A.What is the EOS? 0. Relationship to energy and to nuclear masses 1. Important questions B. What observables are sensitive to the EOS and

A. What is the EOS? 0. Relationship to energy and to nuclear masses 1. Important questions

B. What observables are sensitive to the EOS and at what densities?0. Astrophysical observables (neutron stars) 1. Binding energies 2. Radii of neutron and proton matter in nuclei 3. “Pigmy” and giant resonances 4. Particle flow and particle production: symmetric EOS 5. Particle flow and particle production: symmetry energy

C Summary

A. What is the EOS? 0. Relationship to energy and to nuclear masses 1. Important questions

B. What observables are sensitive to the EOS and at what densities?0. Astrophysical observables (neutron stars) 1. Binding energies 2. Radii of neutron and proton matter in nuclei 3. “Pigmy” and giant resonances 4. Particle flow and particle production: symmetric EOS 5. Particle flow and particle production: symmetry energy

C Summary

Constraining the properties of dense matterConstraining the properties of dense matter

William Lynch, Michigan State University

The EoSThe EoS

• The EoS at finite temperature: – Suppose you want the pressure (P) in nuclear matter as a function of

density (), temperature (T) and asymmetry

• Zero temperature (appropriate for neutron stars)

• Practical approach is to calculate (,0,) theoretically by some density functional approach (e.g. Hartree Fock), by a variational calculation

• or to constrain it experimentally • or both.

• The EoS at finite temperature: – Suppose you want the pressure (P) in nuclear matter as a function of

density (), temperature (T) and asymmetry

• Zero temperature (appropriate for neutron stars)

• Practical approach is to calculate (,0,) theoretically by some density functional approach (e.g. Hartree Fock), by a variational calculation

• or to constrain it experimentally • or both.

T,

FP P( ,T, ) ;F ,T, / A ,T, T ,T,

V

2

T, T,

,0,EP P( ,T, )

V

n p n p

energy/nucleon

entropy/nucleon

More about the EoSMore about the EoS

• Some facts about the EoS may be useful:

• At saturation density, Sint()0.6S(). • Sint() is the uncertain contribution to S().

• Some facts about the EoS may be useful:

• At saturation density, Sint()0.6S(). • Sint() is the uncertain contribution to S().

• A homogenous EoS is approximate in the vicinity of phase transition.

• Matter outside the shock wave in type II supernovae and to the inner crust of a neutron star, is a mixed phase of nuclei in a more dilute gas.

• There, Coulomb forces introduce long range order (solid phase).

• The EoS for homogeneous matter can be in a input to the properties of such systems, not the entire story.

• A homogenous EoS is approximate in the vicinity of phase transition.

• Matter outside the shock wave in type II supernovae and to the inner crust of a neutron star, is a mixed phase of nuclei in a more dilute gas.

• There, Coulomb forces introduce long range order (solid phase).

• The EoS for homogeneous matter can be in a input to the properties of such systems, not the entire story.

f SM

f int

int NM SM

3( ,0,0, ) ( ) V

5and for the symmetry energy of neutron matter S

1S( ) ( ) S

3where

S V V .

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2

S()kin

S()int

(Stiff)

S()int

(Soft)

S()

(M

eV)

/0

O

Relationship to nuclear masses and binding energiesRelationship to nuclear masses and binding energies

• 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.

• One can largely remove Coulomb effects by comparing g.s. and isobaric analog states in the same nucleus (Danielewicz and Lee Int.J.Mod.Phys.E18:892,2009 )

• 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.

• One can largely remove Coulomb effects by comparing g.s. and isobaric analog states in the same nucleus (Danielewicz and Lee Int.J.Mod.Phys.E18:892,2009 )

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,

In the local density approximation: -BA,Z =A<E/A>over nucleus= A<(,0,>over nucleus

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,

In the local density approximation: -BA,Z =A<E/A>over nucleus= A<(,0,>over nucleus

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

2 2 2A,Z p n A,ZM c Zm c A Z m c B

Provides a useful initial constraint on the EoS. Provides a useful initial constraint on the EoS. O

• Nuclear effective interactions do not constrain neutron matter,

• Main uncertainty is the density dependence of the symmetry energy

• Nuclear effective interactions do not constrain neutron matter,

• Main uncertainty is the density dependence of the symmetry energy

EOS: What are the questions?EOS: What are the questions?

