Isoscaling in Multifragmentation

A MODEL FOR PROJECTILE FRAGMENTATIONVariable Energy Cyclotron Centre, INDIACollaborators: S. Mallik, VECC, India S. Das Gupta, McGill University, Canada

1Gargi ChaudhuriDifferent Stages of Projectile FragmentationOur ModelResults Comparison with different experimental dataSummaryCONTENTS 2

PROJECTILE FRAGMENTATION (Different Stages) Collision of the projectile & target nuclei above certain energy (> 100 MeV/n) (COLLISION)

Part of the projectile goes into the participant & remaining part (projectile spectator or PLF) gets sheared off (ABRASION)

Hot, abraded PLF (As, Zs) expands to about 3V0 4V0.(V0-normal nuclear volume) & breaks up into many fragments (MULTIFRAGMENTATION)

The excited fragments de-excite by sequential decay (EVAPORATION)

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3 ProjectileMulti-fragmentationEvaporation TargetAbrasion Pictorial Scenario PLFProjectile fragmentation4

4Abrasion Stage Calculation Abrasion Cross section

We use an impact parameter dependent temperature profile T(b) for the PLF

PLF size for different reactions Overlapping volume V(b) (participant region) of projectile & target using straight-line geometry ( for different impact parameter b )

PLF Size : average number of proton () and neutron ( ) Probability of formation of PLF (Ns ,Zs)

using minimal distributionVs(b)=V0-V(b)5

Ref: S. Mallik, G.Chaudhuri & S. Das Gupta Phys. Rev. C 83 (2011) 044612 T is independent of projectile beam energy T depends on impact parameter (b).Temperature of PLF

T independent of As/AoFor all reactionsAs(b)/A0Simplest parametrizationFrom many sets of experimental data

It depends upon the wound of the original projectile which is (1.0 As/A0)6

Multi-fragmentation StageHigh excitation energy ExpansionDensity fluctuationBreaking into composites and nucleonsThermodynamic Equilibrium@ freeze-outHot primary fragments production PLF(As,Zs) Canonical Thermodynamical Model (CTM)Evaporation Stage :-(based on Monte-Carlo Simulation)Weisskopfs evaporation theory Decay Channels:- p, n, , d, t, 3He,

Hot primary fragmentsEvaporation ModelCold Secondary fragments7

Ref: G.Chaudhuri & S.Mallik Nucl. Phys. A 849 (2011) 190 Ref: C. B. Das , S. Das Gupta et al. , Phys . Rep. 406 (2005) 1 Canonical Thermodynamical Model (CTM)

Baryon & charge conservation constraintsni,j=No of fragments with i neutrons & j protonsCanonical Partition function of PLF AS (ZS ,NS) i,j=Partition function of the fragment ni , j Computationally difficult ! Recursion relation

An exact computational method which avoids Monte Carlo by exploiting some properties of the partition functionMost important feature of our modelPossible to calculate partition function of very large nuclei within secondsCrux of the model8

CTM contdPartition function of the fragment ni ,j

Intrinsic part of the partition function

translational intrinsicLiquid drop modelFermi-gas modelAverage no. of composites {i,j} or Multiplicity

Cross-section after multi-fragmentation stage:- abrasionmultifragmentation9

Model summary..Results Different Target-projectile combinations Incident energy ObservablesComparison with experimental data10

observablesAs(b) Zs(b) CTM +evaporationAbrasion Model PLF sizeFreeze-out volume=3V0

Projectile size (A0,Z0 )& target size (At, Zt )Comparison of theoretical and experimental temperature profile Experimental Temperature ProfileBy isotope thermometry methodGood agreement

solid lines modelSquares with error bars data= ZS - No. of Z=1 fragmentsZboundExperiment:- 600 MeV/nucleon (ALADIN @GSI) 107Sn and 124Sn on natural Sn 11

dashed lines modelsolid lines dataVariation of IMF multiplicity with ZboundIMF size: 3 Z 20Nice agreement with dataExperiment :- 600 MeV/nucleon (ALADIN @GSI) 107Sn and 124Sn on natural Sn 12

Differential Charge Distribution in Projectile Fragmentation Lower Zbound range

higher T of PLF breaks into many fragments of very small charge.

