Towards the Equation of State of Dense Asymmetric Nuclear Matter

Towards the Equation of State of Dense Asymmetric Nuclear Matter

• Present constraints on the EOS.

• Relevance to dense astrophysical objects:

• Probes of the asymmetry term of the EOS.

• Current investigations:–Isoscaling—scaling of isotope ratios

–Sensitivity to EOS

• Probing high density matter.

• Summary and Outlook.

Outline

• The density dependence of asymmetry term is largely unconstrained.

• Pressure, i.e. EOS is rather uncertain even at 0.

Isospin Dependence of the Nuclear Equation of State

E/A (,) = E/A (,0) + 2S()

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

-20

-10

0

10

20

30

0 0.1 0.2 0.3 0.4 0.5 0.6

EOS for Asymmetric Matter

Soft (=0, K=200 MeV)asy-soft, =1/3 (Colonna)asy-stiff, =1/3 (PAL)

E/A

(M

eV)

(fm-3)

=(n-

p)/(

n+

p)

PAL: Prakash et al., PRL 61, (1988) 2518.Colonna et al., Phys. Rev. C57, (1998) 1410.

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

• Microscopic theory provides incomplete guidance regarding the extrapolation away from saturation density.

Constraints on S() from microscopic models

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5

Total Symmetry Energy

ParisBL

AV14

UV14

AV18

S(

) M

eV

(fm-3

)

I.Bom

baci (2000)

BHF

Varia-tional

Nucleus-nucleus collision provide the only terrestrial means to probe nuclear matter under controlled conditions.

What has been learned about the EOS of dense symmetric matter?

What is the relevance to dense objects? How can one probe the asymmetry term?

• Prospects are good for improving constraints further.

• Relevant for supernovae - what about neutron stars?

Determination of EOS from nucleus-nucleus collisionsWhat is known about the symmetric matter EOS?

• Measurements are constraining symmetric matter EOS at >2 0.

– Observables: transverse, elliptical flow.

y

px

y

Apx

/D

anielewicz et al., (2002)

Danielewicz et al., (2002)

Extrapolation to neutron stars

• Uncertainty due to the density dependence of the asymmetry term is greater than that due to symmetric matter EOS.

• Macroscopic properties:– Neutron star radii, moments of

inertia and central densities.– Maximum neutron star masses

and rotation frequencies.• Proton and electron fractions

throughout the star.– Cooling of proton-neutron star.

• Thickness of the inner crust.– Frequency change

accompanying star quakes.• Role of Kaon condensates and

mixed quark-hadron phases in the stellar interior.

Danielewicz et al., (2002) Symmetry term influences:

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

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

Probing the asymmetry term

• “First order”: depend on isospin of detected particles.– Prequilibrium particle

emission.– Asymmetry of bound vs.

emitted nucleons. – Transverse flow.– Isospin dependencies of pion

production.

• “Second order”: do not depend on isospin of detected particles. • Sign of mean field opposite for

protons and neutrons.• Shape is influenced by

incompressibility..

-100

-50

0

50

100

0 0.5 1 1.5 2

Li et al., PRL 78 (1997) 1644 V

asy

(MeV

)

/o

NeutronProton

F1=2u2/(1+u)

F2=u

F3=u

F1F2

F3

u =

stiff

soft

Observables:

Studies of asymmetry term at 0.• Direct measurements of n vs. proton emission rates and transverse flows - Probes the pressure from asymmetry term at saturation density and below.

– Predictions of transport theory:– Similar effects in t vs. 3He flow, etc?

• Measurements will be performed at NSCL in July 2003.

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100

112Sn+112SnF1F3

(dN

n/dE

)/(d

Np/d

E)

E (MeV)

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100

132Sn+132SnE/A=50 MeV

1<b<5 fm

F1F3

(dN

n/dE

)/(d

Np/d

E)

E (MeV)

stiff asymmetry term

soft asymmetry term

Bao-An Li (2000)

Studies of asymmetry term at 0.• Complimentary measurements of the asymmetry of bound matter.

Experimental study:– Central 112Sn+112Sn, 124Sn+124Sn collisions at E/A=50 MeV.

