
THERMAL MULTIFRAGMENTATION, NUCLEAR FOG
AND CRITICAL TEMPERATURE FOR THE LIQUID-GAS PHASE
TRANSITION
V.A.Karnaukhov a, H.Oeschler b, S.P.Avdeyev a, V.K.Rodionov a,
A.Budzanowski c, W.Karcz c, I.Skwirczinska c, O.V.Bochkarev d,
E.A.Kuzmin d, L.V.Chulkov d, E.Norbeck e, A.S.Botvina f
a
b
Joint Institute for Nuclear Research, Dubna
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 distribution of the intermediate mass fragments produced in p(3.6 GeV) + Au and p(8.1 GeV) + Au
collisions are analyzed in the framework of 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. The nuclear equation of the state and thermal multifragmentation
The investigation of the decay properties of the very hot nuclei is currently one of the most
challenging topics of nuclear physics. The excitation energy of the hot nuclei (500-700 MeV) is
comparable with the total binding energy. They disintegrate via a new multibody decay mode –
thermal multifragmentation [1]. This process is characterized by the copious emission of
intermediate mass fragments (IMF, 2 < Z  20), which are heavier than alpha particles but lighter
than fission fragments. Such multibody disintegration is not an exotic but the main decay channel of
a very hot nuclear system.
The development of this field for the last two decades has been strongly stimulated by an idea
that this process is related to the nuclear liquid-gas phase transition. One of the first nuclear models,
suggested by N.Bohr, K.Weizsäcker and Ya.I.Frenkel 65 years ago, is the liquid-drop model, which
is still alive . The liquid-gas phase transition in nuclear matter was predicted much later [2-4] on the
basis of the similarity between van der Waals and nucleon-nucleon interactions. In both cases the
attraction between particles is replaced by repulsion at a small interaction range. As a result, the
equations of the state are similar for very different systems. It is well seen in the phase diagram given
in Ref. [3]. A point of particular interest is the part of the diagram where the isotherms correspond to
negative compressibility. The density here is significantly reduced as compared to the liquid phase.
This is a spinodal region characterized by phase instability. One can imagine that a hot nucleus (at
temperatures 5-7 MeV) expands due to thermal pressure and enters the unstable region. Due to
density fluctuations, a homogeneous system converts into a mixed phase state, consisting of droplets
(IMFs) surrounded by nuclear gas. In fact, the final state of this transition is a nuclear fog (P.J.
Siemens, [4]), which explodes due to Coulomb repulsion and is detected as multifragmentation. The
disintegration time is determined by the time scale of the density fluctuations and is very short. This
is the scenario of nuclear multifragmentation as a spinodal decomposition, considered in a number of
papers (see, for example [1, 5-13] and review papers [14, 15]. It was proven experimentally that
thermal multifragmentation occurs at reduced (3-4 times) densities [16-8], and the disintegration time
is less than 100 fm/c [19-21]. The spinodal decomposition is, in fact, the liquid-fog phase transition
in a nuclear system.
An effective way to produce hot nuclei is by the collision of heavy ions with energies of up to
hundreds of MeV per nucleon. But in this case heating of nuclei is accompanied by compression,
strong rotation and shape distortion, which may essentially influence the decay properties of hot
nuclei. The picture becomes clearer when light relativistic projectiles (protons, antiprotons, pions)
are used. In contrast to heavy ion collisions, fragments are emitted by the only source – the slowly
moving target spectator. Its excitation energy is almost entirely thermal. Light relativistic projectiles
provide, therefore, a unique possibility for investigating thermal multifragmentation, which is
governed by the thermodynamic properties of a hot nuclear system.
2. The critical temperature for the liquid-gas phase transition
An important model parameter of this scenario is the critical temperature for the nuclear liquidgas phase transition Tc at which the isotherm in the phase diagram has an inflection point. 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, 22, 23], for example, it is done by using a
Skyrme effective interaction and the thermal Hartree-Fock theory. The values of Tc were found to be
in the range 10-20 MeV depending upon the chosen Skyrme interaction parameters and the details of
the model.
