Bimodality and latent heat of gold nuclei
Eric Bonnet (GANIL) for the INDRA and ALADIN collaboration
IWM 09, Catania
Outline
First order phase transition in finite system
Definition and related signals starting from microcanonical entropy
Canonical ensemble and Quasi­Projectile 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 2D­distribution Z1­E* (Excitation energy)
Derive the equivalent canonical distribution
Constrain canonical ensemble using comparison with data Extract properties for the hot nuclei de­excitation First order phase transition in finite system : Definition
Presence of a residual convex intruder in the entropy S(X) of the mixed system.
∂2X S(X) >0 define the
spinodal zone
X : order parameter
(X) = ∂X S : conjugate variable
First order phase transition in finite system : Consequences
Bimodal canonical distribution of an order parameter ∂2X S(X) >0 define the
spinodal zone
Pcan(X) = exp (S(X) ­ X)
What information can we get ?
Boundaries of the coexistence zone
∂X Pcan (X) = 0
 latent heat
First order phase transition in finite system : Consequences
Bimodal canonical distribution of an order parameter ∂2X S(X) >0 define the
spinodal zone
Pcan(X) = exp (S(X) ­ X)
What information can we get ?
Boundaries of the coexistence zone
∂X Pcan (X) = 0
 latent heat
Boundaries of the spinodal zone
∂2X Pcan (X) > 0
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)
∂2X S(X) >0 define the
spinodal zone
∂2E S = ­1/CT2 >0  C < 0
Boundaries of the spinodal zone
First order phase transition in finite system : Results
Negative heat capacity (C)
Experimental evidence of abnormal fluctuations of configurational energies
M. Bruno et al, NPA 807 (2008)
M. D'Agostino et al, PLB 473 (2000)
Le Neindre et al, NPA 795 (2007)
INDRA
Multics­Miniball
Au+Au 35 A MeV
Ck = canonical heat capacity
k = configurational energies fluctuations
Ck > As k/T2  C < 0 First order phase transition in finite system : Results
Negative heat capacity (C)
Experimental evidence of abnormal fluctuations of configurational energies
M. Bruno et al, NPA 807 (2008)
M. D'Agostino et al, PLB 473 (2000)
Le Neindre et al, NPA 795 (2007)
INDRA
Multics­Miniball
Au+Au 35 A MeV
Deduce boundaries for the spinodal region
Liquid side = 2.0­2.5 MeV/A
Gas side = 5.5­6.5 MeV/A
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 semi­peripheral 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)
How to deal with canonical ensemble in QP fragmentation data ?
First attempt ...
Canonical ensemble = System in contact with thermal bath
Target "temperature" as control parameter
Au+Au 80 A MeV
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 (Z1­Z2)/(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)
How to deal with canonical ensemble in QP fragmentation data ?
... going to the next step, F. Gulminelli, NPA 791 (2007)
Au+Au INDRA@GSI
Selected QP­events 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 trace­back W(E*,Z1) ?
How to deal with canonical ensemble in QP fragmentation data ?
... going to the next step, F. Gulminelli, NPA 791 (2007)
How to trace­back W(E*,Z1) ?
In canonical ensemble:
Start with a system with a given density of states W(E*,Z1)
Describe hot nuclei
How to deal with canonical ensemble in QP fragmentation data ?
... going to the next step, F. Gulminelli, NPA 791 (2007)
How to trace­back W(E*,Z1) ?
In canonical ensemble:
Start with a system with a given density of states W(E*,Z1)
Derive the canonical probability
P(can)(E*,Z1) = W(E*,Z1)  exp (­  E*)  ℤ  -1
-Boltzmann factor exp (­  E*)
­Partition sum ℤ
Describe hot nuclei
How to deal with canonical ensemble in QP fragmentation data ?
... going to the next step, F. Gulminelli, NPA 791 (2007)
How to trace­back W(E*,Z1) ?