– Measure isotopic distributions of multifragmentation products.

pre-equilbrium nucleons

multifragmentfinal state bound matter

Experimental investigations of isospin effects in fragmentation

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

Provides good coverage of central collisions: 2.0b

ycm

Liu et al., (2003)

Expected dependencies on asymmetry termFor a neutron rich system at :

asy-soft (F3) more symmetricdense region

neutron-rich emitted particles

N/Zres=Ntot/Ztotasy-stiff (F1)

N/ZemNtot/Ztot

EOS Residue N/Z

F_3 (asy-soft) 95/77=1.23

EOS Residue N/Z

F_1 (asy-stiff) 102/71=1.44

BUU predictions for central 124Sn+ 124Sn (N0/Z0=1.48) collisions at E/A=50 MeV

Comparison of isotopic distributions

• ISMM decay of BUU residue: b=1fm, E*/A=4MeV, /0=1/6• There is a sensitivity to the density dependence of the

asymmetry term of the EOS.

124Sn+124Sn Central Collisions at E=50AMeV

Asy-stiff F1 Asy-soft F3(N/Z=1.44) (N/Z=1.23)

exp.final

10-2

10-1

1Li Be

10-2

10-1

1

Yie

ld

B C

10-3

10-2

10-1

1

-2 0 2 4

N-Z

N

-2 0 2 4 6

O

10-2

10-1

1Li Be

10-2

10-1

1

Yie

ld

B C

10-3

10-2

10-1

1

-2 0 2 4N-Z

N

-2 0 2 4 6

O

Tan (2001)

R21=Y2/ Y1

TZN pne/)( Zp

Nn

Scaling behavior of Isotopic Ratios

• Compact parameterization of total isospin dependence.

• Factorization of yields into p & n densities

• Slopes are insensitive to sequential decay calculations- Slopes are sensitive to masses,

which are inaccurately calculated in many models.

Relative Isotope Ratios in SMM models

Xu P

RL

(2000)

Tsang P

RC

(2001)

• What do the isoscaling parameters reveal about the asymmetry term?

Relative Neutron(Proton) Densities

• and are not sensitive to secondary decays.• Sdfsdf increases more rapidly than (N/Z)0 fractionation. • Comparison favors the stiffer asymmetry term.• Similar conclusions obtained from comparisons of mirror nuclei.

n̂ p̂ pn ˆ/ˆ

Tan et al. P

RC

(2001)

Model dependence of fragment isotopic distributions.

• Few precise calculations:– ISMM, EES, AMD and SMF.

• Existing calculations break into two groups:– Bulk multifragmentation models predict that asy-soft EOS favors

more symmetric fragments:

– One important surface emission model (EES model), however, predicts that asy-soft EOS favors neutron rich fragments

This occurs because such models assume that fragments are at similar low freezeout density:

EES model assume that residue is at low density:

EES model assumes that fragments are at normal density:

Predictions of EES model

• Theoretically:

• Separation energies depend on density dependence asymmetry term:

• Strong influence of symmetry term on fragment isotopic ratios.

– trends are opposite to those of the equilibrium SMM approach.

)T/)]ZZ(e

sZsNexp([C)Z,N(R

tot

pn21

0MeV4.23)(S

Use scaling to simplify representation:

Zp21 ˆ/)Z,N(R)N(S

Tsang et al., PRL (2001)


• Theoretically:




)T/)]ZZ(e

sZsNexp([C)Z,N(R

tot

pn21

0MeV4.23)(S


Zp21 ˆ/)Z,N(R)N(S


Data

Test: Measurements of isotopically resolved spectra• Energy spectra reflect the cooling dynamics.

– high energies reflect higher temperatures, earlier times

• Emission rates are temperature (and therefore time dependent).

– More strongly bound particles are preferentially emitted at low temperature. (B=BEpar-BEdau-BEf)

• Data display the expected trends, which were previously observed only for helium and beryllium isotopes.

0.01

0.1

1

10

100

0 10 20 30 40 50

Energy Spectra: Cooling effects

T=1 MeVT=2 MeVT=3 MeVT=4 MeVT=5 MeVT=6 MeVSum

Yie

ld (

arbi

trar

y un

its)

K.E. (MeV)

)exp(

),(

T

EBE

EtR

em

emi

Eem (MeV)

0.01

0.1

1

0 20 40 60 80 100 120 140 160

112Sn+112Sn E/A=50 MeV

11C12C

dM/d

dE

(M

eV-1

)

Ecm

(MeV)

Data

Liu et al., (2003)

New results supporting the surface emission picture

• Expectation: Kinetic energy reflects simply the thermal and collective contributions (neglecting recoil effects).