E/A (,) = E/A (,0) + d2S()

d = (n- p)/ (n+ p) = (N-Z)/A

E/A (,) = E/A (,0) + d2S()

d = (n- p)/ (n+ p) = (N-Z)/A

a/s

2 A/EP

-20

0

20

40

60

80

100

120

0 1 2 3 4

symmetric matter

NL3Bog1:e/aBog2:e/aK=300, m*/m=70Akmal_corr.

E/A

( M

eV)

/0

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,

Brow

n, Phys. R

ev. Lett. 85, 5296 (2001)

neutron matter

0

=1=0

O

EOS, Symmetry Energy and Neutron StarsEOS, Symmetry Energy and Neutron Stars

• Influences neutron Star stability against gravitational collapse

• Stellar density profile• Internal structure:

occurrence of various phases.

• Observational consequences:– Cooling rates of proto-

neutron stars and– Temperatures and

luminosities of X-ray bursters .

– Stellar masses, radii and moments of inertia.

• Can be studied by X-ray observers.

• Influences neutron Star stability against gravitational collapse

• Stellar density profile• Internal structure:

occurrence of various phases.

• Observational consequences:– Cooling rates of proto-

neutron stars and– Temperatures and

luminosities of X-ray bursters .

– Stellar masses, radii and moments of inertia.

• Can be studied by X-ray observers.

• Currently , scientists are working to constrain EoS with data on X-ray bursts and low mass X-ray binaries.

• R vs. M relationship reflects EoS over a range of densities

Þ It is important to obtain complementary laboratory constraints at specific densities.

• Currently , scientists are working to constrain EoS with data on X-ray bursts and low mass X-ray binaries.

• R vs. M relationship reflects EoS over a range of densities

Þ It is important to obtain complementary laboratory constraints at specific densities.

O

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)

Comparisons of <r2>n1/2

and <r2>p1/2

Comparisons of <r2>n

1/2 and <r2>p

1/2

• Approximating the density profile by a Fermi function (r)=0/(1+exp(r-R)/a), we show representative proton and neutron distributions for 208Pb.

• Calculations predict that a stiff symmetry energy will result in a larger neutron skin.

• Why?– The symmetry pressure repels neutrons and

attracts protons. The pressure is larger if S() is strongly density dependent.

• Relation between skin thickness and EoS can be somewhat model dependent

• Approximating the density profile by a Fermi function (r)=0/(1+exp(r-R)/a), we show representative proton and neutron distributions for 208Pb.

• Calculations predict that a stiff symmetry energy will result in a larger neutron skin.

• Why?– The symmetry pressure repels neutrons and

attracts protons. The pressure is larger if S() is strongly density dependent.

• Relation between skin thickness and EoS can be somewhat model dependent

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

(at =0.1 fm-3 )

stiffersofter

,T

2

sym

sym.matt_sym.matt_neut

SP

PPP

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

Pneut. matt. (MeV/fm3)

Measurements of radiiMeasurements of radii

• Parity violating electron scattering may provide strong constraints on <rn

2>1/2- <rp

2>1/2 and on S() for < s. Expected uncertainties were of order 0.06 fm. (see Horowitz et al., Phys. Rev. 63, 025501(2001).)

• Nuclear charge and matter radii can be measured by diffractive scattering.

• For example, <rp2>1/2 has been measured

stable nuclei, by electron scattering to about 0.02 fm accuracy. – (see G. Fricke et al., At. Data Nucl.

Data Tables 60, 177 (1995).)• Strong interaction shifts in the 4f3d

transition in pionic 208Pb also provide sensitivity to the rms neutron radius. (Garcia-Recio, NPA 547 (1992) 473) – <rn

2>1/2 = 5.74.07ran .03sys fm– <rn

2>1/2- <rp2>1/2 = 0.22 .07ran .03sys

fm

• Parity violating electron scattering may provide strong constraints on <rn

2>1/2- <rp

2>1/2 and on S() for < s. Expected uncertainties were of order 0.06 fm. (see Horowitz et al., Phys. Rev. 63, 025501(2001).)

• Nuclear charge and matter radii can be measured by diffractive scattering.

• For example, <rp2>1/2 has been measured

stable nuclei, by electron scattering to about 0.02 fm accuracy. – (see G. Fricke et al., At. Data Nucl.

Data Tables 60, 177 (1995).)• Strong interaction shifts in the 4f3d

transition in pionic 208Pb also provide sensitivity to the rms neutron radius. (Garcia-Recio, NPA 547 (1992) 473) – <rn

2>1/2 = 5.74.07ran .03sys fm– <rn

2>1/2- <rp2>1/2 = 0.22 .07ran .03sys

fm

• Proton elastic scattering is sensitive to the neutron density. (see the talk by T. Murakami.)

