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Phase Transitions in Nuclear Reactions

Phase Transitions in Nuclear Reactions. Definition of phase transitions Non analytical thermo potential Negative heat capacity (1 st order) Liquid gas case C<0 and Volume (order parameter) fluctuations Thermo of nuclear reactions Coulomb reduces coexistence and C<0 region

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Phase Transitions in Nuclear Reactions

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  1. Phase Transitions in Nuclear Reactions • Definition of phase transitions • Non analytical thermo potential • Negative heat capacity (1st order) • Liquid gas case • C<0 and Volume (order parameter) fluctuations • Thermo of nuclear reactions • Coulomb reduces coexistence and C<0 region • Statistical description of radial flow • Conclusion and application to data Ph. Chomaz and F. Gulminelli, Caen France

  2. I -I- Definition of phase transition • Discontinuity in derivatives of thermo potential when • 1st order equivalent to negative heat capacities when

  3. Phase transition in infinite systems Thermodynamical potentials non analytical L.E. Reichl, Texas Press (1980) Caloric curve Order of transition: discontinuity b Temperature Ehrenfest’s definition E1 E2 Energy Ex: first order: EOS discontinuous R. Balian, Springer (1982)

  4. Complex b Im(b) b Re(b) • Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404 1st order in finite systems PC & Gulminelli Phys A (2003) • Order param. free (canonical)

  5. Complex b Im(b) b Re(b) b E distribution at • Zeroes of Z reach real axis Yang & Lee Phys Rev 87(1952)404 • Bimodal E distribution(P(E)) Energy K.C. Lee Phys Rev E 53 (1996) 6558 Caloric curve b Temperature E1 E2 E1 E2 Energy • Back Bending in EOS(T(E)) K. Binder, D.P. Landau Phys Rev B30 (1984) 1477 Ck(3/2) • Abnormal fluctuation (sk(E)) J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153 E1 E2 Energy 1st order in finite systems PC & Gulminelli Phys A (2003) • Order param. free (canonical) • Order param. fixe (microcanonical) sk /T2

  6. II -II- Liquid gas transition • Negative heat capacity in fluctuating volume ensemble • Difference between CP and CV • Negative heat capacity in statistical models • Neg. C and Channel opening

  7. Canonical lattice gas at constant V T=10 MeV 8 Pressure ( MeV/fm3) -4 -2 0 2 4 6 8 6 4 0 0.2 0.4 0.6 0.8 r / r Density 0 Volume: order parameter L-G in a box: V=cst Lattice-Gas Model • Negative compressibility Gulminelli & PC PRL 82(1999)1402

  8. Canonical lattice gas at constant V T=10 MeV 8 Pressure ( MeV/fm3) Thermo limit -4 -2 0 2 4 6 8 6 4 Thermo limit 0 0.2 0.4 0.6 0.8 r / r Density 0 Volume: order parameter L-G in a box: V=cst Lattice-Gas Model • CV > 0 • Negative compressibility Gulminelli & PC PRL 82(1999)1402 See specific discussion for large systems Pleimling and Hueller, J.Stat.Phys.104 (2001) 971 Binder, Physica A 319 (2002) 99. Gulminelli et al., cond-mat/0302177, Phys. Rev. E.

  9. Isobar canonical lattice gas Isobar ensemble P(i) µexp-lV(i) Open systems (no box)Fluctuating volume PC, Duflot & Gulminelli PRL 85(2000)3587 • Bimodal P(V) • Negative compressibility • Bimodal P(E) • Negative heat capacity Cp<0 microcanonical lattice gas

  10. Isochore canonical lattice gas Constrain on Vi.e. on order paramameter • Suppresses bimodal P(E) • No negative heat capacity CV > 0 microcanonical lattice gas See specific discussion for large systems Pleimling and Hueller, J.Stat.Phys.104 (2001) 971 Binder, Physica A 319 (2002) 99. Gulminelli et al., cond-mat/0302177, Phys. Rev. E.

  11. Probability 2 Coulomb interaction VC 1.6 1.2 0.8 0.4 Probability 0 2 4 6 8 10 12 14 Nuclear energy C<0 in statistical modelsLiquid-gas and channel opening • q = 0 => no-Coulomb • Many channel opening • One Liquid-Gas transition Bimodal P(EN)

  12. III -III- Thermo of nuclear reaction • Coulomb reduces coexistence and C<0 region • Statistical treatment of Coulomb • (EN,VC) a common phase diagram for charged and uncharged systems • Statistical treatment of radial flow • Isobar ensemble required • Phase transition not affected

  13. Coulomb reduces L-G transition • Reduces C<0 • Lattice-Gas • OK up to heavy nuclei • Reduces coexistence • Bonche-Levit-Vautherin Nucl. Phys. A427 (1984) 278 A=207 Z= 82 Gulminelli, PC, Comment to PRC66 (2002) 041601

  14. Statistical treatment of Non saturating forces interactions - Effective charge q = 1 charged system - - Effective charge q = 0 uncharged system - • A unique framework: e-bE = e-bNEN -bCVC • Introduce two temperatures bN=band bC =q2b • O two energies ENand VC •  bC = 0 Uncharged bC = bNCharged