• Actual trends suggest cooling phenomena that can be reproduced by EES model calculations in which fragments are emitted from surface of expanding and cooling system.

2.. 2

1collNCoulthermalmc vmAeZEE

112Sn+ 112Sn, E/A=50 MeV central collisions

Liu, F

riedman, S

ouza (2003)

Data

SMMcarbon

EES

Current Status: studies of asymmetry term for 0

• n vs. p emission rates: all models predict consistent qualitative trends. Such measurements provide the least ambiguous information about the

density dependence of the asymmetry term. (expect first results in early 2004)

• Fragment isocaling parameters: there is a model dependence for the sensitivity to the asymmetry term. Present constraints on the asymmetry term are model dependent. The combination of n vs. p rates with isoscaling data will rule out at least

one commonly employed multi-fragmentation model and thereby further our understanding of the liquid-gas phase transition.

)(2

112112124124

112112124124

RR

New observable: isospin diffusion in peripheral collisions

• Vary isospin driving forces by changing the isospin of projectile and target.

• measure the scaling parameter for projectile decay.

• isospin equilibrium is not achieved in peripheral collisions.

• Since, , renormalize to symmetric collisions:

ta rg e t

p ro je c tilesymmetric systemno diffusion

asymmetric systemweak diffusion

asymmetric systemstrong diffusion

neutron richsystem

proton richsystem

Measure scaling parameter

Tsang et al., (2003)

Rami et al., PRL, 84, 1120 (2000)

)(2

112112124124

112112124124

RR





• Since, , renormalized to symmetric collisions:

ta rg e t




neutron richsystem

proton richsystem



Rami et al., PRL, 84, 1120 (2000)

)(2

112112124124

112112124124

RR





• Since, , renormalize to symmetric collisions:

ta rg e t




neutron richsystem

proton richsystem



Rami et al., PRL, 84, 1120 (2000)

asy-stiff

asy-soft

Studies of asymmetry term at >0.• High density behavior of asymmetry terms can be group into two classes: i.e.

increasing or decreasing with density.

stronger density dependence (asy-stiff)

weaker density dependence (asy-soft) typical of many Skyrme EOS’s or some variational calculations (UV14+UV11)

Influences significantly the inner structure of neutron star

Bao-An Li NPA 2002

Observables: pion yields• Softer asymmetry term favors neutron rich dense regions

and larger relative -/+ yield ratios:

dashed:asy-soft

solid:asy-stiff

neut

ron

rich

neut

ron

defi

cien

t

incr

easi

ng

- /+

Bao-An Li NPA 2002Bao-An Li NPA 2002

Conclusion• Significant constraints on symmetric matter EOS have

been obtained.

• Density dependence of symmetry energy can be examined experimentally.

• Observed isoscaling laws

• Conclusions from fragmentation

work are model dependent:

– BUU-SMM favors 2 dependence of S().

– SMF favors 2 dependence but isotope distributions are poorly reproduced

– EES favors 2/3 dependence of S().

• Signals have been identified for measurements of asymmetry term at higher densities 0-3.50 :

– Isospin dependence of n vs. p flow, pion yields.

-1 0 0

-5 0

0

5 0

1 0 0

0 0 .5 1 1 .5 2

L i et a l., P R L 78 (199 7 ) 1 644

Vas

y(M

eV)

N eu tro nP ro to n

3

A sy-stiff F1

A sy-so ft F

Acknowledgements

W.P. Tan, R. Donangelo, W.A. Friedman, C.K. Gelbke, T.X.

Liu, X.D. Liu, Bao-An Li, W.G. Lynch, A. Vander Molen, S.R.

Souza, M.B. Tsang, M.J. Van Goethem, G. Verde, A. Wagner,

H.F. Xi, H.S. Xu, L. Beaulieu, B. Davin, Y. Larochelle, T.

Lefort, R.T. de Souza, R. Yanez, V. Viola, R.J. Charity, L.G.

