Bimodality and latent heat of gold nuclei

Transcription

1 Bimodality and latent heat of gold nuclei Eric Bonnet (GANIL) for the INDRA and ALADIN collaboration IWM 09, Catania

2 Outline First order phase transition in finite system Definition and related signals starting from microcanonical entropy Canonical ensemble and Quasi Projectile fragmentation data Study of the bimodal distribution of the charge of the biggest fragment (Z1) in Au+Au reactions. Trace back the density of states of Gold nuclei Starting from the experimental 2D distribution Z1 E* (Excitation energy) Derive the equivalent canonical distribution Constrain canonical ensemble using comparison with data Extract properties for the hot nuclei de excitation

3 First order phase transition in finite system : Definition Presence of a residual convex intruder in the entropy S(X) of the mixed system. 2 X S(X) >0 define the spinodal zone X : order parameter (X) = X S : conjugate variable

4 First order phase transition in finite system : Consequences Bimodal canonical distribution of an order parameter 2 X S(X) >0 define the spinodal zone P can (X) = exp (S(X) X) What information can we get? Boundaries of the coexistence zone P X can (X) = 0 latent heat

5 First order phase transition in finite system : Consequences Bimodal canonical distribution of an order parameter 2 X S(X) >0 define the spinodal zone P can (X) = exp (S(X) X) What information can we get? Boundaries of the coexistence zone P X can (X) = 0 latent heat Boundaries of the spinodal zone 2 P X can (X) > 0

6 First order phase transition in finite system : Consequences Negative heat capacity (C) Experimental evidence of abnormal fluctuations of configurational energies P. Chomaz et al, NPA 647 (1999) 2 X S(X) >0 define the spinodal zone 2 E S = 1/CT2 >0 C < 0 Boundaries of the spinodal zone

7 First order phase transition in finite system : Results Negative heat capacity (C) Experimental evidence of abnormal fluctuations of configurational energies Le Neindre et al, NPA 795 (2007) INDRA M. Bruno et al, NPA 807 (2008) M. D'Agostino et al, PLB 473 (2000) Multics Miniball Au+Au 35 A MeV C k = canonical heat capacity k = configurational energies fluctuations Ck > As k /T 2 C < 0

8 First order phase transition in finite system : Results Negative heat capacity (C) Experimental evidence of abnormal fluctuations of configurational energies Le Neindre et al, NPA 795 (2007) INDRA M. Bruno et al, NPA 807 (2008) M. D'Agostino et al, PLB 473 (2000) Multics Miniball Au+Au 35 A MeV Deduce boundaries for the spinodal region Liquid side = MeV/A Gas side = MeV/A

9 First order phase transition in finite system : Guideline Canonical ensemble allows to make a direct link between finite size system which passed through a phase transition and the bimodal distribution of an order parameter Take as a guideline for analysis bimodality observed in the QP fragmentation data. Which experimental observable? Charge of the biggest fragment (Z1) as a reliable order parameter F. Gulminelli et al, PRC 71 (2005) What have we studied? Well defined QP of Gold sources produced in semi peripheral collisions Au+Au@80, 100 MeV/A INDRA@GSI Two data selection method : M. Pichon et al, NPA 779 (2006) E. Bonnet et al, NPA 807 (2009)

10 How to deal with canonical ensemble in QP fragmentation data? First attempt... Au+Au 80 A MeV Canonical ensemble = Target "temperature" as control parameter System in contact with thermal bath M. Pichon et al, NPA 779 (2006) M. Bruno et al, NPA 807 (2008) Target "temperature" as control parameter Localize the transition region Bimodality observed in Z1 vs (Z1 Z2)/(Z1+Z2) In this region, «coexistence» of events with two excitation energies at the same canonical temperature? In this approach, an other interpretation is given in A. LeFèvre et al, PRC 80 (2009) (see Arnaud poster)

11 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) Au+Au Selected QP events in the Z1 E* plane and projections Most events with low E* > Dominated by most peripheral collisions P (exp) (E*,Z1) W(E*,Z1) g (exp) (E*) It has been checked that only P (exp) (E*) is biased by experimental conditions. E. Bonnet et al, PRL103 (2009) For a given E*, the Z1 distribution seems to reflect intrinsic density of states W(E*,Z1). How to trace back W(E*,Z1)?

12 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) How to trace back W(E*,Z1)? In canonical ensemble: Start with a system with a given density of states W(E*,Z1) Describe hot nuclei

13 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) How to trace back W(E*,Z1)? In canonical ensemble: Start with a system with a given density of states W(E*,Z1) Derive the canonical probability Describe hot nuclei P (can) (E*,Z1) = W(E*,Z1) exp ( E*) Z -1 -Boltzmann factor exp ( E*) Partition sum Z

14 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) How to trace back W(E*,Z1)? In canonical ensemble: Start with a system with a given density of states W(E*,Z1) Derive the canonical probability Describe hot nuclei P (can) (E*,Z1) = W(E*,Z1) exp ( E*) Z -1 -Boltzmann factor exp ( E*) Partition sum Z Experimental ensemble : P (exp) (E*,Z1) W(E*,Z1) g (exp) (E*) Get rid of experimental bias

