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Symmetry energy from the Canonical Thermodynamic Model of Nuclear Multifragmentation

Symmetry energy from the Canonical Thermodynamic Model of Nuclear Multifragmentation. Gargi Chaudhuri Variable Energy Cyclotr on Centre India. NuSYM11. MOtivation.

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Symmetry energy from the Canonical Thermodynamic Model of Nuclear Multifragmentation

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  1. Symmetry energy from the Canonical Thermodynamic Model of Nuclear Multifragmentation Gargi Chaudhuri Variable Energy Cyclotron Centre India NuSYM11

  2. MOtivation • Examining the different functional relationships between the isoscaling and isotopic observablesand symmetry energy coefficient csym in the framework of thecanonical thermodynamical model. • Study the effects of secondary decay on the • observables sensitive to symmetry energy

  3. Statistical Multifragmentation Models • Formation of highly excited nuclear system (high E* & internal pressure) expansionbreak-up into hotde-excitationcold secondary (density fluctuations)primary fragmentsfragments STATISTICAL MODELS Intensive exchange of mass, charge, energy during expansion Thermodynamic equilibrium prior to break up at r < r0 probability of a break-up channel (final state) Monte Carlo simulations statistical weight in the available phase-space  Canonical Thermodynamical Model (CTM) (Subal Das Gupta et al.) (A grand canonical version is also there) No Monte Carlo (much simpler)

  4. Canonical Thermodynamical Model (CTM) Computationally difficult Canonical partition function of a nucleus A0 (N0,Z0) ni,j No of composites with i neutrons and j protons all partitions Baryon & charge conservation wi,j Partition function of the composite {i,j} Crux of the model Recursion relation An exact computational method which avoids Monte Carlo by exploiting some properties of the partition function Possible to calculate partition function of very large nuclei within seconds Most important feature of our model

  5. CTM contd...… Partition function of the fragment {i,j} translational Intrinsic • Constant Csym as input • Fragments at normal density Csym=23.5 MeV For A > 4 Liquid drop formula Interaction • The available volume is Vf = V(freeze-out volume) – V0(volume of A0) • No interaction between composites except for Coulomb and excluded-volume correction. • Coulomb interaction between composites through Wigner Seitz approximation Ref: C. B. Das et al. , Phys. Rep. 406 (2005) 1

  6. CTM contd… Inputs Canonical Thermodynamical Model A0, Z0, ,T, r/r0 Q N , Z for all N,Z Outputs Average no. of primary fragments Evaporation Model Secondary fragments hot cold • The evaporation model is based on Monte-Carlo Simulation. • Weisskopf’s evaporation theory used. • Decay Channels:- p, n, α, d, t, He3, γ, fission • Outputs:-Secondary fragments Ref: G.Chaudhuri et. al. Nucl. Phys. A 849 (2011) 190

  7. Isoscaling (source) Formula Two reactions 1 & 2 have different isospin asymmetry but nearly same T Formula from EES model (Formula 1) M.B. Tsang et. al., PRC 64, 054615 (2001) Grand canonical model Isoscaling parameter (related to mn)

  8. dmn from canonical model 112/124Sn on 112/124Sn central collisions • Extracting Cs from dmn using Eq. 1 • difficult because of T dependence • The curves with different values of • Cs approach each other as T rises dmn Eq. 1 Eq. 1 dmn increases with T from our model ; it is a constant in formula 1. Formula 1 cannot be exact at finite T

  9. Relating dmn to Csymin canonical model • T 0 leads to formula 1 • At very high T dmn= T ln[N0 (2)/N0 (1)] (not a function of Csym) Similar result from Percolation Model • No simple formula at any general T dmn must be an evolving f(n) of T from T =0 to very high T Formula 1 does not have any explicit T dependence

  10. Isoscaling in CTM 112/124Sn on 112/124 Sn central collisions A1=168 , A2=186 Z1=Z2=75 Primary Isoscaling nicely valid for primary fragments linear slope, a can be extracted easily Secondary Isoscaling approximately valid for secondary fragments; improves for higher Z values

  11. Variation of Isoscaling Parameter With Temperature A1=168 , A2=186 Z1=Z2=75 a calculated from the slope of the ratios • α from slope of the ratios & • from mn match very well Isoscaling parameter a decreases marginally after evaporation from T=4 MeV

  12. Variation of Isoscaling Parameter With Input Symmetry energy for primary and secondary fragments Primary dotted line A1=168 , A2=186 Z1=Z2=75 Secondary solid line T=5 MeV black line T=7 MeV  red line • a vs cinsym almost linear for primary fragments. • Temperature independent afor very low Cinsym. • a becomes less sensitive to Csym • after secondary decay, especially for higher T • Similar results from SMM 5 MeV 7 MeV Extraction of symmetry energy from isoscaling analysis should be done cautiously.

  13. Isoscaling fragment formula (Formula 2) Approximate grand canonical expression connecting csym with Z/<A> of fragments α(z)  isoscaling slope parameter of a fragment of charge z <Ai>  average mass number of a fragment of charge z produced by source i(=1,2) (Formula 2) A. Ono. et al., PRC68, 051601(R) (2003) Assumptions • Isotopic distributions are essentially Gaussian. • Free energies contain only bulk terms. • This formula tested with the canonical thermodynamical model coupled with an evaporation code. • Results presented for both the primary(at beak up stage) fragments and for the secondary fragments (after evaporation).