  15. 2 1.6 1.2 0.8 0.4 0 2 4 6 8 10 12 14 With and without Coulomb a unique S(EN,VC) Probability • (EN ,VC) a unique phase diagram • Coulomb reduces E bimodality • different weight • rotation of E axis 0 4 8 12 Energy Coulomb interaction VC bC =  Uncharged bC = bNCharged E=EN+VC, charged E=EN, uncharged Energy Gulminelli, PC, Raduta, Raduta, submitted to PRL

  16. 0 0 2 2 4 4 6 6 8 8 10 10 12 12 14 14 Correction of Coulomb effects • If events overlap Energy distribution: Entropy Reconstructed Exact Probability 2 2 1.6 1.6 bC = 0 Uncharged Coulomb interaction VC Coulomb interaction VC 1.2 1.2 bC = bN Charged Reconstruction of the uncharged distribution from the charged one 0.8 0.8 Energy Energy Gulminelli, PC, Raduta, Raduta, submitted to PRL

  17. 4 bC = 0 Uncharged 3 Coulomb interaction VC 2 1 bC = bNCharged 0 2 4 6 8 10 12 14 Energy Partitions may differ for heavy nuclei • Channels are different (cf fission) • Re-weighting impossible • However, a unique phase diagram (EN VC) Gulminelli, PC, Raduta, Raduta, submitted to PRL

  18. x Thermal distribution in the moving frame Flow induces negative pressure z Expansion Radial flow at “equilibrium” • Equilibrium = Max S under constrains • Additional constrains: radial flow<pr(r)> • Additional Lagrange multiplier(r) • Self similar expansion<pr(r)> = m a r =>(r) =  r • Requires a confining constrain => isobar ensemble Gulminelli, PC, Nucl-Th (2002)

  19. x z Expansion Radial flow at “equilibrium” • Does no affect r partitioning • Only reduces the pressure • Shifts p distribution • Changes fragment distribution: fragmentation less effective • Requires a confining constrain => isobar ensemble pressure Gulminelli, PC, Nucl-Th (2002)

  20. IV -IV- Conclusion and application to data • Exact theory of phase transition • Negative heat capacity in finite systems • Liquid-gas and role of volume • Negative heat capacity in open systems • Role of Coulomb • C<0 phenomenology qualitatively preserved • Role of flow • Thermodynamics of the isobar ensemble

  21. Canonical: total energy Ztot = Zk Zp sE2 = sk2 + sp2 2b log Zi = Ci /b2 = si2 • Microcanonical: partial energy Wtot = Wk  Wp sE2 = 0, sk2 = sp2 2b log Wtot = -1/CT2 = f(sk2) PC, Gulminelli NPA(1999) Heat capacity from energy fluctuation

  22. Canonical: total energy Ztot = Zk Zp sE2 = sk2 + sp2 2b log Zi = Ci /b2 = si2 • Microcanonical: partial energy Wtot = Wk  Wp sE2 = 0, sk2 = sp2 2b log Wtot = -1/CT2 = f(sk2) PC, Gulminelli NPA(1999) Heat capacity from energy fluctuation

  23. 2/T2 T V=cte p=cte px Tsalis ensemble pz Transparency x z Expansion A robust signal • Depends only on the state PC, Duflot & Gulminelli PRL 85(2000)3587 • Out of equilibrium • With flow (20%) Gulminelli, PC Phisica A (2002) Gulminelli, PC Nucl-th 2002

  24. E = m m + E E • 2 i i i coul coul E =E -E * • 1 2  • <E >=< a > T 2 + 3/2 <M-1> T 1 i i Multifragmentation experiment Sort events in energy (Calorimetry) Reconstruct a freeze-out partition 1- Primary fragments: 2- Freeze-out volume: => <E1>, 1 3- Kinetic EOS: T, C1

  25. Ni+Ni, 32 AMeV Quasi-Projectile 2 D’Agostino et al. Hot Au, ISIS collaboration Fluctuation, Heat Capacity 1 INDRA 0 0 2 4 6 8 Excitation Energy (AMeV) Bougault et al Xe+Sn central ISIS ISIS INDRA Heat capacity from energy fluctuation Multics E1=20.3 E2=6.50.7 Isis E1=2.5 E2=7. Indra E2=6.0.5 MULTICS MULTICS Excitation Energy (AMeV)

  26. Au+Au 35 EF=0 Au+Au 35 EF=1 Comparisoncentral / peripheral peripheral M.D ’AgostinoIWM2001 central Ckin= dEkin/dT Up to  35 A.MeV the flow ambiguity is a small effect

  27. Heat capacity from any fluctuation • From kinetic energy

  28. Heat capacity from any fluctuation • From kinetic energy • From any correlated observable (Ex. Abig) • r = skA /sksA correlation • sA “canonical” fluctuations

  29. Correction of experimental errors • Heat capacity from fluctuations • scan can be“filtered” • (apparatus and procedure)

  30. Correction of experimental errors • Heat capacity from fluctuations • scan can be“filtered” • (apparatus and procedure) Corrected freeze-out fluctuation • sk can be corrected • Iterate the procedure • Freeze-out reconstruction • Decay toward detector Reconstructed freeze-out fluctuation

  31. No Heat Bath E  Etr Order parameters: , Heat Bath  = tr = (SA30- SA2-30)/As

  32. - 1b -Equivalence betweenbimodality of P(b)Zero's of the partition sum Z(bb)

  33. A alternative definition in the "intensive" ensemble • First order phase transition in finite systems is defined by • A uniform density of zero's of the partition sum on a line crossing the real baxis perpendicularly Yang Lee unit circle theorem

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