• Proton elastic scattering is sensitive to the neutron density. (see the talk by T. Murakami.)

cm (deg)

208Pb(p,p)Ep =200 MeV

<rn2>1/2=5.6 fm

<rn2>1/2=5.7 fm

Karataglidis et al., PRC 65 044306 (2002)

O

PDR: Electric dipole excitations of the neutron skinPDR: Electric dipole excitations of the neutron skin

• Coulomb excitation of very neutron rich 130,132Sn isotopes reveals a peak at E*10 MeV.– not present for stable isotopes

• Consistent with low-lying electric dipole strength.

• calculations suggest an oscillation of a neutron skin relative to the core.

• Coulomb excitation of very neutron rich 130,132Sn isotopes reveals a peak at E*10 MeV.– not present for stable isotopes

• Consistent with low-lying electric dipole strength.

• calculations suggest an oscillation of a neutron skin relative to the core.

P. Adrich et al., P

RL

95 (2005) 132501

Relation to symmetry energy Relation to symmetry energy

• Random phase approximation (RPA) calculations show a strong correlation between the neutron -proton radius difference and the fractional strength in the pygmy dipole resonance.

• Random phase approximation (RPA) calculations show a strong correlation between the neutron -proton radius difference and the fractional strength in the pygmy dipole resonance.

• Random phase approximation (RPA) calculations show a strong correlation between the fractional strength and the symmetry pressure

• Random phase approximation (RPA) calculations show a strong correlation between the fractional strength and the symmetry pressure

3

4

5

6

7

8

9

10

11

0.15 0.2 0.25 0.3 0.35 0.4

relative pigmy strength

B(E1)PDR

/B(E1)GDR

B(E

1)P

DR/

B(E

1)G

DR [

%]

<rn

2>1/2 - <rp

2>1/2 [fm]

B 0

sym0 sym 0

B 0

E 3L 3 P

A. Klimkiewicz, et al., PRC76, 051603(R) (2007)A. Carbone,et al., PRC 81, 041301(R) (2010)

Data

% T

RK

ene

rgy

wei

ghte

d su

m r

ule

Giant resonancesGiant resonances

• Imagine a macroscopic, i.e. classical vibration of the matter in the nucleus.– e.g. Isoscaler Giant Monopole (GMR) resonance

• GMR and also ISGDR provide 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 vibration of the matter in the nucleus.– e.g. Isoscaler Giant Monopole (GMR) resonance

• GMR and also ISGDR provide 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

2

2

2

2

116)0,0,( sMeV

range of motion

(fm-3)

Giant resonances 2Giant resonances 2

• Of course, nuclei have surfaces, etc. This motivates a "leptodermous" expansion (see Harakeh and van der Woude, “Giant Resonances” Oxford Science...) :

– Ksym is a function of the first and second derivatives of the symmetry energy (G. Colo, et al., Phys. Rev. C70, 024307 (2004).)

– Measurements of GMR resonance energies over a range of isotopes can provide information the first and second derivatives of the symmetry energy as a function of density! (Actually, the first derivative is more important)

• Value of Ksym=500100 MeV has been reported by T. Liet.al, PRC 81, 034309 (2010) from analysis of (,’) on stable Sn isotopes.

• Of course, nuclei have surfaces, etc. This motivates a "leptodermous" expansion (see Harakeh and van der Woude, “Giant Resonances” Oxford Science...) :

– Ksym is a function of the first and second derivatives of the symmetry energy (G. Colo, et al., Phys. Rev. C70, 024307 (2004).)

– Measurements of GMR resonance energies over a range of isotopes can provide information the first and second derivatives of the symmetry energy as a function of density! (Actually, the first derivative is more important)

• Value of Ksym=500100 MeV has been reported by T. Liet.al, PRC 81, 034309 (2010) from analysis of (,’) on stable Sn isotopes.

AGMR 2

1/3 2 2 4/3A nm surf sym Coul

KE , where

m r

K K K A K K Z / A

3322 0

sym 0 02 3V, ,, ,

81S S SK 9 9

K

O

Constraints on the symmetric matter EoS from laboratory measurements

Constraints on the symmetric matter EoS from laboratory measurements

• The symmetric matter EoS strongly limits what you probe with nuclei– If the EoS is expanded in a Taylor series about 0, the incompressibility,

Knm provides the term proportional to (-0)2. Higher order terms influence the EoS at sub-saturation and supra-saturation densities.