Sobotka

Some examples

• 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.

p+e- n+ n p+e-+

0/ );(. uuFconstS pot

Neutron star radii: Cooling of proto-neutron stars:

S tan d a rd co o lin gD irec t U R C AIso th e rm a l

Test: Measurements of isotopically resolved spectra• Energy spectra reflect the cooling dynamics.


• Emission rates are temperature (and therefore time dependent).– More strongly bound particles are

preferentially emitted at low temperature. (B=BEpar-BEdau-Bem)

0.01

0.1

1

10

100

0 10 20 30 40 50



Yie

ld (

arbi

trar

y un

its)

K.E. (MeV)

)T

EBexp(E

E

EBEE

)E,t(R

em

*parpar

em*pardau

emi

Eem (MeV)

0.01

0.1

1

0 20 40 60 80 100 120 140 160

112Sn+112Sn E/A=50 MeV

11C12C

dM/d

dE

(M

eV-1

)

Ecm

(MeV)

Data

• Theoretically:




)T/)]ZZ(e

sZsNexp([C)Z,N(R

tot

pn21


Zp21 ˆ/)Z,N(R)N(S

=0

=1

Data


0MeV4.23)(S


• Theoretically:




0MeV4.23)(S

)T/)]ZZ(e

sZsNexp([C)Z,N(R

tot

pn21


Zp21 ˆ/)Z,N(R)N(S



Isoscaling laws for isotopic distributions

• Basic trends from Grand Canonical ensemble:– Yields relevant term in partition function.

• Ratios to reduce sensitivity to secondary decays:

• Scaling parameters C, ,

),(),(),(

)/exp(12

),(/),(exp),(*

int

int

ZNfZNYZNY

TEJZwhere

ZNZTZNBZNZNY

HOTCOLD

iii

pnHOT

feeding correction

TZ

pNn

TZTN

C

CZNY

ZNYZNR

/12

//

1

221

e ,ˆ ;ˆˆ

e),(

),(, pn

n̂ p̂

Pressure and collective flow dynamics

• The blocking by the spectator matter provides a clock with which to measure the expansion rate.

pressure contours

density contours

n vs. p flow differences

• The exploration of the density dependent asymmetry will be an important objective at RIA.

y

px

dashed:asy-soft

solid:asy-stiff

differences between neutron and proton flow values

Bao-A

n Li N

PA

2002

Magnitude of the effect?

• Figure shows the predicted ratio of neutron and proton center of mass spectra for two different asymmetry terms.

• There is an uncertainty regarding the expected magnitude of the effect.

• Open question whether these differences stem from differences in the models (QMD vs. BUU), the numerics of the simulations, or differences in the asymmetry terms.

• More theoretical work quantifying the signature is needed.

• Bao-al Li: spectra are in the lab frame and integrated over angle.• Above spectra are in the C.M. frame, angular cut (?)....

2

2.2

2.4

2.6

2.8

3

3.2

0 20 40 60 80 100

S() dependence from QMD

=0.5=1.5

n(Ecm

)/ p(E

cm)

Ecm

(MeV)

QMD calc.

Zhuxia L

i 2002

Current Status: studies of asymmetry term for 0

• n vs. p emission rates: all models predict consistent qualitative trends. Such measurements provide the least ambiguous information about the

density dependence of the asymmetry term. (expect first results in early 2004)

• Fragment isocaling parameters: there is a model dependence for the sensitivity to the asymmetry term. Present constraints on the asymmetry term are model dependent. The combination of n vs. p rates with isoscaling data will rule out at least

one commonly employed multi-fragmentation model and thereby further our understanding of the liquid-gas phase transition.

Isotopic scaling as a general phenomenon

• Additional statistical mechanisms:– Deep-inelastic

– Evaporation:

• Note that these simple scaling laws require the two systems to be at a common temperature.

– There are also generalized scaling laws systems at different temperatures.

)T/]sZsNexp([C)Z,N(R

.systQ

pn21

.s.g

)T/)]ZZ(e

sZsNexp([C)Z,N(R

.Weiss

tot

pn21

Zp21 ˆ/)Z,N(R)N(S


Extrapolation of isotope distributions

• Isoscaling laws provide a straightforward scaling of isospin dependence of isotope distributions :– Other three reactions are

scaled by applying the isoscaling parameters to the 124Sn+124Sn data.