15 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) How to trace back W(E*,Z1)? In canonical ensemble: Start with a system with a given density of states W(E*,Z1) Derive the canonical probability Describe hot nuclei P (can) (E*,Z1) = W(E*,Z1) exp ( E*) Z -1 -Boltzmann factor exp ( E*) Partition sum Z Experimental ensemble : P (exp) (E*,Z1) W(E*,Z1) g (exp) (E*) Get rid of experimental bias Renormalization of 2D distributions in order to have equiprobable E* distribution

16 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) How to trace back W(E*,Z1)? In canonical ensemble: Start with a system with a given density of states W(E*,Z1) Derive the canonical probability Describe hot nuclei P (can) (E*,Z1) = W(E*,Z1) exp ( E*) Z -1 -Boltzmann factor exp ( E*) Partition sum Z Experimental ensemble : P (exp) (E*,Z1) W(E*,Z1) g (exp) (E*) Renormalization of 2D distributions in order to have equiprobable E* distribution P (exp) (E*) = P (exp) (E*,Z1) dz1 = W(E*) g (exp) (E*) P (exp) (E*,Z1) = P (exp) (E*,Z1)/P (exp) (E*) = W(E*,Z1)/W(E*) Get rid of experimental bias Renormalization prescription Focus on the density of states

17 How to deal with canonical ensemble in QP fragmentation data?... going to the next step, F. Gulminelli, NPA 791 (2007) To compare directly canonical / experimental distributions, derive an analytic expression P (can) (E*,Z1) = W(E*,Z1) exp ( E*) Z -1 W(E*,Z1) = exp ( S(E*,Z) ) + double saddle point approximation for entropy P can 1 E, Z1 = i =l, g N i exp 1 det i 2 x i 1 i x i Double gaussian function : 11 parameters 4 parameters for each phase correlation factor Population of the 2 phases Nl, Ng

18 Renormalization and comparison between experimental and canonical P(E,Z1) distributions Au+Au P (exp) (E*,Z1)

19 Renormalization and comparison between experimental and canonical P(E,Z1) distributions Au+Au P (exp) (E*,Z1) (exp) (E) = ( P (exp) (E*,Z1) dz1) 1

20 Renormalization and comparison between experimental and canonical P(E,Z1) distributions Au+Au P (exp) (E*,Z1) (exp) (E) P (exp) (E*,Z1) (exp) (E) = ( P (exp) (E*,Z1) dz1) 1

21 Renormalization and comparison between experimental and canonical P(E,Z1) distributions Au+Au P (exp) (E*,Z1) P (can) (E*,Z1) (exp) (E) (can) (E) P (exp) (E*,Z1) P (can) (E*,Z1) (exp) (E) = ( P (exp) (E*,Z1) dz1) 1 (can) (E) = ( P (can) (E*,Z1) dz1) 1

22 Renormalization and comparison between experimental and canonical P(E,Z1) distributions Au+Au P (exp) (E*,Z1) P (can) (E*,Z1) The two distributions can be compared directly W(E*,Z1)/W(E*)

23 Results: Fitting procedure Projection Mean Value Standard Deviation Fitting procedure has been used to obtain best reproduction of experimental distribution by canonical distributions Fit range E* = [2,7] MeV/A. Assuming that canonical distribution is derived at transition temperature. 4 sets of data have been taken into account : 2 incident bombarding energies 2 QP event selections P (exp) (E*,Z1) P (can) (E*,Z1) The two distributions can be compared directly W(E*,Z1)/W(E*)

24 Results: Fitting procedure : One solution Au+Au@80 A.MeV Projection Mean Value Standard Deviation Experimental ensemble to be reproduced Canonical ensemble fit result Adequacy of the fit

25 Results: Fitting procedure : One solution Au+Au@80 A.MeV Projection Mean Value Standard Deviation Canonical Data Experimental ensemble to be reproduced Canonical ensemble fit result Adequacy of the fit

26 Results: P can 1 E, Z1 = i =l, g N i exp 1 det i 2 x i 1 i x i E. Bonnet et al, PRL103 (2009) Extraction of the latent heat

27 Results: Fitting procedure : One solution Au+Au@80 A.MeV

28 Preliminary Results: Boundaries of the coexistence zone El = MeV/A Eg = MeV/A Boundaries of the spinodal zone 2 P > 0 E MeV/A for liquid side MeV/A for gas side Negative heat capacity signal MeV/A MeV/A M. Bruno et al, NPA 807 (2008) Le Neindre et al, NPA 795 (2007) M. D'Agostino et al, PLB 473 (2000)

29 Conclusion: A system passing through a first order phase transition described in the canonical ensemble seems to be a good tool to understand the decay hot nuclei. The bimodality of the charge of the biggest (Z1) can be related to an evidence of such phase transition. Using a renormalisation prescription for experimental and canonical 2D distributions, we can get rid of experimental bias on the experimental energy distribution. From the comparison of the renormalized 2D distributions, we have extracted parameters related to 2 phases and deduce the latent heat for Gold nuclei. We confirm results obtained from negative heat capacity, and show that the spinodal zone is entirely within the coexistence region.

30 Perspective: Widen this analysis to address the effect of system size on properties like latent heat and spinodal zone. 65, 80 and 100 MeV/A (INDRA@GSI) and study of Xe Quasi Projectile is running An heavier system like U+U@100 MeV/A would be a good tool to bring also some information on the role plays by coulomb. THANK YOU FOR YOUR ATTENTION!

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