– The solid black, dashed brown and dashed blue EoS’s all have Knm=300 MeV → the difference between these EoS'.

• The symmetric matter EoS strongly limits what you probe with nuclei– If the EoS is expanded in a Taylor series about 0, the incompressibility,

Knm provides the term proportional to (-0)2. Higher order terms influence the EoS at sub-saturation and supra-saturation densities.

– The solid black, dashed brown and dashed blue EoS’s all have Knm=300 MeV → the difference between these EoS'.

-20

0

20

40

60

80

100

120

0 1 2 3 4

symmetric matter


E/A

( M

eV)

/0

-20

0

20

40

60

80

100

120

0 1 2 3 4

symmetric matter


E/A

( M

eV)

/0

• Nuclear properties are mainly sensitive to the EoS at 0.50 1/250

• Nuclear properties are mainly sensitive to the EoS at 0.50 1/250

• To probe the EoS at 30, you need to compress matter to 30.

• To probe the EoS at 30, you need to compress matter to 30.

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. .


pressure contours

density contours

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

Flow studies of the symmetric matter EOSFlow studies of the symmetric matter EOS

• Theoretical tool: transport theory:– Example Boltzmann-Uehling-Uhlenbeck eq. (Bertsch Phys. Rep. 160, 189

(1988).) has derivation from TDHF:

• f is the Wigner transform of the one-body density matrix• semi-classically, = (number of nucleons/d3rd3p at ). • BUU can describe nucleon flows, the nucleation of weakly bound light

particles and the production of nucleon resonances. – The most accurately predicted observables are those that can be calculated

from i.e. flows and other average properties of the events.

• Theoretical tool: transport theory:– Example Boltzmann-Uehling-Uhlenbeck eq. (Bertsch Phys. Rep. 160, 189

(1988).) has derivation from TDHF:

• f is the Wigner transform of the one-body density matrix• semi-classically, = (number of nucleons/d3rd3p at ). • BUU can describe nucleon flows, the nucleation of weakly bound light

particles and the production of nucleon resonances. – The most accurately predicted observables are those that can be calculated

from i.e. flows and other average properties of the events.

43212143122

33

111

11112

4ffffffffv

d

ddkd

fUft

f

nn

prrv

),,( tprf

p and

r

),,( tprf

Procedure to study EOS using transport theoryProcedure to study EOS using transport theory

• 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))– Isospin diffusion– Neutron vs. proton emission and flow.– Pion production.

• 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 calculate the EOS.

• 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))– Isospin diffusion– Neutron vs. proton emission and flow.– Pion production.

• 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 calculate the EOS.

symmetric matter EOS

symmetry energy

• quick • detailed





pressure contours

density contours

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

• quick

Directed transverse flowDirected transverse flow

• Event has “elliptical” shape in momentum space, with the long axis in the reaction plane

• Analysis procedure:– Select impact parameter.– Find the reaction plane.– Determine <px(y)> in this plane

– note:

• Event has “elliptical” shape in momentum space, with the long axis in the reaction plane

• Analysis procedure:– Select impact parameter.– 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 (GeV)

Au+Au collisionsEOS TPC data

dyApd x //

y/ybeam (in C.M)

Partlan, PRL 75, 2100 (1995).

totalJ

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 and 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 and in-medium cross sections.

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

O

in plane

out of plane1

10

100

1 1.5 2 2.5 3 3.5 4 4.5 5

symmetric matter

RMF:NL3Fermi gasBogutaAkmalK=210 MeVK=300 MeVexperiment

P (

MeV

/fm3 )

/0

1

10

100

1 1.5 2 2.5 3 3.5 4 4.5 5

Symmetric Matter

GMR (K=240)Flow Experiment

P (

MeV

)/fm

-3)

/0

1

1 0

1 0 0

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

sy m m etr ic m a tter

R M F :N L 3A k m alF e rm i g asF lo w E x p e rim en tF S U A uG M R E x p e rim en t

P(M

eV/f

m-3

)

/0

1

1 0

1 0 0

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

E O S _ D D

R M F :N L 3A k m alF e rm i g asF lo w E x p e rim en tK ao n E x p e rim en tF S U A uG M R E x p erim en t

P(M

eV/f

m-3

)

/0

1

10

100

1 1.5 2 2.5 3 3.5 4 4.5 5

neutron matter

Akmalav14uvIINL3DDFermi GasExp.+Asy_softExp.+Asy_stiff

P (

MeV

/fm3)

/0

Danielew

icz et al., Scie

nce

298,1

59

2 (2

002).

