Nuclear Symmetry energy and

Intermediate heavy ion reactions

R. Wada, M. Huang, W. Lin, X. Liu

IMP, CAS

Symmetry Energy of nuclear matter

What is important for Symmetry energy at T>>1 and dense or diluted nuclear matter? 1. Heavy ion collisions -- Symmetry energy is one of key factors to determine isotope distribution 2. Astrophysics -- Radius and cooling of neutron stars

Esym (ρ) = E(N=A,N=0) – E(N=Z,A) : Energy difference between the symmetric nuclear matter (N=Z) and neutron matter E(ρ,δ)/A = {E(ρ,δ=0) + S(ρ) δ2}/A : δ = (N-Z)/A asymmetry

A

BE=17.87 ̶ ̶ ̶8.95A ̶ ̶ ̶asymI2/A

BE=Mass ̶Formula

BE=17.87 ̶ ̶ ̶8.95A

N=Z

Weizsäcker-Bethe mass formula

M(Z,N) = avA – asA2/3 – acZ(Z-1)/A1/3

– asym (N-Z) 2/A – δ(N,Z)

According to the Modified Fisher Model, the yield is given by

Y(N,Z) = y0A̶ τ · exp(-F/T) (ρn)N(ρp)Z

= y0A̶ τ · exp[(W(N,Z)+μnN+ μpZ)/T] ,

W(N,Z) can be given by the following generalized Weizsäcker-Bethe semi-classical mass formulation at a given T and density ρ,

W(N,Z) = av(ρ,T)A – as(ρ,T)A2/3 – ac(ρ,T)Z(Z-1)/A1/3 – asym(ρ,T) I 2/A – δ(N,Z)

I=N-Z and δ is the paring energy, given by = δ0 (for odd-odd nucleus)

δ(N,Z) = 0 (for even-odd nucleus) = -δ0 (for even-even nucleus).

1. Isotope yield and Symmetry energy

Related ̶fragments

For convenience, Y(N,Z) = Y(I,A) ; I = N-Z , A = N+Z

(M. Huang et al., PRC 81, 044620 (2010) )

Related ̶reaction ̶systemsN=Z

Y(I+2,A) = C A−τ exp{ [avA – asA2/3 – ac(Z-1)(Z-2)/A1/3 – asym (I+2)2/A– δ(N+1,Z-1) +μn(N+1)+μp(Z-1)]/T }

Y(I,A) = C A−τ exp{ [ avA – asA2/3 – acZ(Z-1)/A1/3 – asym I2/A – δ(N,Z) +μnN+μpZ]/T}

R(I+2,I,A) = exp{ [2ac·(Z-1)/A1/3 – asym·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T } · exp[(μn- μp)/T]

1. Yield

2. ̶ ̶Ratio

cancel ̶out

When we focused on isotopes with same A in a given reaction, then

For I= –1, drop out

For even-odd, drop out

Reaction system

Symmetry energy term:R(I+2,I,A) = exp{[2ac·(Z-1)/A1/3 – asym·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T}·exp[(μn- μp)/T]

When I = 1, N – Z = odd, so (N,Z) is even-odd or odd-even. – δ(N+1,Z-1) + δ(N,Z) = 0.

ln[R(3,1,A)] = [2ac·(Z-1)/A1/3 – 8asym/A]/T + (μn- μp)/T

Exp.Isobaric ̶ratioIsoscalingvariance

{Cal ̶: ̶AMD+GeminiCal ̶: ̶AMD ̶(primary)

Primary

Secondary

Reactions used are : 64Zn, 64Ni +58,64Ni, 112,124Sm, Au, 232Th @ 40 A MeV

Two issues:

1. Sequential decay in the cooling process drastically change the isotope distributions and causes a significant mass dependence of the symmetry energy.

2. Temperature (and density) cannot be determine uniquely. Y(N,Z) = y0A̶ τ · exp[(W(N,Z)+μnN+ μpZ)/T] , T relates all parameters by in a/T terms. No density information.