The main source of the experimental information for Tc is the fragment yield. In some statistical
models of nuclear multifragmentation the shape of the IMF charge (or mass) distribution Y(Z) is
sensitive to the ratio T /Tc. The charge distribution is well described by the power law Y( Z) ~ Z- for
a wide range of the colliding systems [24]. In earlier studies on multifragmentation [4, 25] the
power-law behavior of the IMF yield was interpreted as an indication of the proximity of the excited
system to the critical point for the liquid-gas phase transition. This was stimulated by the application
of Fisher's classical droplet model [26], which predicted a pure power-law droplet-size distribution
with the minimal value of  = 2-3 at the critical point.
In Ref. [25], Hirsch et al. estimate Tc to be 5 MeV simply from the fact that the mass
distribution is well described by a power law for IMFs produced in the collision of p (80-350 GeV)
with Kr and Xe targets. In fact, the fragment mass distribution is not exactly described by the power
law. Therefore, Panagiotou et al. [27] suggested the use of the term app, an apparent exponent, to
stress that the exact power-law description takes place only at the critical temperature. In paper [27]
the experimental data were gathered for different colliding systems to get the temperature
dependence of app. As a temperature, the inverse slope of the fragment energy spectra was taken in
the range of the high-energy tail. The minimal value of app was obtained at T = 11-12 MeV, which
was claimed as Tc. The later data smeared out this minimum. Moreover, it became clear that the
«slope» temperature for fragments does not coincide with the thermodynamical one, which is several
times smaller.
A more sophisticated use of Fisher's droplet model for the estimation of Tc has been recently
made by Elliott, et al. [28, 29]. The model was modified by including the Coulomb energy release
when a particle moves from the liquid to the vapor. The multifragmentation data obtained for (8
GeV/c) + Au collisions were analyzed with this model [28]. The extracted critical temperature is
(6.70.2) MeV. The same analysis technique is applied to the data for the multifragmentation in
collisions of Au, La, Kr (at 1.0 GeV per nucleon) with a carbon target [29]. The extracted values of
Tc are (7.60.2), (7.80.2) and (8.10.2) MeV respectively.
It should be noted that in some papers the term «critical temperature» is not used in the strict
thermodynamic sense given above. In Ref. [30] multifragmentation in Au + Au collisions at 35 MeV
per nucleon was analyzed with the so-called Campi plots to prove that the phase transition takes
place in the spinodal region. The characteristic temperature for that process was denoted as Tcrit and
found to be equal to (6.00.4) MeV. In Ref. [31] the bond percolation model is used to interpret 10.2
GeV/c p + Au multifragmentation data. The critical value of the percolation parameter pc = 0.65 was
found from the analysis of the IMF charge distribution. The corresponding «critical temperature» of
(8.30.2) MeV is estimated by using the model relation between the percolation control parameter
«p» and the excitation energy. More appropriate here is the «break-up» or «limiting» temperature, as
suggested in Ref. [32]. This temperature corresponds to the onset of the fragmentation of the nucleus
entering the phase coexistence region. The experimental data on the limiting temperatures was used
to derive the critical temperature of nuclear matter, which was found to be equal to 16.60.86 MeV
[33].
There is only one paper in which Tc is estimated by using data other than the fragmentation ones.
In Ref. [34] it is done by the analysis of the temperature dependence of the fission probability for 4He
+ 184W collisions [35]. It was concluded that Tc > 10 MeV.
Having in mind the shortcomings of Fisher's model [36, 37], we have estimated the nuclear
critical temperature in the framework of the statistical multifragmentation model (SMM) [38] as it
describes well the different properties of the thermal multifragmentation of target spectators
produced in collisions by light relativistic ions. It was found that Tc=203 MeV [39]. Some
additional analysis is presented in the following with emphasis put on the model dependence of the
results.
3. Estimation of Tc using SMM
Within this model one considers a microcanonical ensemble of all break-up channels composed
of nucleons and excited fragments of different masses. It is assumed that an excited nucleus expands
to a certain volume 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
determined from the energy balance. The probability Wj of a decay channel j is proportional to its
statistical weight:
W j  exp Sj(Ex, A0, Z0),
(1)
where Sj is the entropy of the system in a state corresponding to the decay channel j. The excitation
energy, mass and charge of the decaying system are denoted by Ex, A0 and Z0, respectively. The
fragments with mass number A > 4 are treated as heated nuclear liquid drops.