• Note: analysis required additional constraints on m* and NN.

• Flow confirms the softening of the EOS at high density.

• Constraints from kaon production are consistent with the flow constraints and bridge gap to GMR constraints.

• Note: analysis required additional constraints on m* and NN.

• Flow confirms the softening of the EOS at high density.

• Constraints from kaon production are consistent with the flow constraints and bridge gap to GMR constraints.

Constraints from collective flow on EOS at >2 0.Constraints from collective flow on EOS at >2 0.

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

• The symmetry energy dominates the uncertainty in the n-matter EOS.

• Both laboratory and astronomical constraints on the density dependence of the symmetry energy are urgently needed.

• The symmetry energy dominates the uncertainty in the n-matter EOS.

• Both laboratory and astronomical constraints on the density dependence of the symmetry energy are urgently needed.

Danielew

icz et al., Scie

nce

298,1

59

2 (2

002).

O

Probing the symmetry energy by nuclear collisionsProbing the symmetry energy by nuclear collisions

• To maximize sensitivity, reduce systematic errors:– Vary isospin of detected particle– Vary isospin asymmetry =(N-Z)/A of

reaction.• Low densities (<0):

– Neutron/proton spectra and flows– Isospin diffusion– Isotope yields

• High densities (20) :– Neutron/proton spectra and flows – + vs. - production

• To maximize sensitivity, reduce systematic errors:– Vary isospin of detected particle– Vary isospin asymmetry =(N-Z)/A of

reaction.• Low densities (<0):

– Neutron/proton spectra and flows– Isospin diffusion– Isotope yields

• High densities (20) :– Neutron/proton spectra and flows – + vs. - production

E/A(,) = E/A(,0) + d2S() ; d = (n- p)/ (n+ p) = (N-Z)/AE/A(,) = E/A(,0) + d2S() ; d = (n- p)/ (n+ p) = (N-Z)/A

symmetry energy

• Collide projectiles and targets of differing isospin asymmetry

• Probe the asymmetry =(N-Z)/(N+Z) of the projectile spectator during the collision.

• 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

• Collide projectiles and targets of differing isospin asymmetry

• Probe the asymmetry =(N-Z)/(N+Z) of the projectile spectator during the collision.


• 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+112Sn

Systems{Example:

neutron-rich projectile

proton-rich target

measure asymmetry after collision

Sensitivity to symmetry energySensitivity to symmetry energy

Tsang et al., PRL92(2004)

Stronger density dependence

Weaker density dependence

Lijun S

hi, thesis (2003)

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.


Sensitive to S() at 1/20

• See also the talk of Dan Coupland

Experimental measurements of isospin diffusionExperimental measurements of isospin diffusion

• Experimental device:• Experimental device:

Miniball with LASSA arrayExperiment: 112,124Sn+112,124Sn, E/A =50 MeV

• Projectile and target nucleons are largely “spectators” during these peripheral collisions.– Projectile residue have somewhat less than beam velocity.– Target residues have very small velocities.

• Projectile and target nucleons are largely “spectators” during these peripheral collisions.– Projectile residue have somewhat less than beam velocity.– Target residues have very small velocities.

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


• This can be described by the isoscaling parameters and :


)ZNexp(C

Z,NY

Z,NY

1

2

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

no diffusion

Liu et al.PRC 76, 034603 (2007).

Determining Ri()Determining Ri()

• In statistical theory, certain observables depend linearly on =(n-p)/:

• Calculations confirm this• We have experimentally

confirmed this

• In statistical theory, certain observables depend linearly on =(n-p)/:

• Calculations confirm this• We have experimentally

confirmed this

• Consider the ratio Ri(X), where X = , X7 or some other observable:

• If X depends linearly on 2:

• Then by direct substitution:

• Consider the ratio Ri(X), where X = , X7 or some other observable:

• If X depends linearly on 2:

• Then by direct substitution:

richotonPrrichNeutron

richotonPrrichNeutroni XX

2/)XX(X2)X(R

baX 2

2ii RXR

)ZNexp(C

Z,NY

Z,NY

1

2

richotonrichNeutron


Pr

Pr 2/)(2)(

dcBeY/LiYln X

,ba

277

7

2

• In equilibrium statistical theory =(n,2-n,2)/t

• These chemical potentials reflect the both the asymmetry and the symmetry energy in the system.• This sensitivity has been

explored by Yennello and collaborators to probe the symmetry energy.