Steeper Charge distribution

Higher Zbound range

Lower T of PLF fragmentation is less, both low & high Z fragments

U shaped Charge distributiondashed lines modelsolid lines dataExperiment:- 600 MeV/nucleon (ALADIN)(At different Zbound intervals) 13

Largest Cluster in Projectile Fragmentation

Average size of largest cluster dashed lines modelsolid lines dataExperiment :- 600 MeV/nucleon (ALADIN @GSI) 107Sn and 124Sn on natural Sn Nice agreement with experimentProbability that zm is the largest cluster14

Charge Distribution in Projectile Fragmentation58Ni+9Be 140 MeV/nucleon (MSU)136Xe+208Pb 1 GeV/nucleon (GSI)Experimentally Different Beam EnergyTheoretically Same Temperature Profile

58Ni+181Ta 140 MeV/nucleon (MSU)129Xe+27Al 790 MeV/nucleon (GSI)15Ref: S. Mallik, G.Chaudhuri & S. Das Gupta Phys. Rev. C 84 (2011) 054612 The trend is nicely reproduced for all the reactionsdashed lines modelsolid lines dataIsotopic Distribution in Projectile Fragmentation

58Ni+9Be 140 MeV/nucleon (MSU Experiment)Circles joined by dotted lines modelSquares with error bars dataNice agreement with data16

SUMMARY The model for projectile fragmentation is grounded in traditional concepts of heavy-ion reaction (abrasion) plus the well known model of multifragmentation (Canonical Thermodynamical Model).

The model is in general applicable and implementable above a certain beam energy.

Universal temperature profile (depending on impact parameter) is introduced as input for different target-projectile combinations & widely varying energy of the projectile.

The model is able to successfully reproduce a wide variety of experimental observables like charge & mass distribution, isotopic distributions, IMF multiplicity, size of largest cluster .

Microscopic BUU calculations is being done in order to estimate the size & excitation of the initial PLF at different impact parameters. The work is in progress.

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18Thank YouFluctuation in number of IMFs for small Projectile like fragments:-Black solid lines dataRed dotted lines direct calculationZbound=ZS- No. of Z=1 fragmentZbound=Non-integer

Experiment :-Zbound=Integer (Due to event by event measurement)Theoretical Calculation :-(Due to average no. of fragment calculation)

Sn107+Sn119Sn124+Sn119Zbound=3=1Zbound=5=1Zbound=4MIMF=1MIMF=0 is calculated by modifying the CTM with experimental decay scheme of different energy levels.Fluctuation contdBlack solid lines dataRed dotted lines direct calculationBlue triangles modified calculationSn107+Sn119Sn124+Sn119

AX3 BX3 +neutron(s)AX4 BX3 +neutron(s)+protonAX5 BX3 +neutron(s)+2 protonsAX5 BX5 +neutron(s)AX5 BX3 +CHe2 +neutron(s)AX4 BX4 +neutron(s)AX4 CHe2 +DHe2 +neutron(s)Block diagram of the evaporation model :- ni (Ai, Zi ) & Ei* (Ai, Zi ) t=0 (Excited Fragments from CTM)1st Monte-Carlo Simulation(Evaporation/fission or not)2nd Monte-Carlo Simulation (which type of evaporation or fission)3rd Monte-Carlo Simulation(Ek of evaporated particle) Adjustment of A, Z & E*nf(Af, Zf )(Secondary Fragments)NOYESCalculation of different decay widths (Weisskopf Formalism)

Energetically further evaporation/fission or notNOt=t+tttotYES2122Partial Decay Width for the emission of a light particle Bohr-Wheeler Fission Width

Fission Barrier

Variation of ZS and Zbound with impact parameter:-2324