A BUDZANOWSKI

A SIMOMENKO

E KUZMIN

H OESCHLER

I SKWIRCZYNSKA

S AVDEYEV

V KARNAUKHOV

V KIRAKOSYAN

V RODIONOV

W KARCZ

English

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Multifragmentation and nuclear phase transitions (liquid-fog and liquid-gas) 
 
V.A. Karnaukhova , H. Oeschlerb , S.P. Avdeyeva , V.K. Rodionova ,  A.V. Simomenkoa,   
V.V. Kirakosyana, A. Budzanowskic, W. Karczc, I. Skwirczynskac, E.A. Kuzmind,   
E. Norbecke, A.S. Botvinaf 
  
a Joint Institute for Nuclear Research, Dubna  (karna@nusun.jinr.ru) 
 
b Institut für Kernphysik, Darmstadt University of Technology, Darmstadt,  Germany 
 
c H.Niewodniczanski Institute of Nuclear Physics, Cracow, Poland 
 
d Kurchatov Institute,  Moscow 
 
e University of Iowa, Iowa City, USA 
 
f GSI, Darmstadt, Germany 
 
  Thermal multifragmentation of hot nuclei is interpreted as the nuclear liquid-fog phase 
transition.  The  charge  distributions of  the  intermediate   mass   fragments   produced       in 
p(3.6 GeV) + Au and p(8.1 GeV) + Au collisions are analyzed within the statistical 
multifragmentation model with the critical temperature for the nuclear liquid-gas phase 
transition Tc as a free parameter. The analysis presented here provides strong support for a 
value of Tc > 15 MeV.  
  
1. INTRODUCTION 
 
  The investigation of the decay properties of very hot nuclei is currently one of the most 
challenging topics of nuclear physics. The excitation energy of the hot nuclei is comparable 
with the total binding energy. They disintegrate via a new multibody decay mode–thermal 
multifragmentation [1], which is characterized by the copious emission of intermediate mass 
fragments (2 < Z ≤ 20). The development of this field has been strongly stimulated by an idea 
that this process is related to the nuclear liquid-gas phase transition, which was predicted on 
the basis of the similarity between van der Waals and nucleon-nucleon interactions [2-4]. The 
equations of state for the two cases are similar. For both systems there is a spinodal region 
characterized by phase instability. The density here is reduced as compared to the liquid 
phase. One can imagine that a hot nucleus (at T = 5-7 MeV) expands due to thermal pressure 
and enters the unstable region [5]. Due to density fluctuations, a homogeneous system 
converts into a mixed phase state consisting of droplets (IMF) surrounded by nuclear gas. The 
final state of this transition is a nuclear fog [4], which explodes due to Coulomb repulsion and 
is detected as multifragmentation. The disintegration time is very short. This is the scenario of 
a spinodal decomposition. It was proven experimentally that thermal multifragmentation 
occurs at reduced (3-4 times) densities, and the disintegration time is less than 100 fm/c. The 
spinodal decomposition is, in fact, the liquid-fog phase transition in a nuclear system. 
  An important model parameter of this scenario is the critical temperature Tc for the nuclear 
liquid-gas phase transition. The surface tension vanishes at Tc, and only the gas phase is 
possible above this temperature. There are many calculations of Tc for finite nuclei. In Refs. 
[2,3,6,7] it is done by using a Skyrme interaction and the thermal Hartree-Fock theory. The 
values of Tc were found to be in the range 10-20 MeV depending on the details of the model. 
The main source of the experimental information for Tc is the fragment yield. In statistical 
models of nuclear multifragmentation the shape of the IMF charge distribution Y(Z) is 
sensitive to the ratio T /Tc. The charge distribution is well described by the power law Y(Z) ~ 
Z -
τ  for a wide range of colliding systems. In earlier studies  the power-law behavior of the 
IMF yield was interpreted as an indication of the proximity of the excited system to Tc. This 
was stimulated by the application of Fisher's classical droplet model [8], which predicts a pure 
power-law droplet-size distribution with τ = 2-3 at the critical point. A more sophisticated use 
of this model for the estimation of Tc has been made in Refs. [9,10]. The data obtained for π 
(8 GeV/c) + Au collisions were analyzed giving Tc= (6.7±0.2) MeV. The same analysis was 
applied to the data for collisions of Au, La, Kr (at A·1.0 GeV) with  carbon. The extracted 
values of Tc were (7.6±0.2), (7.8±0.2) and (8.1±0.2) MeV respectively.   
  Having in mind the shortcomings of Fisher's model [11,12], we have estimated the nuclear 
critical temperature using the statistical multifragmentation model (SMM) [13] as it describes 
well the different experimental data. It was found that Tc=20±3 MeV [14]. Some additional 
analysis is given below with emphasis put on the model dependence of the results. 
  
