Zhuxia Li (China Institute of Atomic Energy)

Collaborators: Yinxun Zhang (CIAE), Qingfen Li (ITP),

Ning Wang(ITP)

Probing the density dependence of the symmetry potential

2004.8 Weihai 2

Outline

1) Equation of state for asymmetric nuclear matter2) Probing the density dependence of the symmetry potential at low densities3) Probing the density dependence of the symmetry potential at high densities

2004.8 Weihai 3

I. Equation of State for asymmetry nuclear matter

)0,(),( 0 EE ),()( 42 OEsym

Empirical parabolic law:

Esym(ρ)=E(ρ,neutron matter) -E(ρ,symmetric matter)

MeV

MeVEK symsym 460

400)(9

0

2

222

icrelativistMeV

icrelativistnonMeVEa sym

4035

3827

2

1

0

2

22

4

pn

pn

2004.8 Weihai 4

EOS for Asymmetric Nuclear Matter

EOS of Neutron matter for 18 Skyrme Parameter sets( B. Alex Brown, PRL85 5296)

Esym(ρ)<0 when ρ>3ρ0

extreme variation is observed Other interactions such as Gogny,density dependent M3Y also give either positive or negative symmetry energies at high densitiesThe sign of symmetry energy at ρ>3ρ0 is very uncertain. At ρ~0.5ρ0 Esym is variant. Even at normal density the values of Esym(symmetry energy coefficient) are different for different interactions.

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The implication of the Esym(ρ) in astrophysics:

a) Nucleosynthesis in pre-supernova evolution of massive starb) Mechanism of supernova explosionc) Composition of protoneutron stard) Cooling mechanism of protoneutron starse) Kaon condensation of neutron starsf) Quark-hadron phase transition in neutron starsg) Mass-radius correlation of neutron starsh) Isospin separation instability and structure of neutron stars Refs. H.A.Bethe, Rev.of Mod. Phys. 62(1990)801 C.J. Pethick and D.G. Ravenhall, Annu.Rev.Nucl.Part.Sci.85(95)429

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Obtaining more accurate information of the symmetry potential is highly requisite

By nuclear structure:the accurate measurements of of Pb,Snisovector giant resonance…

pn rr

MeVaMeV 3634 4

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Find sensitive observables to the density dependence of symmetry potential

Study dynamical effect of symmetry potentialon the reaction mechanism

By heavy ion collisions: The matter of

various density and isospin asymmetry are

produced---test the density dependence of

the symmetry potential

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II. Probing the symmetry potential at low densities

Central collisions at intermediate energies ：multifragmentation- isospin distillationin L-G phase transition

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Isoscaling effect (Tsang,et.al, PRL,2001)

Nucl-ex/0406008

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M.B.Tsang,et.al.PRL 92 (2004)

Peripheral reaction ----Isospin diffusion

α

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Probing the equilibrium with respect to isospin sensitive observables in HIC

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

Exp.

Zr+Ru Ru+Zr E(AMeV) b(fm) 100 0 400 0 400 5

RZ

Y

The normalized proton counting number as function of rapidity.Rz=1, for Zr+Zr,Rz=-1, for Ru+Ru,Rz=0, for Zr+Ru and Ru+Zr, if equilib.is reached

Results show protons are not from an equilibrium source and the reaction is half transparent

Li, Li,PRC64(01)064612

) /( ) 2(1 1 2 2 1 1 2 2 2 1 Z Z Z Z Z Rz

2004.8 Weihai 12

Probing the symmetry potential at low densities by peripheral HIC

Products in peripheral collisions at Fermi energies ： Calculations are performed by means of ImQMD model (Wang,Li,et.al., PRC, 65(2002)064648, 69(2004)034608)

2

2

1 uCv ssym 0/ u

nuclear potential energy density functional :

2004.8 Weihai 13pn

pn

0

u

0 1 2-20

-10

0

10

20

e (

Me

V)

u

?

0

0.20.40.60.8

1.0

uuF For low densities we take the density dependence ofSymmetry potential:

2)(2

)( uFC

v ssym

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Density dependence of the mean field contributing from symmetry potential

2/ 122

uuC

U Spnsym

0 1 2-20

0

20

=0. 5 =1. 0 =1. 5

Pr ot ons

Neut r ons

Usym

u

When > 0 neutrons are more bound for =0.5 than for stiff symmetry pot. When < 0 neutrons are less bound for =0.5 than for stiff symmetry pot.It is just opposite for protons

np

symsymnp

VU

,,

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0.0 0.5 1.0 1.5-35

-30

-25

-20

-15

-10

-5

0C

S:

:

Dash Dotted CS=27 MeV

Line: =0.5

u(=/0)

Pure Line: =1.5Dashed Line: =0.5

Protons

Neutrons

=16/96 p

,n (

MeV

)

Density dependence of chemical pot.

