Presentación de PowerPointscoccola/iguazu/NTD/07_NTD7-Dorso.pdf · p p p p q q p p A Z Z A ......

of 34 /34
SLAFNAP6 Iguazú, Argentina. October 3 to 7, 2005

Embed Size (px)

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 npfAZμμ

    ζ_

    exp),(),(

    ⎥⎦⎤

    ⎢⎣⎡ −=

    TTfBA

    Vg AZAZ

    T

    fAZAZ

    )(exp2/33λζ

    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(,...),,( QQNNZZzat

    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()( ZAZz

    at

    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

    which are nn ofa “proton”as a function of T

  • Symmetry?

    Binding energies forZ=50 , 40

  • 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 btfor 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

    Conclusions