nuclear symmetry energy and intermediate heavy ion reactions r. wada, m. huang, w. lin, x. liu imp,...
TRANSCRIPT
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
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).
For N=Z=A/2 (I = 0)
( )
BE=17.87 ̶ ̶ ̶8.95A ̶ ̶ ̶asymI2/A
BE=Mass ̶Formula
BE=17.87 ̶ ̶ ̶8.95A
Exp.Cal.
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
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.
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
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
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
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) ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