Presentación de PowerPointscoccola/iguazu/NTD/07_NTD7-Dorso.pdf · p p p p q q p p A Z Z A ......
Transcript of Presentación de PowerPointscoccola/iguazu/NTD/07_NTD7-Dorso.pdf · p p p p q q p p A Z Z A ......
SLAFNAP6
Iguazú, Argentina. October 3 to 7, 2005
IsoscalingGeometry andCorrelations
C.O.Dorso
Depto. Fisica
FCEN-UBA
Argentina
Isoscaling
),(),(),(
1
221 NZY
NZYNZR =
)exp(),(21 NZCNZR βα −=
Isoscaling is the property that fragment yields of similarbut isotopically different reactions have a exponentialdependence on N and Z
Isoscaling and Statistical models
Tnn)1()2( μμα −
=
⎥⎦
⎤⎢⎣
⎡=
TNZ
VTZNY npfAZ
μμζ
_exp),(),(
⎥⎦⎤
⎢⎣⎡ −
=T
TfBAV
g AZAZ
T
fAZAZ
)(exp2/33λ
ζ
In the Grand Canonical ensemble
Tpp)1()2( μμ
β−
=
with
then
Outline Outline •N-species percolation problemin k dimensions•Nuclear Percolation Model
•Lattice Animals
•Numerical calculations
•Non equilibrium
•Correlations
•Lattice gas model
Start with an empty grid
Nuclear Percolation model
Populate all nodes
according to
p
q=1-p
Nuclear Percolation model
Break bonds with probability
b
Nuclear Percolation model
Identify clusters
Nuclear Percolation model
This cluster isCharacterized byA=6 nodes
N=3,Z=3a=6 active bondst=11 broken bonds
One dimensional case, 2 colors
Given a linear chain(1-b) activation probability
nA = limL ∞ [NA/L]
For this case the cluster have a unique value of theperimeter t, t=2
NA = (1-b)A-1 b2 L
Activated bond
Broken bond
b
Number of clusters of size Aper bond
We address the problem of the analysis of the Relative yields from two lattices of sizes A1 and A2. The A1 and the A2 nodes are randomly assigned colorsdenoted by CN,CZ,CQ,…with probabilities pN , pZ,pQ ,…
The number of nodes with color CN will be denoted byN
The probabilities are normalized and independent.
In order to produce fragments, bonds are broken with probability b.
We then calculate R21(N,Z,Q,…)
N colors Percolation model in d dimensions
Percolation model in d dimensions
If we only look at the size of the clusters withoutTaking care about the colors, in the ∞ limit
dA
LA L
Nn lim∞→
=
∑ −=ta
taAatA bbgn
,)1(
Number of clusters of size A per node (d Dim.)
Which can be written as
gAat is the number of ways of building a lattice animal ofmass A , with a bonds and a perimeter t
Lattice animalsin 2D with A=4(square lattice)
84931034 )1()1(4)1(18 bbbbbbn −+−+−=
Percolation model
N-colors Percolation model
total mass A=Z+N+Q+P+……1= pZ + pN + pQ + …
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−= ∑∑
= Az
NN
ZZz
at
taAatA pppbbgn
,0...)1( α
[ ]!...!!/! QZNZ AAAA=α
with
then [ ]...)1(,...),,( QQ
NN
ZZz
at
taAatA pppbbgQNZn α⎥
⎦
⎤⎢⎣
⎡−= ∑
We now include the color assignation
g_isoscaling1
2
1
221 ,...),,(
,...),,(,...),,(,...),,(
QZNnQZNn
QZNYQZNYR
A
A==
N-colors Percolation model
[ ][ ]
1
2
11
22
1
2
...
...
)1(
)1(
,...),,(,...),,(
NN
ZZ
NN
ZZ
at
taAat
at
taAat
A
A
ppp
ppp
bbg
bbg
QNZnQNZn
⎥⎦
⎤⎢⎣
⎡−
⎥⎦
⎤⎢⎣
⎡−
=
∑
∑
If b1 = b2
[ ][ ]
1
2
1
2
...
...,...),,(,...),,(
NN
ZZ
NN
ZZ
A
A
pppppp
QNZnQNZn
=
.........