• It has little or no impact on isospin diffusion, however.

Probing the asymmetry of the SpectatorsProbing the asymmetry of the Spectators





)ZNexp(C

Z,NY

Z,NY

1

2

Tsang et. al.,PRL 92, 062701 (2004)1.0

0.33

-0.33

-1.0

Ri()

Liu et al.PRC 76, 034603 (2007).

Quantitative valuesQuantitative values

• Reactions:– 124Sn+112Sn: diffusion– 124Sn+124Sn: neutron-rich limit– 112Sn+112Sn: proton-rich limit

• Exchanging the target and projectile allowed the full rapidity dependence to be measured.

• Gates were set on the values for Ri() near beam rapidity. – Ri() 0.470.05 for 124Sn+112Sn

– Ri() -0.44 0.05 for 112Sn+124Sn

• Obtained similar values for Ri(ln(Y(7Li)/ Y(7Be))– Allows exploration of dependence

on rapidity and transverse momentum.

• Reactions:– 124Sn+112Sn: diffusion– 124Sn+124Sn: neutron-rich limit– 112Sn+112Sn: proton-rich limit

• Exchanging the target and projectile allowed the full rapidity dependence to be measured.

• Gates were set on the values for Ri() near beam rapidity. – Ri() 0.470.05 for 124Sn+112Sn

– Ri() -0.44 0.05 for 112Sn+124Sn

• Obtained similar values for Ri(ln(Y(7Li)/ Y(7Be))– Allows exploration of dependence

on rapidity and transverse momentum.

Liu et al., (2006)

v/vbeam

Liu et al.P

RC

76, 034603 (2007).

richotonrichNeutron


Pr

Pr 2/)(2)(

iR

Comparison to QMD calculationsComparison to QMD calculations

• IQMD calculations were performed for i=0.35-2.0, Sint=17.6 MeV.

• Momentum dependent mean fields with mn*/mn =mp*/mp =0.7 were used. Symmetry energies: S() 12.3·(ρ/ρ0)2/3 + 17.6· (ρ/ρ0) γi

• IQMD calculations were performed for i=0.35-2.0, Sint=17.6 MeV.

• Momentum dependent mean fields with mn*/mn =mp*/mp =0.7 were used. Symmetry energies: S() 12.3·(ρ/ρ0)2/3 + 17.6· (ρ/ρ0) γi

• Experiment samples a range of impact parameters – b5.8-7.2 fm.– larger b, smaller i

– smaller b, larger i

• Experiment samples a range of impact parameters – b5.8-7.2 fm.– larger b, smaller i

– smaller b, larger i

mirror nuclei

Tsang et al, PRL. 102, 122701 (2009)

Go to constraints

Expansion around r0: Symmetry slope L & curvature Ksym

Symmetry pressure Psym

...183

2

0

0

0

0

BsymB

osym

KLSE sym

B

sym PE

LB

00

33

0

Diffusion is sensitive to S(0.4), which corresponds to a contour in the (S0, L) plane.

Diffusion is sensitive to S(0.4), which corresponds to a contour in the (S0, L) plane.

fits to IASmasses

fits to

ImQMDCONSTRAINTS

Tsang et al., PRL 102, 122701 (2009).

Danielewicz, Lee,

NPA 818, 36 (2009)

Danielewicz, Lee,

NPA 818, 36 (2009)

GDR: Trippa,

PRC77, 061304

GDR: Trippa,

PRC77, 061304

PDR: Carbone et al, PRC 81,

041301 (2010) ;

A. Klimkiewicz,

PRC 76, 051603 (2007).

PDR: Carbone et al, PRC 81,

041301 (2010) ;

A. Klimkiewicz,

PRC 76, 051603 (2007).

Sn neutron skin thickness:

Chen et al., PRC C 82, 024321 (2010)

Sn neutron skin thickness:

Chen et al., PRC C 82, 024321 (2010)

• 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.

• 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.

• Densities of 20 can be achieved at E/A400 MeV.– Provides information about neutron star radii, 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 greater in neutron-rich dense region is formed for stiffer S().

• Densities of 20 can be achieved at E/A400 MeV.– Provides information about neutron star radii, 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 greater in neutron-rich dense region is formed for stiffer S().