)

,(

,,

npnp

u

Cs=35MeV

Esym-stiff.

Esym-soft

ε is the energy density

μn(ρ)-μp(ρ)=4Esym(ρ)δ

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neutrons move to the neck region faster than protons, neck area experiences weakcompression, expansion and finally rupturesPLF and TLF are at normal densitynucleons and light charged particles are emitted from neck

)(symv directly influences the motion of nucleons towards to neck region influences the emission rate of the neutrons and protons

)(, pnsym

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mass and charge distribution

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Time evolution of N/Z ratio for particles at neck region

Neutron skin effectN/Z increases with b

plateau

matter at neck area is neutron -rich

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The spectrum of N/Z ratio

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N/Z ratio of free nucleons as function of impact parameters for peripheral reactions of

KrSn 86124,112

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Yields of and3H 3He

EES model

Ni/Zi ,N/Z ratio of particles at neck area(emission source)

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Yields of 3H and 3He as function of b

stiff

soft

stiff

soft

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)(

)(3

3

HeY

HY

124Sn+86Kr

112Sn+86Kr

Soft-sym Stiff-sym

2.5 1.9

1.98 1.54

36Ar+58Ni

exp

1.4

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Conclusions I(low densities)

1) The N/Z ratio of emitted nucleons is enhanced with soft symmetry potential, while the slope of N/Z ratio of free nucleons vs impact parameters is enhanced with stiff symmetry potential for peripheral reactions.

2) The yields of H3 ,He3 and the ratio depend on Esym(ρ) sensitively. The reducing slope of yield of H3 with impact parameters for peripheral reactions is very sensitive to the Esym(ρ) and asymmetry of the reaction system, while that of He3 is not.

)(

)(3

3

HeY

HY

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III. Study the density dependence of the symmetry potential at high density

π-/π+ ratio is sensitive to the Esym(ρ) at ρ>ρ0 B.A. Li, NPA

2002

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Stiff symmetry potential

B.A. Li, NPA,2002

Soft symmetry potential

The density dependence ofEsym strongly influence the structure of neutron star

Direct URCL limitProton fraction 1/9

2004.8 Weihai 28B.A. Li, NPA 2002

π-/π+ ratio is sensitive to the Esym at ρ>ρ0

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Probing the density dependence of symmetry potential by -/+ and Σ-Σ + ratio by means of UrQMD-V1.3

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The production rate of and at different densities

UrQMDwithoutsymmetry potential

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Symmetry potential for resonances

(Δ++,+,0,-,N*) and ,Σ+-,0

For resonances: are determined by isospin C-G coef. in B*+N

For Σ+-,0, assuming charge independence of the baryon-baryon interaction, in the linear approximation in y= (Z-N)/A

V (Σ+-)=V0 (Σ+-)12V1 (Σ+-) y

V1, Lane potential

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-/+ and Σ-Σ + ratio by UrQMD with symmetry potential

Stiff Esym(a)

Soft Esym(b)

1.5AGeV

3.5AGeV

Sensitity to Esym (ρ) reduces as energy increase for -/+

2.5AGeV

b aa

b

b

a

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At low energy case pions are produced mainly through , the -/+ ratio is determined by n/ p.

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N/Z|132Sn

=1.64

Red lines for soft-Esym and black lines for stiff-Esym

b

a

b

a

ba

b

a

ba

ba

ba

ba

b

a

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Δ+ +, Δ+, Δ0 , Δ– production strongly depends on ρn/ρp

For E~1AGeV or less pions are mainly produced by Δ therefore π-/π+~ (N/Z)2

For E>>1AGeV many channels open. The situation becomes more complicatedΣ-/ Σ+ is more complicated than π-/π+

2004.8 Weihai 38

Σ is baryon, as soon as it is produced it will be under of the mean field of nuclear matter.

The ratio of Σ+/ Σ- therefore is also depends on the symmetry potential of Σ in nuclear matter, in addition to those of particles which produce Σ

2004.8 Weihai 39

Soft-sym

Stiff-sym

similar with -/+

without the symmetry potential of Σ

b

ab

a

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The effect of the symmetry potential of Σ in nuclear matter can not be neglected! The strength of this effect depends on V1

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Conclusions II(high densities)

1) A strong dependence of the ratios of -/+ and Σ -/ Σ + on Esym(ρ) which provide good means for st

udy Esym at ρ> ρ0 .

2) The ratio of -/+ n/ p for E=1.5 AGeV case but not 3.5 AGeV case. The sensitivity of -/+ ratio to Esym(ρ) reduced as energy increases.

2004.8 Weihai 42

3) The ratio depends on the symmetry potential of in addition to those of particles which produce ’s.

Therefore a more complicated situation appears for the ratio, a reversion is appeared from E= 1.5 AGeV to E=3.5 AGeV, which may provide a useful probe to obtain the information of Lane potential V1.