2. ESTIMATION OF TC USING SMM 
 
  Within this model one considers a microcanonical ensemble of all break-up channels 
composed of nucleons and excited fragments. It is assumed that an excited nucleus expands 
and then breaks up into nucleons and hot fragments. It is also assumed that at the break-up the 
nucleus is in thermal equilibrium characterized by the channel temperature T. The charge 
yield depends on the contribution of the surface free energy of fragments SAZF  to the entropy 
of a given final state. The surface energy depends on the critical temperature: 
,
4/5
22
22
)0()(,3/2)(










+
−
==
TcT
TcT
saTsaATsa
s
AZF
                                                                     (1) 
with as(T) = )(4 20 Tr σπ , where σ(T) is a surface tension coefficient. This equation was 
obtained in [15]. It is successfully used by the SMM for describing the decay of hot finite 
nuclei. The comparison of the measured and calculated IMF charge yields is the way to 
estimate the Tc value.  
  The reaction mechanism is usually divided into two stages. The first one is a fast energy-
depositing stage. We use the intranuclear cascade model (INC) [16] for describing this stage. 
The second stage is described by the  SMM,  which  considers decay of a hot target  spectator.  
But such an approach fails to explain the IMF multiplicities. An expansion stage (Exp.) is 
inserted between the two parts of the calculation. The excitation energies and the residual 
masses are then fine tuned [1,5] to get agreement with the measured IMF multiplicities. 
Figure 1a presents the measured fragment charge distribution for p(8.1 GeV)+Au collisions 
and the calculations performed with  Tc as a free parameter.  Experiments were done using the 
                                                                                                                                                                              
4π-setup FASA [5]. The 
lines  show  the  calculated 
distributions for Tc = 7, 11 
and 18 MeV. The SMM 
parameter k is set to 2.  It 
corresponds to the break-
density ρf=1/3ρο.  The  level 
density parameter is taken to 
be a=A/8. The statistical 
errors of the data do not 
exceed  the  size  of  the 
dots. The calculations  are  
close to the data for Tc = 18 
MeV.  The estimated mean 
temperature of the  system is 
around  6 MeV,    the    mean        Fig. 1. IMF charge distributions for for p + Au at 8.1 GeV:                          
charge  and  mass     numbers        data and the INC+Exp+SMM model predictions (a), 
are 67 and 158. The theoreti-         power law fits (b). 
cal  curves  deviate from  the  
data    with    decreasing   Tc.  Figure1b gives the results of the power-law fits for the data and 
model calculations. The Be yield was corrected in the fitting procedure for the loss of unstable  
 8Be.   Similar  results  are  obtained  for 
p+Au collisions at 3.6 GeV. Compa-
risons   of   the experimental power-law 
exponents and model predicted ones for 
different values of Tc are shown in 
Fig.2. In contrast to our paper [14] the 
power-law fits are done in the range 
Z=4-11 to exclude the influence of  
nonequibrium emission of  Li [1]. The 
measured power-law exponents are  
given  as  a  band  with a width 
determined by the statistical error. The 
size of the symbols for the calculated 
values of τapp is of the order of the error 
bar.  From the best fit of the data for 
both beam energies and the calculations 
the critical  temperature is  (17±2) MeV. 
Calculations were also performed with 
the surface tension coefficient linearly 
dependent on T/Tc as in [9,10] (middle 
panel, solid  circles).      The    predicted   
values of τapp in this case are remarkably 
lower   than  the  data.  The results  only   
slightly depend on the level density 
parameter. This is seen  from  the  bottom               Fig.2. Power-law exponents for the IMF 
panel  where upper line is calculated with                charge distributions for p+Au collisions. 
a = A/10. The lower line in Fig.2 is obtained on the assumption of k =5, that is 
ρf = 1/6 ρο. The model prediction does not agree with the data. Note that our choice of k = 2 is 
motivated by analysis of the shape of the IMF energy spectra, which is sensitive to the size of 
the fragmenting nucleus [5]. 
     
 
3. SUMMARY 
 
  Thermal multifragmentation of hot nuclei is interpreted as the liquid-fog phase transition. 
The critical temperature for the nuclear liquid-gas phase transition Tc (at which the surface 
tension vanishes) is estimated by using the statistical multifragmentation model. For that 
purpose, the IMF charge distributions for p + Au collisions at 3.6 and 8.1 GeV have been 
analyzed within the SMM with Tc as a free parameter. The value Tc = (17 ± 2) MeV obtained 
from the best fit to the data should be considered as some effective value of Tc averaged over 
all the fragments produced in the collision. This value is larger than those found in [9,10] 
using Fisher's droplet formalism, but it is close to the values obtained in [17,18]. Although our 
value for Tc is model dependent, as is any other estimate of the critical temperature, the 
analysis presented here provides strong support for a value of Tc > 15 MeV. 
  The research was supported in part by Grant 03-02-17263 from the Russian Foundation for 
Basic Research, by the Grant of the Polish Plenipotentiary to JINR, and by Contract 06DA453 
with BFT(Germany). 
 
 
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Multifragmentation and nuclear phase transitions (liquid-fog and liquid-gas)

Multifragmentation and nuclear phase transitions (liquid-fog and liquid-gas)

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