Asymmetry term studies at 20

(Unique contribution from collisions investigations)

Asymmetry term studies at 20

(Unique contribution from collisions investigations)

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

Comparisons of neutron and proton observables :Comparisons of neutron and proton observables :

• Most models predict the differences between neutron and proton flows and t and 3He flows to be sensitive to the symmetry energy and the n and p effective mass difference.

• In this prediction, the ratio of neutron over proton spectra out of the reaction plane displays a significant sensitivity the symmetry energy.

• Comparisons of elliptical flow of neutrons and protons have been recently published for Au+Au collisions at 400 MeV/u by P. Russotto et al., PLB 697 (2011) 471–476. See talk by Lemmon.

• Most models predict the differences between neutron and proton flows and t and 3He flows to be sensitive to the symmetry energy and the n and p effective mass difference.

• In this prediction, the ratio of neutron over proton spectra out of the reaction plane displays a significant sensitivity the symmetry energy.

• Comparisons of elliptical flow of neutrons and protons have been recently published for Au+Au collisions at 400 MeV/u by P. Russotto et al., PLB 697 (2011) 471–476. See talk by Lemmon.

B.-A

. Li et al., P

hys. Rep. 464 (2008) 113.

soft

soft

• Such measurements have been done recently at GSI:− P. Russotto et al.

• Such measurements have been done recently at GSI:− P. Russotto et al.

High density probe: pion productionHigh density probe: pion production

• Larger values for n/ p at high density for the soft asymmetry term (x=0) causes stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1).

• - /+ means Y(-)/Y(+)– In delta resonance model,

Y(-)/Y(+)(n,/p)2

– In equilibrium,

(+)-(-)=2( p-n)

• The density dependence of the asymmetry term changes ratio by about 10% for neutron rich system.

• Larger values for n/ p at high density for the soft asymmetry term (x=0) causes stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1).

• - /+ means Y(-)/Y(+)– In delta resonance model,

Y(-)/Y(+)(n,/p)2

– In equilibrium,

(+)-(-)=2( p-n)

• The density dependence of the asymmetry term changes ratio by about 10% for neutron rich system.

soft

stiff

Li et al., arXiv:nucl-th/0312026 (2003).

stiff

soft

• Investigations are planned with stable or rare isotope beams at the MSU and RIKEN.– Sensitivity to S() occurs primarily

near threshold in A+A

• Investigations are planned with stable or rare isotope beams at the MSU and RIKEN.– Sensitivity to S() occurs primarily

near threshold in A+A

t (fm/c)

- /+

Zhang et al., arX

iv:0904.0447v2 (2009)

Choice of beams for pion ratiosChoice of beams for pion ratios

Xiao, et al., arX

iv:0808.0186 (2008)

Reisdorf, et al., N

PA 781 (2007) 459.

• Sensitivity to symmetry energy is larger for neutron-rich beams• Sensitivity increases with decreasing incident energy.• Data have been measured for Au+Au collisions.• It would be interesting to measure with rare isotope beams such as 132Sn and

108Sn.– Interesting comparison because the Coulomb interaction is the same for

both to first order. Coulomb also strongly influences the pion ratios.

• Sensitivity to symmetry energy is larger for neutron-rich beams• Sensitivity increases with decreasing incident energy.• Data have been measured for Au+Au collisions.• It would be interesting to measure with rare isotope beams such as 132Sn and

108Sn.– Interesting comparison because the Coulomb interaction is the same for

both to first order. Coulomb also strongly influences the pion ratios.

Zhigang Xiao, et al, PRL, 102, 062502 (2009)

Au+Au

Analysis of Au+Au data at E/A=400 MeV suggest very weak density dependence at >30

Analysis of Au+Au data at E/A=400 MeV suggest very weak density dependence at >30

?MSU

GSI

Isospin diffusion, n-p flow

Pion production

Xiao, et al., arX

iv:0808.0186 (2008)

Reisdorf, et al., N

PA 781 (2007) 459.

• The SAMURAI TPC would be used to constrain the density dependence of the symmetry energy through measurements of:– Pion production – Flow, including neutron flow

measurements with the nebula array. • The TPC also can serve as an active target

both in the magnet or as a standalone device.– Giant resonances.– Asymmetry dependence of fission

barriers, extrapolation to r-process.

• The SAMURAI TPC would be used to constrain the density dependence of the symmetry energy through measurements of:– Pion production – Flow, including neutron flow

measurements with the nebula array. • The TPC also can serve as an active target

both in the magnet or as a standalone device.– Giant resonances.– Asymmetry dependence of fission

barriers, extrapolation to r-process.

