1. A MODEL FOR PROJECTILE FRAGMENTATION Collaborators: S. Mallik, VECC, India S. Das Gupta, McGill University, Canada 1 Gargi Chaudhuri

2.  Different Stages of Projectile Fragmentation  Our Model  Results  Comparison with different experimental data  Summary CONTENTS 2

3. 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 (A s, Z s ) expands to about 3V 0 – 4V 0.(V 0 -normal nuclear volume) & breaks up into many fragments (MULTIFRAGMENTATION)  The excited fragments de-excite by sequential decay (EVAPORATION) 3

4. Projectile Multi-fragmentation Evaporation Target Abrasion Pictorial Scenario PLF Projectile fragmentation 4

5. Abrasion 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 (N s,Z s ) using minimal distribution V s (b)=V 0 -V(b) 5 Ref: S. Mallik, G.Chaudhuri & S. Das Gupta Phys. Rev. C 83 (2011) 044612

6.  T is independent of projectile beam energy  T depends on impact parameter (b). Temperature of PLF T independent of A s /A o For all reactions A s (b)/A 0 Simplest parametrization From many sets of experimental data It depends upon the wound of the original projectile which is (1.0 – A s /A 0 ) 6

7. Multi-fragmentation Stage High excitation energy Expansion Density fluctuation Breaking into composites and nucleons Thermodynamic Equilibrium @ freeze-out Hot primary fragments production PLF(A s,Z s ) Canonical Thermodynamical Model (CTM) Evaporation Stage :- (based on Monte-Carlo Simulation) Weisskopf’s evaporation theory Decay Channels:- p, n, α, d, t, 3 He, γ Hot primary fragments Evaporation Model Cold Secondary fragments 7 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

8. Canonical Thermodynamical Model (CTM) Baryon & charge conservation constraints n i,j =No of fragments with i neutrons & j protons Canonical Partition function of PLF A S (Z S,N S ) ω i,j =Partition function of the fragment n i, j Computationally difficult ! Recursion relation An exact computational method which avoids Monte Carlo by exploiting some properties of the partition function Most important feature of our model Possible to calculate partition function of very large nuclei within seconds Crux of the model 8

9. CTM contd… Partition function of the fragment n i,j Intrinsic part of the partition function translationalintrinsic Liquid drop model Fermi-gas model Average no. of composites {i,j} or Multiplicity Cross-section after multi-fragmentation stage:- abrasionmultifragmentation 9

10. Model summary……….. Results……… Different  Target-projectile combinations  Incident energy  Observables Comparison with experimental data 10 observables A s (b) Z s (b) CTM + evaporation Abrasion Model PLF size Freeze-out volume=3V 0 Projectile size (A 0,Z 0 ) & target size (A t, Z t )

11. Comparison of theoretical and experimental temperature profile Experimental Temperature Profile By isotope thermometry method Good agreement solid lines  model Squares with error bars  data = Z S - No. of Z=1 fragments Z bound Experiment:- 600 MeV/nucleon (ALADIN @GSI) 107 Sn and 124 Sn on natural Sn 11

12. dashed lines  model solid lines  data Variation of IMF multiplicity with Z bound IMF size: 3 ≤ Z ≤ 20 Nice agreement with data Experiment :- 600 MeV/nucleon (ALADIN @GSI) 107 Sn and 124 Sn on natural Sn 12

13. Differential Charge Distribution in Projectile Fragmentation  Lower Z bound range higher T of PLF breaks into many fragments of very small charge. Steeper Charge distribution  Higher Z bound range Lower T of PLF fragmentation is less, both low & high Z fragments “U” shaped Charge distribution dashed lines  model solid lines  data Experiment:- 600 MeV/nucleon (ALADIN) (At different Z bound intervals) 13

14. Largest Cluster in Projectile Fragmentation Average size of largest cluster dashed lines  model solid lines  data Experiment :- 600 MeV/nucleon (ALADIN @GSI) 107 Sn and 124 Sn on natural Sn Nice agreement with experiment Probability that z m is the largest cluster 14

15. Charge Distribution in Projectile Fragmentation 58 Ni+ 9 Be 140 MeV/nucleon (MSU) 136 Xe+ 208 Pb 1 GeV/nucleon (GSI) Experimentally Different Beam Energy Theoretically Same Temperature Profile 58 Ni+ 181 Ta 140 MeV/nucleon (MSU) 129 Xe+ 27 Al 790 MeV/nucleon (GSI) 15 Ref: S. Mallik, G.Chaudhuri & S. Das Gupta Phys. Rev. C 84 (2011) 054612 The trend is nicely reproduced for all the reactions dashed lines  model solid lines  data

16. Isotopic Distribution in Projectile Fragmentation 58 Ni+ 9 Be 140 MeV/nucleon (MSU Experiment) Circles joined by dotted lines  model Squares with error bars  data Nice agreement with data 16

17. 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……. 17

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19. Fluctuation in number of IMFs for small Projectile like fragments:- Black solid lines  data Red dotted lines  direct calculation Z bound =Z S - No. of Z=1 fragment Z bound =Non-integer Experiment :- Z bound =Integer (Due to event by event measurement) Theoretical Calculation :- (Due to average no. of fragment calculation) Sn 107 +Sn 119 Sn 124 +Sn 119

20. Z bound =3 =1 Z bound =5 =1 Z bound =4 M IMF =1 M IMF =0 is calculated by modifying the CTM with experimental decay scheme of different energy levels. Fluctuation contd… Black solid lines  data Red dotted lines  direct calculation Blue triangles  modified calculation Sn 107 +Sn 119 Sn 124 +Sn 119 A X 3  B X 3 +neutron(s) A X 4  B X 3 +neutron(s)+proton A X 5  B X 3 +neutron(s)+2 protons …… A X 5  B X 5 +neutron(s) A X 5  B X 3 + C He 2 +neutron(s) …… A X 4  B X 4 +neutron(s) A X 4  C He 2 + D He 2 +neutron(s) ……

21. Block diagram of the evaporation model :- n i (A i, Z i ) & E i * (A i, Z i ) t=0 (Excited Fragments from CTM) n i (A i, Z i ) & E i * (A i, Z i ) t=0 (Excited Fragments from CTM) 1 st Monte-Carlo Simulation (Evaporation/fission or not) 1 st Monte-Carlo Simulation (Evaporation/fission or not) 2 nd Monte-Carlo Simulation (which type of evaporation or fission) 3 rd Monte-Carlo Simulation (E k of evaporated particle) 3 rd Monte-Carlo Simulation (E k of evaporated particle) Adjustment of A, Z & E * n f (A f, Z f ) (Secondary Fragments) n f (A f, Z f ) (Secondary Fragments) NO YES Calculation of different decay widths (Weisskopf Formalism) Energetically further evaporation/fission or not NO t=t+Δ t≤t tot t=t+Δ t≤t tot YES 21

22. 22 Partial Decay Width for the emission of a light particle ν Bohr-Wheeler Fission Width Fission Barrier

23. Variation of Z S and Z bound with impact parameter:- 23

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