Statistical theory provides: • Statistical theory provides:

• Consider the isoscaling ratio Ri(X), where X = or

• When X depends linearly on 2:

• By direct substitution:

– true for known production models

– linear dependences confirmed by data.

• Consider the isoscaling ratio Ri(X), where X = or

• When X depends linearly on 2:

• By direct substitution:

– true for known production models

– linear dependences confirmed by data.

richotonPrrichNeutron

richotonPrrichNeutroni XX

2/)XX(X2)X(R

baX 2

2ii RXR

12

sympn

symp

symn

:where

,T/C4T

and

,T/1C2T

T/1C2T

)ZNexp(C

Z,NY

Z,NY

1

2

richotonrichNeutron

richotonrichNeutroniR

Pr

Pr 2/)(2)(

Probing the asymmetry of the SpectatorsProbing the asymmetry of the Spectators



• The the shift can be compactly described by the isoscaling parameters and obtained by taking ratios of the isotopic distributions:

• The the shift can be compactly described by the isoscaling parameters and obtained by taking ratios of the isotopic distributions: )ZNexp(C

Z,NY

Z,NY

1

2 Liu et al., (2006)

Tsang et. al.,PRL 92, 062701 (2004)

no diffusion1.0

0.33

-0.33

-1.0

Ri()

Constraints from Isospin Diffusion DataConstraints from Isospin Diffusion Data

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:

Esym() ~ 32(/0) [(n - p) /(n +p)]

2

for cases A, B, C)

A

B

C124Sn+112Sn data

•Interpretation requires assumptions about isospin dependence of in-medium cross sections and effective masses

•Interpretation requires assumptions about isospin dependence of in-medium cross sections and effective masses

O

Final Summary Final Summary


– It can be calculated by various techniques. Skyrme parameterizations are relatively easy.



– 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.


• 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.


– It can be calculated by various techniques. Skyrme parameterizations are relatively easy.



– 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.


• 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.


Influence of production mechanism on isoscaling parametersInfluence of production mechanism on isoscaling parameters

• Statistical theory:

– Final isoscaling parameters are often similar to those of the primary distribution

– Both depend linearly on

R()=R()

• 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.

• 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.

-0.1

0

0.1

0.2

0.3

0.4

0.1 0.12 0.14 0.16 0.18 0.2

Statistical Multifragmentation ModelSn+Sn collisions

Primary_stiffFinal_stiff

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

AMD - Gogny: Ca+Ca collisions

primarysecondary

Primary: Before decay of excited fragments, Final: after decay of excited fragments

final

R

Test of linearity using central collisionsTest of linearity using central collisions

• Data analyzed in well-mixed region at 70cm110.

• Data analyzed in well-mixed region at 70cm110.

• Linearity is demonstrated for , and ln(Y(7Li)/Y(7Be))-

• Linearity is demonstrated for , and ln(Y(7Li)/Y(7Be))-

Liu et al., (2006)

Liu et al., (2006)

R

• Densities of 20 can be achieved at E/A400 MeV.

– Provides information about direct URCA cooling in proto-neutron stars, stability and phase transitions of dense neutron star interior.





• S() influences diffusion of neutrons from dense overlap region at b=0.

– Diffusion is reduced, neutron-rich dense region is formed for soft S().



• S() influences diffusion of neutrons from dense overlap region at b=0.

– Diffusion is reduced, neutron-rich dense region is formed for soft S().

Asymmetry term studies at 20Asymmetry term studies at 20

Yong et al., Phys. Rev. C 73, 034603 (2006)

R

First observable: pion productionFirst observable: pion production

• 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 + and attracts -.

• 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 + and attracts -.

• The density dependence of the asymmetry term changes ratio by about 15% for neutron rich system.

– How does one reduce sensitivity to systematic errors?

• The density dependence of the asymmetry term changes ratio by about 15% for neutron rich system.

– How does one reduce sensitivity to systematic errors?


soft

stiff

Double ratio: pion productionDouble ratio: pion production

• Double ratio involves comparison between neutron rich 132Sn+124Sn and neutron deficient 112Sn+112Sn reactions.

• Double ratio maximizes sensitivity to asymmetry term.

– Largely removes sensitivity to difference between - and + acceptances.

• Double ratio involves comparison between neutron rich 132Sn+124Sn and neutron deficient 112Sn+112Sn reactions.

• Double ratio maximizes sensitivity to asymmetry term.

– Largely removes sensitivity to difference between - and + acceptances.

112112112112

124132124132

112112124132/

Y/Y

Y/Y

SnSn/SnSnR


soft

stiff

R

Independent observable: n/p spectraIndependent observable: n/p spectra

• 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 constructing a double ratio:

• Removes sensitivity to calibration and efficiency problems

• 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 constructing a double ratio:

• Removes sensitivity to calibration and efficiency problems

112112112112

124132124132

112112124132p/n

pY/nY

pY/nY

SnSn/SnSnR

Li et al., arX

iv:nucl-th/0510016 (2005)soft

stiff

Alternate observable: n-p differential transverse flowAlternate observable: n-p differential transverse flow

• Transverse directed flow is usually obtained by plotting the mean transverse momentum <px> vs. the rapidity y.

• The neutron-proton differential flow is defined here to be:

• Sensitivity to acceptance effects might be minimized by constructing the difference:

• Transverse directed flow is usually obtained by plotting the mean transverse momentum <px> vs. the rapidity y.

• The neutron-proton differential flow is defined here to be:

• Sensitivity to acceptance effects might be minimized by constructing the difference:

F

(prot.) neut. (-1); 1w

pwyN

1F

i

)y(N

i

xii

xpn

SnSnF

SnSnFD112112x

pn

124132xpn

xpn

Li et al., arXiv:nucl-th/0504069 (2005)

Constraints on momentum dependence of mean fields and in-medium cross sections

Constraints on momentum dependence of mean fields and in-medium cross sections

• We need calculations of the corresponding double ratios.

– Not clear that we have a good way to distinguish momentum and density dependencies.

• We need calculations of the corresponding double ratios.

– Not clear that we have a good way to distinguish momentum and density dependencies.

• Important to control the number of n-p collisions, p-p and n-n collisions

– compare 37Ca+112Sn to 37Ca+124Sn

– compare 52Ca+112Sn to 52Ca+124Sn.

• Important to control the number of n-p collisions, p-p and n-n collisions

– compare 37Ca+112Sn to 37Ca+124Sn

– compare 52Ca+112Sn to 52Ca+124Sn.

0

0.1

0.2

0.3

0.4

0.5

0.6

-1 -0.5 0 0.5 1

=1 np

= pp

= <>

=1 free cross sections=2

np =

pp = <>

=2 free cross sections

=(N

-Z)/

(N+

Z)

(y/ybeam

)cm

(normal kinematics)

40Ca+100Zn

E/A=200 MeV

Li et al., P

hys. Rev. C

71, 054603 (2005)

Li et al., Phys. Rev. C 69, 011603(R) (2004)

R

Documents

Constraining the properties of dense matter