Break-up of nucleus Projectile Target Energy E* ~ few MeV/n < 1 MeV/n (p, α, π, heavy ions) Accelerators fraction of MeV/n to several GeV/n

Nuclear Multifragmentation 4 E*  1 MeV/n  sequential decay or fission E* > 3 MeV/n  continuous flux of nucleons & light clusters from decaying nucleus E*  MeV/n  multiple emission of fragments of different mass “multifragmentation” nuclear system decays to a final configuration that contains multiple fragments of charge 3 ≤ Z ≤ 20 (IMF) (explosion like process leading to total disintegration of nucleus ) - Discovered in cosmic rays - Later observed in accelerator experiments  emission >  relaxation  emission   relaxation  emission <  relaxation Time scales (J.P.Bondorf) 1976

5 Intermediate mass fragments (IMF) Enhanced IMF yield Multiple IMF emission W(M)  Probability of emitting M IMF in a single break-up event Rise and fall of IMF with Impact parameter b unambiguous signals of multifragmentation yield

6 Some typical experimental data 1 GSI (2003)11.5 ANL (1975) 300 (1974) Mass distribution and charge distribution

7 Multifragmentation Experimental approaches for multifragmentation studies  Central heavy-ion collisions at incident energies of E/A ≤ 100 MeV  Larger impact parameter heavy ion collisions at E/A > 100 MeV (Projectile/Target spectator fragmentation)  Central light ion induced reactions at E ≥ 5 GeV e.g,  - /p + Au Break-through in multifragmentation studies in intermediate-energy heavy-ion reactions made by 4  detector systems  ALADIN and FOPI at GSI  MINIBALL at MSU  INDRA at GANIL  DUBNA (Russia)  CATAINA (Italy)  RIKEN (Japan)  VECC (India) (Under preparation)

Basic Motivation How to calculate the multiplicity of fragments of different mass and charge ?

Scenario of Multi-fragmentation Projectile Target Multi- fragmentation Evaporation Collision Pre-equilibrium Emission Three Main Stages 1.Formation of Excited System ( A, N, Z, E*) 2. Formation of the fragments 3. Secondary decay of the fragments Dynamical models Dynamical or Statistical models Statistical Evaporation Models 123

collision compression expansion multifragmentation secondary de-excitation (Schematic View) Objective To have a solvable model which describes the physics of the situation at freeze-out when one averages over many nucleus- nucleus collisions

high E*  expansion  density  high density regions freeze (V~3-6V 0 ) & pressure fluctuation form composites out  Intensive exchange of mass, charge, energy Statistical equilibrium prior to break up T, S Laws of equilibrium thermodynamics applicable Statistical Multifragmentation Models  large no. of initial states  large no. of nucleon- nucleon collisions  huge number of open decay channels almost all the relevant phase space available Statistical Ensembles Complex system

Statistical Models  Microcanonical model ( S. Koonin & J. Randrup)  Microcanonical Model (D.H.E Gross et al.)  Statistical Multifragmentation Model SMM (J. P. Bondorf et al.)  Microcanonical Multifragmentation Model MMM (A. Raduta et al.)  Canonical & Grand Canonical Models (S. Das Gupta et al.) Monte Carlo simulations Canonical Thermodynamical Model much simpler No Monte Carlo ! Grand Canonical Thermodynamical Model Finite nuclei calculationsNuclear matter calculations

Canonical Thermodynamical Model (CTM) :- Basic Assumption:- The system remains in thermo-dynamical equilibrium at temperature T & freeze-out volume V f =(Few times original nuclear volume). One Component System :- Partition function for a system of x identical particles Partition function for a spin less particle without any internal structure Basic Formulae:-

If a system of A particles breaks into n i fragments (having i particles) then where Considering particle conservation Actual Expression for partition function Probability of occurrence of a given channel CTM contd…

Canonical Thermodynamical Model (CTM ) probability of a break-up channel (final state) Statistical weight in available phase-space In canonical ensemble where f(y) = -T ln Q(y) & Q(y) is canonical partition f(n) in channel y probability of a break-up channel (final state) Canonical partition function of a nucleus A 0 (N 0,Z 0 ) all partitions Baryon & charge conservation N o of composites with neutron i & proton j Partition function of the composite {i,j} composites(i,j) n i,j  i,j

No of partitions P 0 for medium and heavy P 0 (100)= nucleus enormous and grows rapidly with A P 0 (200)= Calculation of the partition function Computationally difficult ! (baryon & charge conservation constraints) Recursion relation Possible to calculate partition function of very large nuclei within seconds No Monte-Carlo ! Q N0,Z0 = f(n) of all lower Q’s Probability of a given channel Average no. of composites {i,j}

Partition function of the fragment {i,j} Calculation of the partition function translationalintrinsic The intrinsic part Liquid-drop Model Fermi-gas model   The available volume is V = V f (freeze-out volume) – V 0 (volume of A 0 )  No interaction (except Coulomb) between the fragments is considered at freeze-out.  Coulomb interaction between composites via Wigner Seitz approximation.  Drip lines calculated using LDM and all isotopes within the boundaries are included. Assumptions

High excitation of compound nuclear system Expansion Density fluctuation Breaking into composites and nucleons Thermodynamic freeze-out Hot primary fragments production Canonical Thermodynamical Model (CTM) Decay Channels:- p, n, α, d, t, 3 He, γ Hot primary fragments Evaporation Model Cold Secondary fragments Block diagram

