heaviest fragment of partitions => order parameter
DESCRIPTION
The prominent role of the heaviest fragment in multifragmentation and phase transition for hot nuclei. heaviest fragment of partitions => order parameter Size/charge of the heaviest fragment => good estimator of E* and of the freeze-out volume - PowerPoint PPT PresentationTRANSCRIPT
IWND09 Bernard Borderie
The prominent role of the heaviest fragment in multifragmentation
and phase transition for hot nuclei heaviest fragment of partitions => order parameter
Size/charge of the heaviest fragment => good estimator of E* and of the freeze-out volume
bimodal behavior of the heaviest fragment distribution
=> generic signal of first order phase transition for finite systems
IWND09 Bernard Borderie
INDRA@GANIL and INDRA-ALADIN@GSI
IWND09 Bernard Borderie
Two ways to heat nuclei in H.I. collisions at int. energies
IWND09 Bernard Borderie
The heaviest fragment of multifragmentation partitions is recognized as order parameter
Universal fluctuations: Δ-scaling lawsR. Botet and M. Ploszajczak Lecture Notes in Physics vol 65 (2002)
Gaussian shape Gumbel shape
J.D. Frankland et al., PRC 71 (2005) 034607
IWND09 Bernard Borderie
The size of the heaviest fragment
Its size/charge estimates E* but only for heavy hot nuclei (Z>=60)B. B., MF Rivet, PPNP 61 (2008) 551
QP: Au + Au 80 AMeV <Zs> : 79 - 65
QF: Xe + Sn 25-50 AMeV <Zs> : 90 - 80
E.Bonnet et al.,NPA 816 (2009) 1
Fragment formation stage: we can think that the size of the heaviest fragment is correlated with the particle density and can give information on freeze-out density/volume
IWND09 Bernard Borderie
A complete simulation to derive information at
freeze-out• built event by event from all the available experimental information
(LCP spectra, average and standard deviation of frag. velocity spectra and calorimetry)
• F.O. partitions are built by dressing fragments with particles• Excited fragments and particles at F.O. undergo propagation
(Coulomb+ thermal kin. E) during which fragments evaporate particles
• 4 free parameters to recover the data: - percentage of particles evaporated from primary frag. - radial collective energy - minim. distance between the surfaces of products at F.O. - limiting temperature for fragments (vanishing of the level density at high E*- S.E. Koonin and J. Randrup A474 1987,173)
IWND09 Bernard Borderie
Comparison data-simulation (asymptotic values)
A limiting temperature of 9 MeV is mandatory to reproduce the measured widths
S. Piantelli et al., NPA 809 (2008) 111
QF: Xe + Sn 32-50AMeV frag.-frag. correlations
IWND09 Bernard Borderie
The normalized heaviest fragment Z1/ZS is used to calibrate the F.O. volume F.O.Volumes for QF sources (Xe+Sn 32-50 AMeV) taken from the simulation Piantelli et al. (NPA 809, 2008, 111)Calibrate F.O. volumes with
the relation V/V0 = f(Z1/ZS) for QF, and derive freeze-outvolumes for QP’s
At a given E*, QP volumes are smaller than QF volumes
E.Bonnet et al.,NPA 816 (2009) 1
QP: Au + Au 80 AMeVQF: Xe + Sn 25-50AMeV
IWND09 Bernard Borderie
QP: Au + Au 80,100 AMeV
QF: Xe + Sn 25-45 AMeV
At fixed reduced Mfrag and fixed E*
the size of Z1 is determined
E. Bonnet et al., PRL to be submitted
IWND09 Bernard Borderie
Finite syst. and first order phase transition X extensive variable
(E, N, V)
Conjugate intensive variable (X)=S / X(1/T, - μ/T, P/T)
NEGATIVE HEAT CAPACITY
μ canonical sampling(Fixed value of X)
BIMODALITY
Canonical-Gaussian sampling
P(X) exp(S(X)- X)
SPINODAL INSTABILITY Ph. Chomaz et al., Phys. Rep. 389
IWND09 Bernard Borderie
Bimodal behavior of the heaviest fragment distribution ?