Channels are characterized by the multiplicities, NAZ, of fragments AZ. The channel entropy is
obtained by summing the entropies of all the particles in a given channel:
 F 
S j   N AZ S AZ , S AZ   AZ  .
 T V
(2)
The fragment free energy FAZ is a sum of volume, surface, symmetry, Coulomb and translational
terms:
V
S
sym
C
t
(3)
FAZ  FAZ
 FAZ
 FAZ
 FAZ
 FAZ
.
S
The surface energy term, FAZ
, depends upon the critical temperature, so the fragment charge
S
distribution is sensitive to the value of Tc. The following expression is used in the SMM for FAZ
:
S
AZ
F
 a s (T ) A
2/3
T 2 T 2 

, a s (T )  a s (0) c2
2 
 Tc  T 
5/ 4
,
(4)
with as(T) = 4r02 (T ) , where (T) is a temperature-dependent coefficient of the surface tension.
This equation was obtained in Ref. [40], devoted to the theoretical study of thermodynamic
properties of a plane interface between two phases of nuclear matter (liquid and gas) in equilibrium.
Fig. 1. The calculated temperature-dependent coefficient of the surface tension as a function of T / Tc :
solid line is according to eq. (4), dashed and dotted lines are for linear and quadratic
parameterizations of as(T)/ as(0) as function of T / Tc .
This parameterization is successfully used by the SMM for describing the multifragment decay of
hot finite nuclei. Fig.1 shows as(T)/ as(0) as a function of T / Tc according to (4), points are
calculated in Ref.[2]. Two other parameterizations are presented also: linear one ~ (1- T / Tc ), which
is used in analysis with Fisher droplet model [28, 29, 31], and quadratic one ~ (1- T / Tc )2 [15].
The comparison of the measured and calculated fragment charge distributions is the way to
estimate the critical temperature Tc.
The reaction mechanism for the light relativistic projectiles is usually divided into two stages.
The first one is a fast energy-depositing stage, during which very energetic light particles are emitted
and a nuclear remnant is excited. We use the intranuclear cascade model (INC) [41] for describing
the first stage. The second stage is described by the SMM, which considers multibody decay of a hot
and expanded nucleus. But such a two-stage approach fails to explain the observed 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 [1,17] to get agreement with the
measured IMF multiplicities, i.e., the values for the residual (after INC) masses and their excitation
energies are scaled on an event-by-event basis.
Fig. 2. Fragment charge distribution for p + Au at 8.1 GeV (dots): a) the lines are calculated by
INC+Exp.+SMM model, assuming Tc = 7 MeV, 11 MeV and 18 MeV; b) the power-law fits.
Fig.2a presents the measured fragment charge distribution for p (8.1 GeV) + Au collisions and
the calculations performed with Tc as a free parameter. Experiments have been done using the 4setup FASA installed at Dubna Synchrophasotron [17]. For all values of Tc the calculations with the
INC + Exp. + SMM model have been properly adjusted to get the mean IMF multiplicity close to the
measured one. The lines show the calculated charge distributions for Tc = 7, 11 and 18 MeV using
eq. (4). he very important SMM parameter k is taken k=2. It corresponds to the break up density b
= The level density parameter is taken to be a = A/8. The free surface energy was calculated
according to eq.(4). The statistical errors of the measurements do not exceed the size of the dots. The
data are corrected for the counting rate loss caused by the cutoff ( 1.2 MeV/nucleon) in the lowenergy part of the IMF spectra. This correction is the largest ( 15%) for the heavier IMFs. The
calculations are close to the data for Tc = 18 MeV. The estimated mean temperature of the
fragmenting system is around 6 MeV, the mean charge and mass numbers are 67 and 158
respectively. The theoretical curves deviate from the data with decreasing Tc. Figure 2b gives the
results of the power-law fits for the data and model calculations (in the range Z = 3-11). 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. Comparisons of the experimental
power-law exponents (for both proton energies) and model predicted ones for different assumed
values of Tc are presented in Fig.3 (upper and middle panels). 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 app is of the order of the error bar. From the best fit to the data and calculations
one concludes that critical temperature is (203) MeV for both beam energies (90% C.L.).