Device: SAMURAI TPC (U.S. Japan Collaboration) T. Murakamia, Jiro Muratab, Kazuo Iekib, Hiroyoshi Sakuraic, Shunji Nishimurac, Atsushi Taketanic, Yoichi Nakaic, Betty Tsangd, William Lynchd, Abigail Bickleyd,

Gary Westfalld, Michael A. Famianoe, Sherry Yennellog, Roy Lemmonh, Abdou Chbihii, John Franklandi, Jean-Pierre Wieleczkoi , Giuseppe Verdej, Angelo Paganoi,

Paulo Russottoi, Z.Y. Sunk, Wolfgang Trautmannl

aKyoto University, bRikkyo University, cRIKEN, Japan, dNSCL Michigan State University, eWestern Michigan University, gTexas A&M University, USA, hDaresbury Laboratory, iGANIL,

France, UK, jLNS-INFN, Italy, kIMP, Lanzhou, China, lGSI, Germany

Device: SAMURAI TPC (U.S. Japan Collaboration) T. Murakamia, Jiro Muratab, Kazuo Iekib, Hiroyoshi Sakuraic, Shunji Nishimurac, Atsushi Taketanic, Yoichi Nakaic, Betty Tsangd, William Lynchd, Abigail Bickleyd,

Gary Westfalld, Michael A. Famianoe, Sherry Yennellog, Roy Lemmonh, Abdou Chbihii, John Franklandi, Jean-Pierre Wieleczkoi , Giuseppe Verdej, Angelo Paganoi,

Paulo Russottoi, Z.Y. Sunk, Wolfgang Trautmannl

aKyoto University, bRikkyo University, cRIKEN, Japan, dNSCL Michigan State University, eWestern Michigan University, gTexas A&M University, USA, hDaresbury Laboratory, iGANIL,

France, UK, jLNS-INFN, Italy, kIMP, Lanzhou, China, lGSI, Germany

Nebula scintillators

SAMURAI dipole

TPC

Q

MSU:Active Target Time Projection ChamberD Bazin, M. Famiano, U. Garg, M. Heffner, R. Kanungo, I. Y. Lee,

W.Lynch, W. Mittig, L. Phair, D Suzuki,G. Westfall

MSU:Active Target Time Projection ChamberD Bazin, M. Famiano, U. Garg, M. Heffner, R. Kanungo, I. Y. Lee,

W.Lynch, W. Mittig, L. Phair, D Suzuki,G. Westfall

• Two alternate modes of operation• Fixed Target Mode with target wheel inside chamber:

– 4 tracking of charged particles allows full event characterization– Scientific Program » Constrain Symmetry Energy at >0

• Active Target Mode:– Chamber gas acts as both detector and thick target (H2, D2, 3He, Ne, etc.) while

retaining high resolution and efficiency– Scientific Program » Transfer & Resonance measurements, Astrophysically

relevant cross sections, Fusion, Fission Barriers, Giant Resonances

• Two alternate modes of operation• Fixed Target Mode with target wheel inside chamber:

– 4 tracking of charged particles allows full event characterization– Scientific Program » Constrain Symmetry Energy at >0

• Active Target Mode:– Chamber gas acts as both detector and thick target (H2, D2, 3He, Ne, etc.) while

retaining high resolution and efficiency– Scientific Program » Transfer & Resonance measurements, Astrophysically

relevant cross sections, Fusion, Fission Barriers, Giant Resonances

120 cm

SummarySummary

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

– Isoscalar giant resonances , kaon production and high energy flow measurements have placed constraints on the symmetric matter EOS, but the EOS at large isospin asymmetries is 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, pigmy dipole , monopole and isovector giant

resonances, • isospin diffusion, • proton vs. neutron emission and - vs. + emission• neutron-proton correlations.

– 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 probably have not been discovered.

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

– Isoscalar giant resonances , kaon production and high energy flow measurements have placed constraints on the symmetric matter EOS, but the EOS at large isospin asymmetries is 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, pigmy dipole , monopole and isovector giant

resonances, • isospin diffusion, • proton vs. neutron emission and - vs. + emission• neutron-proton correlations.

– 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 probably have not been discovered.

(,0,) = (,0,0) + d2S() ; d = (n- p)/ (n+ p) = (N-Z)/A(,0,) = (,0,0) + d2S() ; d = (n- p)/ (n+ p) = (N-Z)/A

Documents

A.What is the EOS? 0. Relationship to energy and to nuclear masses 1. Important questions B. What observables are sensitive to the EOS and