Recent observations: bimodal behavior ofthe distribution of the asymmetry betweenthe charges of the two heaviest fragments M.Pichon et al., NPA 779 (2006) 267 M. Bruno et al., NPA 807 (2008) 48
and for the heaviest fragment Z1/Zs (related to F.O. volume) ?
QP: Au+Au 60,80,100 AMeV
IWND09 Bernard Borderie
Bimodal behavior of the heaviest fragment distribution in Quasi-Projectile
fragmentationTo select QPs with negligibleneck contribution (mid-rapidityemission)
2 different procedures
(I): eliminating events withsize hierarchy (heaviest fragmentthe most forward, PRC 67 064603)
(II): keeping compact events invelocity space, NPA 816 1
IWND09 Bernard Borderie
How to compare data to predictions of the canonical
ensemble
F. Gulminelli, NPA 791 (2007) 165
comparison of normalizedcorrelations (Z1 versus E*)
IWND09 Bernard Borderie
Normalized distributions
measured ones =>
normalized ones =>
IWND09 Bernard Borderie
Fit procedure to extract parameter values (common E*
range)
(II) as exemple
Correlation coeff.ρ=σZ1E*/σZ1σE*
Zi, σZi, Ei, σEii=L,G
IWND09 Bernard Borderie
Bimodal behavior of the heaviest fragment distribution as signature of a
first order phase transition in finite systems
latent heat of the phase transition(EG-EL) for heavy nuclei Z~ 70
8.1 (±0.4)stat (+1.2 -0.9)syst AMeV
syst. error: different QP selectionsE.Bonnet, D. Mercier et al, PRL August
2009
Z1 versus E* using the deduced parameter values
IWND09 Bernard Borderie
Summary
THE PROMINENT ROLE OF THE HEAVIEST FRAGMENT IN MULTIFRAGTATION AND PHASE TRANSITION OF HOT NUCLEI IS ESTABLISHED
- early recognized as order parameter (universal fluctuation theory)
id large class of transitions involving complex clusters from percolation to gelation, nucleation,aggregation
- representative of E* and of the F.O. volume
- bimodal behavior of its distribution => generic signal expected in
finite systems for a first order phase transition => estimate of the latent heat of the transition for Z~70 and possibly for other sizes
of nuclei in the future
IWND09 Bernard Borderie
Selection of QF nuclei: compact single sources in velocity space using the kinetic
energy tensor and the flow angle (Өflow >= 60o)
Өflot≥60°
IWND09 Bernard Borderie
Selection of QP nuclei : compactness criterion in velocity space
IWND09 Bernard Borderie
Other estimators (normalized dist.)
IWND09 Bernard Borderie
Finite syst. and first order phase transition X extensive variable
(E, N, V)
Conjugate intensive variable (X)=S / X(1/T, - μ/T, P/T)
BIMODALITY
Canonical-Gaussian sampling
P(X) exp(S(X)- X)
IWND09 Bernard Borderie
Observation of a fossil signal with a confidence level of 3-4 σ (QF Xe+Sn 32-50
AMeV)
B.B. et al. PRL 86 (2001) 3252
IWND09 Bernard Borderie
X extensive variable (E, N, V)
Conjugate intensive variable (X)=S / X(1/T, - μ/T, P/T)
NEGATIVE HEAT CAPACITY
μ canonical sampling(Fixed value of X)
Finite syst. and first order phase transition
IWND09 Bernard Borderie
Caloric curves, heat capacity and config. energy fluctuations
IWND09 Bernard Borderie
Negative microcanonical heat capacity
QF QP
N. Le Neindre et al., NPA 699 (2002) 795
N. Le Neindre et al., NPA 795 (2007) 47