In canonical ensemble:
Start with a system with a given density of states W(E*,Z1)
Derive the canonical probability
P(can)(E*,Z1) = W(E*,Z1)  exp (­  E*)  ℤ  -1
-Boltzmann factor exp (­  E*)
Get rid of experimental bias
­Partition sum ℤ
Experimental ensemble :
P(exp)(E*,Z1) ∝ W(E*,Z1)  g(exp)(E*)
Describe hot nuclei
How to deal with canonical ensemble in QP fragmentation data ?
... going to the next step, F. Gulminelli, NPA 791 (2007)
How to trace­back 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*)
Get rid of experimental bias
­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
How to deal with canonical ensemble in QP fragmentation data ?
... going to the next step, F. Gulminelli, NPA 791 (2007)
How to trace­back 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*)
Get rid of experimental bias
­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*)
Renormalization prescription  Focus on the density of states
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
1

E∗, Z1=i =l , g N i
exp− 
x i −1
i xi
2
 det  i
Double gaussian function : 11 parameters
­ 4 parameters for each phase
­  correlation factor
­ Population of the 2 phases Nl, Ng
Renormalization and comparison between experimental and canonical P(E,Z1) distributions
Au+Au INDRA@GSI
P(exp)(E*,Z1)
Renormalization and comparison between experimental and canonical P(E,Z1) distributions
Au+Au INDRA@GSI
P(exp)(E*,Z1)
(exp)(E) = (∫ P(exp)(E*,Z1) dZ1)­1
Renormalization and comparison between experimental and canonical P(E,Z1) distributions
Au+Au INDRA@GSI
P(exp)(E*,Z1)
(exp)(E)
P(exp) (E*,Z1)
(exp)(E) = (∫ P(exp)(E*,Z1) dZ1)­1
Renormalization and comparison between experimental and canonical P(E,Z1) distributions
Au+Au INDRA@GSI
P(exp)(E*,Z1)
(exp)(E) = (∫ P(exp)(E*,Z1) dZ1)­1
P(can)(E*,Z1)
(exp)(E)
(can)(E)
P(exp) (E*,Z1)
P(can) (E*,Z1)
(can)(E) = (∫P(can)(E*,Z1) dZ1)­1
Renormalization and comparison between experimental and canonical P(E,Z1) distributions
Au+Au INDRA@GSI
P(exp) (E*,Z1)
P(can) (E*,Z1)
The two distributions can be compared directly
W(E*,Z1)/W(E*)
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*)
Results:
Fitting procedure : One solution ­ Au+Au@80 A.MeV
Projection
Experimental ensemble
to be reproduced
Mean Value
Canonical ensemble
fit result
Standard Deviation
Adequacy of the fit
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
Results:
P
can
1
1
−1

E∗, Z1=i =l , g N i
exp− x i i 
xi
2
 det  i
E. Bonnet et al, PRL103 (2009)
Extraction of the latent heat
Results:
Fitting procedure : One solution ­ Au+Au@80 A.MeV
Preliminary Results:
Boundaries of the coexistence zone
● El = 1 ­ 1.7 MeV/A
● Eg = 8.5 ­10.4 MeV/A
Boundaries of the spinodal zone
∂2E P > 0
1.5 ­ 3.7 MeV/A for liquid side
● 6.5 ­ 7.0 MeV/A for gas side
●
Negative heat capacity signal
2.0 ­ 2.5MeV/A
5.5 ­ 6.5 MeV/A
M. Bruno et al, NPA 807 (2008)
Le Neindre et al, NPA 795 (2007)
M. D'Agostino et al, PLB 473 (2000)
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 2D­distributions,
we can get rid of experimental bias on the experimental energy distribution.
From the comparison of the renormalized 2D­distributions, 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.
Perspective:
Widen this analysis to address the effect of system size on properties
like latent heat and spinodal zone.
Xe+Sn @ 65, 80 and 100 MeV/A (INDRA@GSI)
and study of Xe Quasi­Projectile 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 !