/

/

2004.8 Weihai 43

Thanks for the patience

2004.8 Weihai 44

II) In-Medium Nucleon-Nucleon Elastic Scattering cross Section

The dynamics in heavy ion collisions at Fermi energies is

dominated by both mean field and collision terms. The isospin dependence of two-body scattering cross s

ections and its medium correction plays an important role in the reaction dynamics.

Empirically, the form of medium correction is taken as: σ= σ0 (1-αρ/ρ0), α is taken as a parameters and is isospin independent

2004.8 Weihai 45

Our study is based on the formalism of the closed time Green’s function . With this approach, both mean field and two-body scattering cross sections can be obtained with the same effective interactions (self-consistently).The analytical expressions of the in-medium two-body scattering cross sections are obtained by computing the collisional self-energy part up to Born terms.Refs: Mao, Li, Zhuo, et.al, PRC.49(1994), Phys.lett. B327(1994)183, PRC53(1996), PRC55(1997)387, … Li, Li, Mao, PRC 64(2001)064612 Li, Li, PRC , accepted

2004.8 Weihai 46

The effective Lagrangian density of density dependentrelativistic hadron field theory:

The energy density is:

The coupling constants are of the functional of densityRef: PRC64(2001)034314

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M*(x)=M0+ΣHσ(x)+ Σ Hδ(x)

Mp

Mn

2004.8 Weihai 48

(Mao,Li, et.al, PRC.49(1994), Li,Li,Mao, PRC 64(2001)064612)

The Feynman diagrams for computing the in-medium nucleon-nucleon elastic scattering cross section

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The isospin dependence of in-medium cross sections is contributed from ρ and δ meson

The contributions from σ and ω exchange

The density dependence of σn

p,

σnn(pp) at Yp=0.5 and Yp=0.3

σnp

σnn(pp)

σnp

σpp

σnn

σnp/σnn(pp)

2004.8 Weihai 50

The contributions to σnn(pp), σnpfrom the ρ and δ related terms(total 7 terms)

There exist strong cancellation effect. The final results are the delicate balance between 7 terms

σ-δ

σρ

σρ

σρ

ωρ

ωδ

ωδ

ωρ

2004.8 Weihai 51

The density and temperature dependence of σnn,σpp,σnp for Y=0.3 Ek=10MeV

Clear isospin dependence for in-medium cross section is seen. The density dependence is stronger than temperature dependence. The isospin dependence of cross section will influence the reaction dynamics strongly.

Y=Z/A

2004.8 Weihai 52

III. Isospin effect in HIC Multifragmentationmultifragmentation in intermediate HIC relating to possible liquid-gas phase transition (M.Fisher,Physics(N.Y.)3(1967)255,PRL88,042701,PRL88,022701, PRC52,2072,…) We study multifragmentation through central collisions in intermediate HIC. isospin distillation ,isoscaling effect, …..N/Z of free nucleons, IMF, light charged particlesstrongly depends on the symmetry energy Flow effectsneutron,proton flow, light charged particle flow, differential flow,… (various kind flow)

Probing the density dependence of Esym at ρ<ρ0

2004.8 Weihai 53

The momentum distribution of Nnpof nucl. and IMF

a)The effect of Cs on Nnp of nucleons is more pronounced at large momentum and that of is more pronounced at small momentum (because nucleons with large momentum mainly emitted at early time and that of small momentum emitted at later stage) .b) Nnp(IMF) for =0.5 enhances at p/pproj<0.25 (Coulomb effect) and at large p/pproj >1.0 ( density dependence of symmetry pot. ?) comparing with sym- stiff case.

0.5 1.0 1.50.0

0.1

0.2

0.3

0.4

0.5 1.0 1.5

0.9 1.00.00

0.05

0.10

Nucleons

CS=0 MeV

CS=35 MeV :

=0.5 =1.0 =1.5

96Zr+96ZrE=100 AMeV, b=0 fm

Nn

p

IMF

=1.0 : C

S=27 MeV

CS=50 MeV

p/pproP/Ppro

j

2004.8 Weihai 54

Probing the equilibrium with respect to isospin sensitive observables in HIC

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

Exp.

Zr+Ru Ru+Zr E(AMeV) b(fm) 100 0 400 0 400 5

RZ

Y

The normalized proton counting number as function of rapidity.Rz=1, for Zr+Zr,Rz=-1, for Ru+Ru,Rz=0, for Zr+Ru and Ru+Zr, if equilib.is reached

Results show protons are not from an equilibrium source and the reaction is half transparent

Li, Li,PRC64(01)064612

2004.8 Weihai 55

Density dependence of the mean field contributing from symmetry potential

2/ 122

uuC

U Spnsym

When > 0 neutrons are more bound for =0.5 than for symmetry-stiff case. When < 0 neutrons are less bound for =0.5 than for symmetry-stiff case.It is just opposite for protons

np

symsymnp

VU

,,