64Zn+112Sn @ 40 A MeV ( 40Ca+40Ca @ 35 A MeV ) have been studied in detail.

Thermometers a. Kinematic energy slope b. Excited-state population c. Double isotope ratio d. Kinematic fluctuation

1. Sequential decay issues : Kinematical focusing

Correlated LP Correlated LP

(M. Rodrigues et al., PRC 88, 034605 (2013) )

Uncorrelated LP

v

4.5≤VIMF<5.5 cm/ns

3.5≤VIMF<4.5 cm/ns

5.5≤VIMF<6.5 cm/ns

data

Total

Uncorr(kMn(Li))

Corr(Mn(23Na))

θIMF-n

4.5≤VIMF<5.5 cm/ns

35o15o 25o45o

Extracted Multiplicities

Neutrons

SecondaryReconstructed isotope Primary

Reconstructed multiplicity distribution and multiplicity distribution of the AMD primary fragments

Exp(cold)Reconstructed

AMD ̶primary

(W. Lin et al., PR C 90, 044603 (2014) )

2. Temperature, density and symmetry energy

( )

BE=17.87 ̶ ̶ ̶8.95A ̶ ̶ ̶asymI2/A

BE=Mass ̶Formula

BE=17.87 ̶ ̶ ̶8.95A

(X. Liu et al., NPA in Press, 2014)

3. Extract asym/T values using all available isotopes with parameters defined in step (1) and (2).

Δμ/T ̶= ̶0.6ac ̶/T ̶= ̶0.18

)

For I = –1

ln [R(1, -1, A)]

ln [

R(1

, -1,

A)]

A

For N=Z=A/2 (I = 0)

( )

BE=17.87 ̶ ̶ ̶8.95A ̶ ̶ ̶asymI2/A

BE=Mass ̶Formula

BE=17.87 ̶ ̶ ̶8.95A

Exp.Cal.

Symbols Exp.Lines: Cal. Exp.

AMDg0g0ASg0ASS

Exp.

AMDg0g0ASg0ASS

g0/g0ASS

g0/g0ASg0/Exp

Temperature Extraction

40Zn+112Sn @ 40 AMeV AMD : 40Ca+40Ca @ 35 AMeV

asym (MeV)

(X. Liu et al., PR C 90, 014605 (2014) )

40Zn+112Sn @ 40 AMeV

All Fragments have the same T : Modified Fisher Model assume thermal and chemical equilibriums in the fragmenting system

T=T0(1- kA)

k is determined iteratively; I.) In the first round k=k1=0. II.) Use step 1-3 to calculate all parameters and extract ρ, T and asym values. III.) Determine a new slope parameter k’ from the temperature distribution. if k’=0, then stop and extract T0. Otherwise set a new k as k2=k1+1/2(k’). Repeat the procedure II.)

1. For I = ─ 1

2. For N = Z

3. For N ≠ Z

AMD : 40Ca+40Ca @ 35 AMeV40Zn+112Sn @ 40 AMeV

T=5.0MeV

k=0.0

k=0.007

T

Mass Dependent apparent Temperature

1. Distribute the thermal energy to each fragemt by a Maxwellian distribution as

2. Require the momentum conservation.

larger fragment has less probable to have large momentum larger fragments have smaller temperature

Fluctuation thermometer results in a flat temperature

AMD : 40Ca+40Ca @ 35 AMeV

3. A smaller system has a larger mass dependence

40Zn+112Sn @ 40 AMeV

Summary: 1. Sequential decay modified the isotope distribution of the final products. This makes difficult to extract the properties of the fragmenting system at the freeze-out density.

5. The mass-dependent apparent temperature originates from the momentum conservation in the fragmentation system and it is system-mass dependent. Smaller systems show a larger mass dependence.

4. Modified Fisher Model is extended with an apparent mass-dependent temperature and ρ/ρ0 = 0.65 +/- 0.2, asym= 23.1 +/- 0.6 MeV, T=5.0 +/- 0.4 MeV are determined.