,...),,(1
2
1
2
1
2
111
22221
Q
Q
QN
N
N
Z
Z
ZQQ
NN
ZZ
NN
ZZ
pp
pp
pp
pppppp
QZNR ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛==
( )...exp
...lnlnlnexp,...),,(1
2
1
2
1
221
+++=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
QZN
pp
Ppp
ZPp
NQZNRQ
Q
Z
Z
N
N
γβα
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2lnN
N
pp
α ⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2lnZ
Z
pp
βwith
then
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2lnP
P
pp
γ
N-colors Percolation model
Nuclear Percolation model
We now work with two colors i.e.“isospin” degree of freedom
⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡−= ∑∑
=
−
Az
ZAZz
at
taAatA ppbbgn
,0
)()1()1( α
!!/! ZNAZ =α
1
2
1
221 )(
)(),(),(
ZnZn
ZNYZNYR
A
A==Isoscaling
with
then)()1()1()( ZAZ
zat
taAatA ppbbgZn −−⎥
⎦
⎤⎢⎣
⎡−= ∑ α
Nuclear Percolation model
NZ
NZ
NZ
pp
qpqpZNR ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛==
1
2
1
2
11
2221 ),(
( )ZNppZ
qqNZNR βα +=⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= explnlnexp),(
1
2
1
221
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2lnqqα ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
1
2lnppβ
with
then
Nuclear Percolation model
We now approximate R21 for finite systems as :
11
22
1
221 )(
)()()(),(
ZnAZnA
ZNZNZNR
A
A
A
A ≈=
NZNZ
t
t qq
pp
pp
pp
AZ
ZA
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛≈⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2
1
2
2
1
1
2
1
2
1
2
2112
21
//
)exp(),(
ppAAC
ZNCZNR
≈=
+= βαWhich renders
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2lnqqα
Finite systems
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2lnppβ
Nuclear Percolation model
a) A2= 7x7x7 , A1=6x6x6, Z=108
b) A=6x6x6p1=0.5, p2=0.33
c) Same as b), with A=5x5x5
d) A=6x6x6 p1=0.5, p2=0.42
No fits
Nuclear Percolation model with correlations(2-species percolation problem)
We now study the effect of particle-particle Correlations.For this purpose the isospin is assigned to the nodesby performing a lattice gas calculation ( fragments Keep on being determined by percolation )
Lattice gasVnp=-5.33Vnn=Vpp=-a
N
R21
Lattice gasVnp=-5.33Vnn=Vpp=0
A1=16x16A2=20x20b=0.3
β=0.5=0.39
Typical resultNuclear Percolation model with correlations
Temperature dependenceof α and β
IndependentProb. limit
Nuclear Percolation model with correlations
β
α
<Number of “protons”>which are nn ofa “proton”as a function of T
Symmetry?
Binding energies forZ=50 , 40<N<60Vnn=Vpp=0Vnn=Vpp=-3Vnn=Vpp=-4
Isoscaling forVnn=Vpp=0 (open)Vnn=Vpp=-4 (full)
Nuclear Percolation model with correlations
Nuclear Percolation model with correlations
If we increase the Temperature the effectis washed away
Non equilibrium effects (?)
• What if b1 is not equal to b2Different temperatures
• What if p2 is not homogeneous
Nuclear Percolation model(2-species percolation problem)
116125135
107126145
114105124
9384103
8282
6
4
)1(14)1(30)1(144)1(2)1(40)1(1746
)1(24)1(8)1(555
)1(4)1()1(184
)1(8)1(23
)1(22
1
bbbbbbbbbbbbA
bbbbbbA
bbbbbbA
bbbbA
bbA
bA
−+−+−+
+−+−+−→=
−+−+−→=
−+−+−→=
−+−→=
−→=
→=Lattice animals
Nuclear Percolation model (2D)
gAat(1-b)a bt
for 2D squareLattice up toA=6
( )( )
⎥⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡
−+−+−−+−+−
=
−
=
=
2
1
1
2
)(
1
2111
31
101
51
121
41
112
32
102
52
122
42
1,5
2,5
)1(24)1(8)1(55)1(24)1(8)1(55
pp
pp
bbbbbbbbbbbb
NN
NNA
NA
NA
Nuclear Percolation model
b2 b1
Nuclear Percolation model
Simulation and lattice animals
b2=0.6b1=0.7
Nuclear Percolation model
b2=0.6b1=0.7
Nuclear Percolation model
But we can try to fit the b1 ≠b2 data
b1=b2 valuesα0=0.307 β0=0.44 C0=1.56
α=0.36 β=0.4 C=0.82
Nuclear Percolation model
Non homogeneousp2
p2
0
1
4
2
Isoscaling is a quite general property of fragmenting systems
No dynamical calculation satisfies it exactly Neither experimental results
In percolation it appears naturally as a fair sampling effect (homogeneous probabilities ⇒ least bias) and can be solved exactlyIs independent of dimensionality and topology and the number ofspecies that populate the lattice
“strong isoscaling”
Particle-particle correlations enhance the effect
Departures from exponential behavior might be traced to a combination of “non equilibrium” effects
Conclusions