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

QQ

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(

,...),,(,...),,(

QQ

NN

ZZ

QQ

NN

ZZ

at

taAat

at

taAat

A

A

ppp

ppp

bbg

bbg

QNZnQNZn

⎥⎦

⎤⎢⎣

⎡−

⎥⎦

⎤⎢⎣

⎡−

=

∑

∑

If b1 = b2

[ ][ ]

1

2

1

2

...

...,...),,(,...),,(

QQ

NN

ZZ

QQ

NN

ZZ

A

A

pppppp

QNZnQNZn

=

.........

,...),,(1

2

1

2

1

2

111

22221

Q

Q

QN

N

N

Z

Z

ZQQ

NN

ZZ

QQ

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

qq

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

qq

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

qq

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