3. Modified Fisher Model and a self-consistent method is applied for the reconstructed isotope distributions and AMD primary isotope distributions with different interactions in the symmetry energy density dependence. a. Density, temperature and symmetry energy values are extracted. b. Extracted temperature values show a mass dependence.

2. Kinematical focusing technique is applied to re construct the primary isotope distribution at the time of the fragment formation.

Thank you for your attention !

It is interesting to see how the density and temperature change as a function of incident energy.Unfortunately there are no available experimental data of the reconstructed isotope yield distributions in different incident energies.

We performed AMD simulations at 35 - 300 AMeV for a 40Ca+40Ca system.10,000 events were generated for b=0 fm at 35, 50, 80, 100, 140, 300 with g0,g0AS and g0ASS.

Analyzed by the self-consistent method. (X. Liu et al., PRC in submission Oct. 2014)

AMD with MFM and self-consistent method

0 fm/c

10 fm/c

30 fm/c

60 fm/c

100 fm/c

-10 0 10Zcm

Density

Temperature

50 A MeV

Statistical nature in AMD

3. Mass distribution can be described by an statistical ensemble. (Furuta et al., PRC79, 014608 (2009))

Statistical ensembles are made by AMD to enclose 36nucleons in a spherical volumewith given T and ρ (Volume).Mass distributions are evaluated as a long time average. Same code is usedfor AMD simulations and the statistical ensemble generation.

40Ca+40Ca @ 35 A MeV

AMD: 64Zn+197Au : b ~ 2fm

AMD Gemini

N.Marie ̶et ̶al., ̶PRC ̶58, ̶256, ̶1998S.Hudan ̶et ̶al., ̶PRC ̶67, ̶064613, ̶2003 ̶

Gemini

Exp

p

d

t

h

α

32 ̶A ̶MeV

39 ̶A ̶MeV

45 ̶A ̶MeV

50 ̶A ̶MeV

200

64Zn ̶47 ̶A ̶MeV

Experiment

IMF ̶20o

129-300-1000-1000 ̶μm

Projectiles: ̶ ̶ ̶ ̶ ̶ ̶64Zn,64Ni,70Zn ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶at ̶40 ̶A ̶MeV

Target ̶ ̶ ̶ ̶ ̶: ̶58,64Ni, ̶112,124Sn, ̶197Au, ̶ ̶232Th

64Zn+112Sn ̶at ̶40 ̶A ̶MeV

Exp. ̶vs ̶AMD-Gemini ̶Semi-violent ̶collisions ̶

16O

“Multi-fragmentation”

Time

vPLF

vIV

vTLF

64Ni+124Sn ̶at ̶40A ̶MeV

E ̶(MeV)

E ̶(M

eV)

Z=14 ̶ ̶ ̶ ̶15 ̶ ̶ ̶ ̶16 ̶ ̶ ̶ ̶17 ̶ ̶ ̶ ̶18 ̶ ̶ ̶ ̶ ̶19 ̶ ̶ ̶ ̶ ̶20 ̶ ̶ ̶

7

8

9 ̶

10 ̶ ̶11 ̶ ̶ ̶12 ̶ ̶13 ̶

Black ̶Histogram: ̶Exp.Red: ̶individual ̶isotopeGreen ̶: ̶linear ̶BGBlue: ̶total

Isotope ̶Identification ̶and ̶yield ̶evaluation

I = N – Z = T 1 : even-odd: ̶

R(1,-1,A) = exp{ 2ac·(Z-1)/A1/3/T } · exp[(μn- μp)/T]

ln[R(1,-1,A)] T Δμ/T = 2ac·(Z-1)/A1/3/T + (μn- μp)/T

1. ̶Coulomb term and Chemical potential

(μn- μp)/T= 0.71

< > : averaged values over all A <ln[R(1,-1,A)]> =< 2ac·(Z-1)/A1/3/T> + <(μn- μp)/T>