– Parameters depend linearly on =(N-Z)/A

Y2 =R21Y1

1T/)ZN( Ye pn

1Z

pN

n YC

1ZN Ye

R21=Y2/ Y1

TZN pne/)( Zp

Nn


• Note: you can reduce the isotopic lines to a single line by removing the Z dependence as follows:

• Could do the same thing for the isotonic dependence if desired.

Xu P

RL

(2000)

Zp21 ˆ/)Z,N(R)N(S

Yield Ratios of Mirror Nuclei

• Comparison favors the stiffer asymmetry term.

Tan et al., PRC (2001)

Ratios of Mirror nuclei

excited

afterdecay

R21=Y2/ Y1

TZN pne/)( Zp

Nn


• Note: you can reduce the isotopic lines to a single line by removing the Z dependence as follows:

• Could do the same thing for the isotonic dependence if desired.

Xu P

RL

(2000)

Zp21 ˆ/)Z,N(R)N(S


• Theoretically:


• Strong influence of symmetry term on fragment isotopic ratios.– trends are opposite to those

of the equilibrium SMM approach.


Zp21 ˆ/)Z,N(R)N(S

0MeV4.23)(S


)T/)]ZZ(e

sZsNexp([C)Z,N(R

tot

pn21

=0

=1

Data

One Key observable: n.vs.p emission rates

N/Zres=Ntot/Ztot more symmetricdense regionasy-stiff (F1) asy-soft (F3)

N/ZemNtot/Ztot neutron-rich emitted particles

For a neutron rich system:

Probing the asymmetry term

• “First order”: depend on isospin of detected particles.

– Prequilibrium particle emission• Asy_soft: <En> > <Ep> *

• Asy_stiff: <En> <Ep> *– Asymmetry of bound vs. emitted

nucleons. • Asy_soft: (N/Z)EM>N/Z)BND.

• Asy_stiff: (N/Z)EM(N/Z)BND.

– Transverse flow• Asy_soft: Fp Fn *

• Asy_stiff: Fp > Fn *

• “Second order”: do not depend on isospin of detected particles. • Sign of mean field opposite for

protons and neutrons.• Shape is influenced by

incompressibility..

-100

-50

0

50

100

0 0.5 1 1.5 2

Li et al., PRL 78 (1997) 1644 V

asy

(MeV

)

/o

NeutronProton

F1=2u2/(1+u)

F2=u

F3=u

F1F2

F3

u =

stiff

soft

Observables: (neutron rich systems)

* Different for neutron deficient systems

One Key observable: n.vs.p emission rates

N/Zres=Ntot/Ztot more symmetricdense regionasy-stiff (F1) asy-soft (F3)

N/ZemNtot/Ztot neutron-rich emitted particles

For a neutron rich system:

What may be wrong with the SMF calculations? • Neglect of impact parameter averaging?• Neglect of dynamical cluster emission?

• What is needed in terms of the primary distributions?– Comparison with SMM calculations suggests that widths of SMF primary distributions need to be broader and the excitation energies lower.

Expanding Emitting Source (EES) Model

• Fragment emission rates dramatically increase for /where fragment separation energies sF become negative. – Separation energies (and

therefore) rates depend on the density dependence of asymmetry term.

W.A. Friedman Phys. Rev. C 42 (1990) 667.

10 MeV

11 MeV

12 MeV

13 MeV

14 MeV15 MeV

Decay of Au Nucleus

Weisskopf emission rates:

)T/)sE(exp(E

EconstdEdt

dN

F

parent

daughterINV

Problems with masses in standard SMM

• Effects appear small but are sufficient to change scaling parameters for primary fragments by nearly a factor of two and secondary fragments by 35%.

disagreement withknown masses is large

Tan et al., (2002)

Tan et al., (2002)

• The density dependence of asymmetry term is largely unconstrained.

Isospin Dependence of the Nuclear Equation of State

E/A (,) = E/A (,0) + 2S()

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

-20

-10

0

10

20

30

0 0.1 0.2 0.3 0.4 0.5 0.6

EOS for Asymmetric Matter

Soft (=0, K=200 MeV)asy-soft, =1/3 (Colonna)asy-stiff, =1/3 (PAL)

E/A

(M

eV)

(fm-3)

=(n-

p)/(

n+

p)

PAL: Prakash et al., PRL 61, (1988) 2518.Colonna et al., Phys. Rev. C57, (1998) 1410.

