Towards the Equation of State of Dense Asymmetric Nuclear Matter
description
Transcript of 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)
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)
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
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, , renormalized 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
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)
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.
– 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-Bem)
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)
)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:
• 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
Use scaling to simplify representation:
Zp21 ˆ/)Z,N(R)N(S
=0
=1
Data
Predictions of EES model
0MeV4.23)(S
Tsang et al., PRL (2001)
• 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.
0MeV4.23)(S
)T/)]ZZ(e
sZsNexp([C)Z,N(R
tot
pn21
Use scaling to simplify representation:
Zp21 ˆ/)Z,N(R)N(S
Tsang et al., PRL (2001)
Predictions of EES model
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
Tsang et al., PRL (2001)
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
Scaling behavior of Isotopic Ratios
• 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
Scaling behavior of Isotopic Ratios
• 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
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.
Use scaling to simplify representation:
Zp21 ˆ/)Z,N(R)N(S
0MeV4.23)(S
Tsang et al., PRL (2001)
)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.
– 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-Bem)
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)
)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