Fig.3. The power-law exponents for the IMF’s charge distributions for p+Au collisions at 3.6 and 8.1 GeV.
The bands correspond to the measured values and their error bar. Upper and middle panels: solid
squares are obtained by calculations using eq.(4) for different values of Tc, freeze out density is taken
to be 1/3  o, level density parameter is A/8; solid circles are calculated for the linear dependence of
the surface tension on T /Tc. Bottom panel: open squares are obtained for a reduced level density
parameter, open circles are calculated for a more dilute fragmenting nucleus ( b ).
The calculations have been performed also using the surface tension coefficient linearly
dependent on T / Tc as in Ref. [28, 29, 31]. It was done for p + Au collisions at 8.1 GeV. The
calculated values of app in this case are remarkably lower than the measured one (Fig.3, middle).
Similar results have been obtained for the quadratic parameterization of as (T) / as(0).
It has been checked how results are sensitive to the level density parameter, having in mind that
the IMF charge distribution is very much influenced by the secondary decay of hot primary
fragments. The calculations with a = A/10 are shown in bottom panel of Fig.3: results are only
slightly changed. The lower line in Fig.3 is obtained assuming k =5, that is b =Such value
of freeze-out density is used in the MMMC model by D.H.E. Gross [42]. The model prediction does
not agree with data. Note that our choice of k = 2 is motivated by analysis of the shape of the IMF’s
energy spectra, which is sensitive to the size of fragmenting nucleus [17, 20].
Fig.4. Fragment charge distributions for p +Au at 8.1 GeV(1), 4He+Au at 4.0 and 14.6 GeV (2, 3) and
for 12C+Au at 22.4 GeV (4): a) the lines are calculated by the INC+Exp.+SMM model for
Tc = 18 MeV; b) the power-law fits with  parameters given in the insert as a function of projectile
energy (in GeV).
Fig.4 (left panel) presents the fragment charge distributions measured by the FASA
Collaboration for collisions of p (8.1 GeV), 4He (4 and 14.6 GeV) and 12C (22.4 GeV) with Au
targets [1] along with the calculated charge distributions, assuming Tc obtained in the present paper.
The agreement between the data and the model prediction is very good. The right panel shows the
power-law fit of the distributions with the  parameter given in the insert as a function of the beam
energy. The corresponding thermal excitation energy range is 3-6 MeV/nucleon. The value of the
power-law exponent has a minimum at the point corresponding to the mean break-up temperature Tb
 6 MeV. This is the so-called “critical behavior” expected for systems under the conditions of
phase transition. If so, we can definitely state that this is the liquid-fog phase transition, which takes
place in the spinodal region at the temperatures Tb  Tc.
The general consideration of the problem of the critical temperature in the framework of SMM is
given in Ref.[43].
4. Conclusion
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 has been analyzed within the SMM
with Tc as a free parameter. The value Tc = (20  3) MeV (90% CL) obtained from the best fit to the
data should be considered as some effective value of the critical temperature averaged over all the
fragments produced in the collision. This value is significantly larger than those found in Refs. [28,
29, 31] by the analysis of the multifragmentation data in terms of Fisher's droplet formalism.
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.
Acknowledgments. The authors are grateful to A.Hrynkiewicz, A.N.Sissakian, S.T.Belyaev,
A.I.Malakhov, N.A.Russakovich for support and to W.Reisdorf, W.Trautmann, A.V.Ignatyuk and
I.N.Mishustin for illuminating discussions. The research was supported in part by Grant No. 03-0217263 from the Russian Foundation for Basic Research, by the Grant of the Polish Plenipotentiary to
JINR, by Grant No.1P03 12615 from the Polish State Committee for Scientific Research, by Contract
No. 06DA453 with Bundesministerium für Forschung und Technologie, and by the US National
Science Foundation.
References
[1] S.P. Avdeyev, et al., Nucl. Phys. A 709 (2002) 392.
[2] G. Sauer, H. Chandra, U. Mosel, Nucl. Phys. A 264 (1976) 221.
[3] H. Jaqaman, A.Z. Mekjian, L. Zamick, Phys. Rev. C 27 (1983) 2782.
[4] P.J. Siemens, Nature 305 (1983) 410; Nucl. Phys. A 428 (1984) 189c.