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

• Microscopic theory provides incomplete guidance regarding the extrapolation away from saturation density.

Constraints on S() from microscopic models

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5

Total Symmetry Energy

ParisBL

AV14

UV14

AV18

S(

) M

eV

(fm-3)

I.Bom

baci (2000)

BHF

Varia-tional

Pressure and collective flow dynamics

• The blocking by the spectator matter provides a clock with which to measure the expansion rate.

pressure contours

density contours

Some examples

• 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.p+e n+ n p+e+

0/ );(. uuFconstS pot

Neutron star radii: Cooling of proto-neutron stars:

S tan d a rd co o lin gD irec t U R C AIso th e rm a l

• Time sequence:– System expands.– Fragments form.– Fragments decouple.

• Time dependencies:– Initial compression and energy

deposition.– Expansion.– Disassembly and freezeout.

• Different approaches– Equilibrium at freezeout density

(BUU-SMM). – Rate equations (EES)– Dynamical (BNV)

Multifragmentation Scenario

EOS relevant

Experimental investigations of isospin effects in fragmentation

• Experimental requirements:– 4 detector with significant

efficiency for isotope resolution.

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

Extrapolation of isotope distributions

• Isoscaling laws provide a straightforward scaling of isospin dependence of isotope distributions :– Other three reactions are

scaled by applying the isoscaling parameters to the 124Sn+124Sn data.

– Parameters depend linearly on =(N-Z)/A

Y2 =R21Y1

1T/)ZN( Ye pn

1Z

pN

n YC

1ZN Ye

R21=Y2/ Y1

ZNe ZpN

n

Removal of Z dependence

• Use Isotope and Isotone dependence to determine and .

• Divide out Z dependence.

N

nN

Zp21

ˆe

ˆ/)Z,N(R)N(S

Cooling effects in weakly expanding systems?• Weakly bound fragment is more

abundantly produced at high energies, early times cooling

effects.

0

2

4

6

8

10

0 5 10 15 20 25

TEMPERATURE

THeH

THeLi

TCLI

Tis

o (M

eV

)E/A

cm (MeV)

isotopetemperatures

100

1000

104

105

106

0 5 10 15 20

Carbon Isotopes

11C12C

Yie

ld (

arbi

trar

y un

its)

E/Acm

(MeV)

10

100

1000

104

105

106

107

0 10 20 30 40 50 60

Helium Isotopes

3He3

He

6He6

Yie

ld (

arbi

trar

y un

its)

E/Acm

(MeV)

3He4He6He

• Cooling via expansion and radiation prior to breakup:

• Basic mechanisms are contained transport codes and in EES model of Friedman.

– Question: is the system thermalized at breakup?

Various Cooling stages

• Cooling of hot fragments in isolation via secondary decay:– Can be accommodated

within equilibrium framework.

• Cooling between breakup and freezeout:

1

10

100

0.01 0.1 1

Friedman. PRL 60, 2125 (1988)

Tmax

=25 MeV

Tmax

=20 MeV

Tmax

=15 MeV

Tm

ax=

25 M

eV

/0

t increasing

Non-equilibrium features of C.N. decay• Energy spectra reflect the cooling dynamics.


• Emission rates are temperature (and therefore time dependent).– More strongly bound particles are

preferentially emitted at low temperature. (B=BEpar-BEdau-Bem)

0.01

0.1

1

10

100

0 10 20 30 40 50



Yie

ld (

arbi

trar

y un

its)

K.E. (MeV)

)T

EBexp(E

E

EBEE

)E,t(R

em

*parpar

em*pardau

emi

Eem (MeV)

3He-4He relative emission rates:• B(3He)>>B(4He) relative emission rates are temperature dependent.

Comparison with compound nucleardecay calculations reveal this slope difference to be a direct signature of evaporative cooling dynamics.

Evaporative decay

dtHe3He4HeH Y/Y/Y/YR

Towards the Equation of State of Dense Asymmetric Nuclear Matter

Documents

asymmetry of bound matter

studies of asymmetry

symmetric matter eos

eos of dense symmetric

asymmetry termfor

n p n p

determination of eos

asymmetry termfirst