  14. Fluctuation formula (Formula 3) • Yield from grand canonical model • Gaussian approximation on grand canonical expression • Isospin variance related to Csym Formula 3 Ad. R. Raduta. et al., PRC 75,044605 (2007) • Csym is the symmetry energy of fragment Z • σ(Z) is width of isotopic distribution

  15. Variation with Z (Input Csym=23.5 MeV) Csym/T=a/4D T =5.0 MeV Source Formula  (1)  black line Fragment Formula  (2)  red line Fluctuation Formula (3) blue line Csym/T vs Z a & 4D vs Z Primary dotted line a Secondary solid line 4D For primary fragments, Csym/T from formulae 1, 2 & 3 close to each other. • & 4D (fromformula 2)both decrease after evaporation. Large increases after evaporation for formulae 2 & 3; Small decrease after evaporation for formula 1

  16. Variation with Z (Input Csym=15.0 MeV) Csym/T=a/4D Source Formula  (1)  black line Fragment Formula  (2)  red line Fluctuation Formula (3) blue line T =5.0 MeV a & 4D vs Z Csym/T vs Z Primary dotted line Secondary solid line a 4D • Similar trends of results as in the case of input Csym=23.5 MeV for formula (2) & (3) • For formula (1), Csym/T increases after evaporation in contrary to Csym=23.5 MeV • increases while • D (fromformula 2)decreases after evaporation

  17. Csym / T from projectile fragmentation reactions Source Formula  (1)  black line Fragment Formula  (2)  red line Fluctuation Formula (3) blue line Comparison with experimental data (Solid line) M.Mocko , P.hD Thesis MSU 58/64Ni on Be at 140 MeV/n Model Abrasion + CTM + evaporation T =4.25 MeV Before evaporation The results from primary from all the three formulae are close to each other cold Formulae 2 & 3 after evaporation hot • Csym/T increases from primary. • Results close to experimental values. • Csym deduced is about 2-3 times input Csym cold PRC 81,044620(2010) Formula 1 after evaporation • Csym/T decreases from primary. Similar results from AMD + Gemini codes from isobaric yield ratios. • and is away from experimental values.

  18. Variation of Output Csym with Input Csym Primary dotted line Source Formula  (1) Fragment Formula  (2) Fluctuation Formula (3) Secondary solid line 7 MeV  red line 5 MeV  black line Formula (1) Formula (2) & (3) 7 MeV 5 MeV • Linear correlation between input and output Csym for hot fragments. • For hot fragments, output Csym is almost equal to that of input Csym. • For formula 1,Csymout becomes less sensitive to input Csym after secondary decay. • For formula 2 & 3, • For the cold fragments, output Csym is 2-3 times of the input Csym . • As temperature increases, disagreement between input and output Csym increases .

  19. aTfrom different pairs of sources Primary Black dotted line Source sizes used 168, 177, 186; Z=75 Secondary Red solid line T=5 MeV Formula 1 Formula 2 Primary Csym=22.61 MeV Primary Csym=29.57 MeV Secondary Csym=21.74 MeV Secondary Csym=62.06 MeV Csym from hot & cold fragments very close Csym from primary close to input Csym Secondary results almost twice of primary

  20. Summary Isoscaling parameter a • Isoscaling nicely valid for the hot fragments, only approximately valid for the cold. • After evaporation, the isoscaling parameter a increases or decreases depending on value of input Csym. • a becomes less sensitive to input Csym after secondary decay.

  21. Summary contd... Isoscaling (source) formula (1) • dmnanalysis reveal that Formula 1 is not good for finite temperature. • Linear correlation between input and outputCsym for the hot fragments. • Csymout becomes less sensitive to Csym inafter secondary decay. • Need to decrease Csyminin the model in order to match experimental data.

  22. Summary contd... Isoscaling (fragment ) formula (2) & the fluctuation formula (3) • Results from both the formulae close to each other for hot and cold fragments. • For the hot fragments, Csym deduced close to that of the input Csym. • For the cold fragments, Csym/T values agree with that from the experiment. • Csym deduced is more than twice that of the input value for the cold fragments . These formulae not good for extraction of Csym from cold fragments.

  23. Comments...... • At high temperature, in the multifragmentation regime, no formula gives a • satisfactory reproduction of the input csym • In multifragmentation models, fragments formed at normal density....hence • not advisable to extract density dependence from such models.. • The different models have to be carefully compared to a large number of • independent observables before one can safely draw any conclusion. Collaborators 1. Prof. Francesca Gulminelli, LPC Caen, France. 2. Prof. Subal Das Gupta, McGill University, Montreal, Canada. 3. Swagata Mallik , VECC, Kolkata, INDIA.

  24. Thank You

  25. ni (Ai, Zi) & Ei* (Ai, Zi) t=0 (Excited Fragments from CTM) Calculation of different decay widths (Weisskopf Formalism) YES NO Energetically further evaporation/fission or not 1st Monte-Carlo Simulation (Evaporation/fission or not) t=t+Δ t≤ttot Adjustment of A, Z & E* YES 2nd Monte-Carlo Simulation(which type of evaporation or fission) NO 3rd Monte-Carlo Simulation (Ek of evaporated particle) nf(Af, Zf) (Secondary Fragments) Block diagram of the evaporation model

  26. Reactions • Reaction 1 Sn112 +Sn112central collisions Dissociating system N0= 93, Z0 = 75, A0 = 168 • ForA=5 & 6,includeZ= 2 & 3 ; A=7, includeZ= 2, 3 & 4 • For higher masses drip lines computed using liquid-drop formula • and include all isotopes within these boundaries • Reaction 2 Sn124 +Sn124central collisions Dissociating system N0= 111, Z0 = 75, A0 = 186

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