< exp{ 2ac·(Z-1)/A1/3/T } > : same for all reactions

1. Chemical potential between different reactions

(μn- μp)/T= [(μn- μp)/T]0 + Δμ (Z/A)/T Δμ (Z/A)/T=c1(Z/A)+c2 (c1=-13.0 c2=8.7)

ac/T = 0.35

(Z/A)sys

R(I+2,I,A) = exp{ [2ac·(Z-1)/A1/3 – asym·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T } · exp[(μn- μp)/T]

Reactions used are : 64Zn, 64Ni +58,64Ni, 112,124Sm, Au, 232Th @ 40 A MeV

Symmetry energy term:R(I+2,I,A) = exp{[2ac·(Z-1)/A1/3 – asym·4(I+1)/A– δ(N+1,Z-1) + δ(N,Z)]/T}·exp[(μn- μp)/T]

When I = 1, N – Z = odd, so (N,Z) is even-odd or odd-even. – δ(N+1,Z-1) + δ(N,Z) = 0.

: ̶Fixed

ln[R(3,1,A)] = [2ac·(Z-1)/A1/3 – 8asym/A]/T + (μn- μp)/T

Exp.Isobaric ̶ratioIsoscalingvariance

{Cal ̶: ̶AMD+GeminiCal ̶: ̶AMD ̶(primary)

Y(N,Z) = y0A̶ τ · exp[(W(N,Z) +μnN+ μpZ)/T] , W(N,Z) = avA – asA2/3

– acZ(Z-1)/A1/3–asymI2/A – δ(N,Z)

Published ̶works:A. ̶Isoscaling ̶and ̶Symmetry ̶energy ̶ ̶ ̶1. ̶ ̶Z. ̶Chen ̶et ̶al., ̶“Isocaling ̶and ̶the ̶symmetry ̶energy ̶in ̶the ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶multifragmentation ̶regime ̶of ̶heavy-ion ̶collisions”, ̶Phys. ̶Rev. ̶C ̶81, ̶064613 ̶(2010) ̶ ̶ ̶2. ̶M. ̶Huang ̶et ̶al., ̶“A ̶novel ̶approach ̶to ̶Isoscaling: ̶the ̶role ̶of ̶the ̶order ̶parameter ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶m=(N_f-Z_f)/A_f”, ̶Nuclear ̶Physics, ̶A ̶847, ̶233 ̶(2010)B. ̶ ̶Isobaric ̶yield ̶ratio ̶and ̶Symmetry ̶energy ̶ ̶ ̶ ̶3. ̶ ̶ ̶M. ̶Huang ̶et ̶al., ̶“Isobaric ̶yield ̶ratios ̶and ̶the ̶symmetry ̶energy ̶in ̶heavy-ion ̶ ̶ ̶ ̶ ̶ ̶ ̶reactions ̶near ̶the ̶Fermi ̶energy”, ̶Phys. ̶Rev. ̶C ̶81, ̶044620 ̶(2010) ̶C. ̶Landau ̶formulation ̶of ̶isotope ̶yield ̶and ̶critical ̶phenomena ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶4. ̶A. ̶Bonasera ̶et ̶al., ̶“Phys. ̶Rev. ̶Lett. ̶101,122702 ̶(2008), ̶ ̶ ̶5. ̶M. ̶Huang ̶et ̶al., ̶“Isospin ̶dependence ̶of ̶the ̶nuclear ̶equation ̶of ̶state ̶near ̶the ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶critical ̶point”, ̶Phys. ̶Rev. ̶C ̶81, ̶044618 ̶(2010) ̶D. Power ̶Law ̶distribution ̶and ̶critical ̶phenomena ̶ ̶ ̶6. ̶M. ̶Huang ̶et ̶al., ̶“Power ̶law ̶behavior ̶of ̶isotope ̶yield ̶distribution ̶in ̶the ̶ ̶ ̶ ̶ ̶ ̶ ̶multifragmentation ̶ ̶regime ̶of ̶the ̶heavy ̶ion ̶reactions”, ̶Phys. ̶Rev. ̶C82,054602 ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶(2010) ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