heaviest fragment of partitions => order parameter

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


INDRA@GANIL and INDRA-ALADIN@GSI


Two ways to heat nuclei in H.I. collisions at int. energies


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


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


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)


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


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


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


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


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


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


How to compare data to predictions of the canonical

ensemble

F. Gulminelli, NPA 791 (2007) 165

comparison of normalizedcorrelations (Z1 versus E*)


Normalized distributions

measured ones =>

normalized ones =>


Fit procedure to extract parameter values (common E*

range)

(II) as exemple

Correlation coeff.ρ=σZ1E*/σZ1σE*

Zi, σZi, Ei, σEii=L,G


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


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


Selection of QF nuclei: compact single sources in velocity space using the kinetic

energy tensor and the flow angle (Өflow >= 60o)

Өflot≥60°


Selection of QP nuclei : compactness criterion in velocity space


Other estimators (normalized dist.)


Finite syst. and first order phase transition X extensive variable

(E, N, V)


BIMODALITY

Canonical-Gaussian sampling

P(X) exp(S(X)- X)


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


X extensive variable (E, N, V)


NEGATIVE HEAT CAPACITY

μ canonical sampling(Fixed value of X)

Finite syst. and first order phase transition


Caloric curves, heat capacity and config. energy fluctuations


Negative microcanonical heat capacity

QF QP

N. Le Neindre et al., NPA 699 (2002) 795

N. Le Neindre et al., NPA 795 (2007) 47

Documents

heaviest fragment of partitions => order parameter