[5] A. Guarnera, XXXIII Winter Meeting on Nucl. Phys., Bormio, 1995;
Preprint GANIL P95-05, 1995, Caen.
[6] S.J.Lee, A.Z. Mekjian, Phys. Rev. C 56 (1997) 2621.
[7] V. Baran, et al., Nucl. Phys. A 632 (1998) 287.
[8] M. D'Agostino, et al., Phys. Lett. B 473 (2000) 219.
[9] L. Beaulieu, et al., Phys. Rev. Lett. 84 (2000) 5971.
[10] O. Lopez, Nucl. Phys. A 685 (2001) 246c.
[11] B. Borderie, et al., Phys. Rev. Lett. 86 (2001) 3252.
[12] N.T. Porile, et al., Phys. Rev. C 39 (1989) 1914.
[13] E. Norbeck, et al., Nucl. Phys. A 607 (1996) 105.
[14] A. Bonasera, et al., La Rivista del Nuovo Cimento 23 (2000) 1.
[15] J. Richert , P. Wagner, Phys. Rep. 350 (2001) 1.
[16] Bao-An Li, et al., Phys. Lett. B 335 (1994) 1.
[17] S.P. Avdeyev, et al., Eur. Phys. J. A 3 (1998) 75.
[18] K. Kwiatkowski, et al., Phys. Rev. Lett. 74 (1995) 3756.
[19] V. Lips,et al., Phys. Lett. B 338 (1994) 141.
[20] S.Yu. Shmakov, et al., Yad. Fiz. 58 (1995) 1735 (Phys. of Atomic Nucl. 58 (1995) 1635).
[21] G. Wang, et al., Phys. Rev. C 53 (1996) 1811;
G. Wang, et al., Phys. Rev. C 57 (1998) R2786.
[22] P. Bonche, et al., Nucl. Phys. A 436 (1985) 265.
[23] Zhang Feng-Shou, Z. Phys. A 356 (1996) 163.
[24] V.A. Karnaukhov, et al., Yad. Fiz. 62 (1999) 272 (Phys. of Atomic Nucl. 62 (1999) 237).
[25] A.S Hirsch, et al., Phys. Rev. C 29 (1984) 508.
[26] M.E. Fisher, Physics. 3 (1967) 255.
[27] A.D. Panagiotou, M.W. Curtin, D.K. Scott, Phys. Rev. C 31 (1985) P. 55.
[28] J.B. Elliott, et al., Phys. Rev. Lett. 88 (2002) 042701.
[29] J.B. Elliott, et al., nucl-ex /0205004v1 (2002).
[30] M. D'Agostino, et al., Nucl. Phys. A 650 (1999) 329.
[31] M. Kleine Berkenbusch, et al., Phys. Rev. Lett. 88 (2002) 022701.
[32] A.S. Botvina, A.S. Il’inov, I.N. Mishustin, Yad. Fyz. 42 (1985) 1127; Sov. J. Nucl. Phys. 42
(1985) 712; J. Bondorf, et al., Nucl. Phys. A 444 (1985) 460.
[33] J.B. Natowitz, et al., Phys. Rev. Lett. 89 (2002) 212701.
[34] V.A. Karnaukhov Yad. Fiz. 60 (1997) 1780 (Phys. of Atomic Nucl. 60 (1997) 1625).
[35] L.G. Moretto, et al., Phys. Lett. B 38 (1972) 471.
[36] J. Schmelzer, G. Ropke, F.P. Ludwig , Phys. Rev. C 55 (1997) 1917.
[37] P.T. Reuter, K.A. Bugaev, Phys. Lett. B 517 (2001) 233.
[38] J. Bondorf, et al., Phys. Rep. 257 (1995) 133.
[39] V.A. Karnaukhov, et al., Phys. Rev. C 67 (2003) 011601(R).
[40] D.G. Ravenhall, C.J. Pethick, J.M. Lattimer, Nucl. Phys. A 407 (1983) 571.
[41] V.D. Toneev, et al., Nucl. Phys. A 519 (1990) 463c.
[42] D.H.E. Gross, Rep. Progr. Phys. 53 (1990) 605.
[43] R. Ogul, A.S. Botvina, Phys. Rev. C 66 (2002) 051601(R).