20 Canonical Thermodynamical Model (CTM ) observables A 0, Z 0 T,  /  0 CTM + evaporation fixed freeze out density fixed freeze out density  Average multiplicity of the composites and various moments of multiplicity distribution.  Total charge and mass distribution, Isotopic/Isobaric distributions  IMF Multiplicity  Size and Charge of the largest clusters and their RMS fluctuation Observables

21 Classification of nuclear disintegration mechanisms U-shapepower-lawexponential Change of N A with E* key feature of multifragmentation Mass distributions

Some results from CTM Studied Reaction : p+Au 197 Variation of emitted IMF with temperature Mass Distribution

23 Dependence on Freeze-out volume (V f /V 0 ), source size(A 0 ) and source isospin (N 0 /Z 0 ) Total multiplicity

Total Number of Fragments 1.Higher Mass Fragment (Z > Z 0 /3) 2.Inter-mediate Mass Fragment (2 < Z ≤ Z 0 /3) 3.Lower Mass Fragment (Z ≤ 2) Higher Mass FragmentLower Mass FragmentIntermediate Mass Fragment Variation of Fragment Production with Temperature

25 Black solid lines  58 Ni+ 9 Be Red dotted lines  64 Ni+ 9 Be Blue dashed lines  68 Ni+ 9 Be Projectiles:- 58 Ni, 64 Ni, 68 NiTarget:- 9 Be Useful Guideline for upcoming facilities Isotopic Distribution

26 Cross-sections of neutron rich nuclei from Projectile fragmentation Si isotopes from 48 Ca + 9 Be Cu isotopes from 86 Kr + 9 Be  Agreement is fair except at the tails Data from MSU data Data from RIKEN  Model gives the rapid decrease of cross section for large A  Cross sections useful in planning experiments with n-rich isotopes produced from projectile fragmentation 140 MeV/n 64MeV/n

Assumptions  Freeze out volume V is taken to be multiplicity independent. (Excellent agreement with SMM model which has this dependence)  Since the approximation of non-interacting composites gets worse as density increases, hence the model is expected to be valid at low and moderate densities (  /  0 3 MeV). We have studied …..  Bimodality Cross-sections of n-rich nuclei Isoscaling & Symmetry Energy  Projectile & target dependence of fragmentation cross-section  Isospin fractionation  Hypernuclei production and others…………

A different view of multifragmentation 28 Thermodynamic properties of hot nuclear matter and finite nuclei Phenomenological approach Variational methods Hartree-Fock formalism Thomas-Fermi approximation Relativistic mean field Quasiparticle approximation Equation of state has Vander-Waals behavior Liquid-Gas Phase Transition Physical picture At  <  0 T < T c homogeneous distribution of nuclear matter thermodynamically unstable w.r.t splitting into liquid (dense) and gaseous (dilute) phases multifragmentation

29 The nuclear liquid-gas phase transition IsothermsIsobars  c   0 T c  MeV region below dotted line (spinodal region) has –ve incompressibility Single homogeneous phase unstable Breaking up into a gas phase and a liquid phase at equilibrium Spinodal decomposition possible mechanism for multifragmentation

Nuclear Matter in Grand Canonical model intrinsic partition function neutron chemical potential proton chemical potential Multiplicity of a composite ( N,Z) sum over composites allowed by drip lines Start with a chosen value of  1 / T, V f freeze out volume A max size of largest composite N 0 /Z 0 asymmetry parameter Eq(n)s to determine  n and  p Pressure P = T  / V f (No Coulomb) (Same prescription as in canonical) A max ≤ A 0

 As temperature increases, phase coexistence finally disappears Phase coexistence  Rise of pressure at small ρ followed by a flattening with increasing ρ signature of 1 st order phase transition coexistence phase gas phase Pressure-density isotherms Nuclear matter calculation with Grand Canonical Model

 At low temperature small as well as large fragments are formed (Coexistence Phase).  With the increase of temperature production of large fragments decreases.  At higher temperature only light mass fragments are produced. (Gas Phase) Liquid Gas Coexistance :-

33 Mass Distribution

Projectile Multi-fragmentation Evaporation Target Abrasion PLF Projectile fragmentation Three Main Stages of Projectile Fragmentation:- 1.Abrasion 2. Multi-fragmentation 3. Evaporation Projectile Fragmentation:-

Botlzmann Uehling Uhlenbeck (BUU) Model:- Boltzmann-Uehling-Uhlenbeck(BUU) equation describes the time evolution of the phase space density =Collision term For numerical implementation of test particle method,

Studied Reaction: 58 Ni+ 9 Be Energy=140 MeV/nucleon Impact Parameter=4fm Time Evolution:- Red Points → Projectile Test Particles Green Points → Target Test Particles

37  Largest cluster distribution  Isospin Fractionation  Isospin effects  Isoscaling & symmetry energy  Projectile and target dependence of the fragments  Cross sections of neutron rich nuclei from projectile fragmentation  Hypernuclei production  Equation of State (EOS) of nuclear matter and finite nuclei  Signatures of Phase Transition from Thermodynamic Variables (Bimodality of order parameter, Caloric Curves etc)  Conditions of equivalence between different thermodynamical ensembles.  Comparison between different theoretical models.  Nuclear astrophysics  Hypernuclei  Collective Flow Theoretical Studies... Experimental Interests…

References J. P. Bondorf et al., Physics Reports 257 (1995) 133. C. B. Das et al., Physics Reports 406 (2005) 1. S. Das Gupta et al., Advances in Nuclear Physics 26 (2001) 91.