ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 12, с. 2101-2107
ЯДРА
CRITICAL TEMPERATURE FOR THE NUCLEAR LIQUID-GAS PHASE TRANSITION (FROM MULTIFRAGMENTATION AND FISSION)
© 2008 V. A. Karnaukhov1), H. Oeschler2), A. Budzanowski3), S. P. Avdeyev1), A. S. Botvina4), E. A. Cherepanov1), W. Karcz3), V. V. Kirakosyan1), P. A. Rukoyatkin1), I. Skwirczynska3), E. Norbeck5)
Received February 2, 2008
Critical temperature Tc for the nuclear liquid—gas phase transition is estimated both from the mul-tifragmentation and fission data. In the first case, the critical temperature is obtained by analysis of the intermediate-mass-fragment yields in p (8.1 GeV) + Au collisions within the statistical model of multifragmentation. In the second case, the experimental fission probability for excited 188Os is compared with the calculated one with Tc as a free parameter. It is concluded for both cases that the critical temperature is higher than 15 MeV.
PACS: 25.70.Pq, 24.10.Lx, 24.10.Pa, 25.55.Ci
1. CRITICAL TEMPERATURE FROM MULTIFRAGMENTATION DATA
The critical temperature Tc for the liquid—gas phase transition is a crucial characteristic related to the nuclear equation of state (EOS). According to [1, 2], the nuclear EOS can be presented as follows:
p = ap + bp2 + cp3,
(1)
where a = kBT, b = -kBTc/pc and c = 2kBTc/6p2. The coefficients b and c depend directly on the value of the critical temperature Tc and the critical density pc. This EOS is similar to Van-der-Waals equation suggested in 1875.
There are many calculations of Tc for finite nuclei. In [2—6], it is done by using the 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 interaction parameters and the details of the model. In [7, 8] the thermostatic properties of nuclei are considered employing the semiclassical nuclear model, based on the Seyler—Blanchard interaction. In [8] critical temperature is estimated to be Tc = 16.66 MeV.
As the temperature of a nucleus increases, the surface tension decreases and then vanishes at Tc.
1)1 Joint Institute for Nuclear Research, Dubna, Russia.
2)Institut fur Kernphysik, Darmstadt University of Technology, Germany.
Niewodniczanski Institute of Nuclear Physics, Cracow, Poland.
4)Institute for Nuclear Research, Moscow, Russia.
5)University of Iowa, USA.
This vanishing defines the critical temperature [9]. For temperature below the critical one, two distinct nuclear phases coexist — liquid and gas. Beyond Tc there is not a two-phase equilibrium, only the nuclear vapor exists.
Figure 1 shows the different approximations used in the literature for the surface-tension coefficient as a function of T/Tc. Curve 1 corresponds to the well known equation for a(T):
a(T) = a(Q)
Г2 - Г2 T2 + T2
5/4
(2)
This equation was obtained in [10] devoted to the consideration of thermodynamic properties of a plane interface between liquid and gaseous phases of nuclear matter in equilibrium. This parametrization is successfully used by the statistical model of multi-fragmentation (SMM) for describing the multifragment decay of hot nuclei [11]. Curve 2 was calculated within the framework of the semiclassical model based on the Seyler—Blanchard interaction [7, 8]. An analytical expression for a(T) obtained in [8] is
a(T) = ff(Q) 1 + 1.5
T
'TP.
1 -
T Tr
1.5
(3)
Two other parametrizations of a(T) are also presented in Fig. 1: linear, a(T) - (1 - T/Tc), which is used in the analysis of the multifragmentation data with the Fisher droplet model [12, 13], and quadratic, a(T) - (1 - T/Tc)2 [14]. The symbols are taken from calculations [3], which were different from those used
2101
2102 KARNAUKHOV et al.
o(7)/o(0)
T/ Tc
Fig. 1. The calculated coefficient of the surface tension as a function of T/Tc. Curves 1 and 2 are obtained according to Eqs. (2) and (3), curves 3 and 4 are for linear and quadratic parametrizations of a(T). The symbols are calculated in [3].
in [10]. The agreement of such different calculations supports the reliability of Eq. (2).
Nuclear multifragmentation is the main source of the experimental information about the critical temperature. In some statistical models the shape of the charge distribution for the intermediate-mass fragments (IMF) is sensitive to the ratio T/Tc. It was noted in earlier papers that the fragment charge distribution is well described by the power law, Y(Z) ~ ~ Z-T [15], as predicted by the Fisher prescription for classical droplets in the vicinity of the critical point [16]. In [15] the critical temperature was estimated to be MeV simply from the fact that the IMF mass distribution is well described by a power law for the collision of p (80-350 GeV) with Kr and Xe. In [17] experimental data were gathered for different colliding systems to get the temperature dependence of the power-law exponent. The temperature was derived from the inverse slope of the fragment energy spectra in the range of the high-energy tail. The minimal value of t was obtained at T = 11-12 MeV, which was claimed to be Tc. The later data smeared out this minimum. Moreover, it became clear that the "slope temperature" for fragments does not coincide with the thermodynamic one, which is significantly smaller.
A more sophisticated use of Fisher's model has been made in [12]. The data for multifragmentation in n (8 GeV/c) + Au collisions were analyzed yielding a critical temperature of 6.7 ± 0.2 MeV. The same analysis technique was applied to collisions of Au, La, and Kr (at 1.0 GeV per nucleon) with a carbon
target [13]. The extracted values of Tc are 7.6 ± ± 0.2, 7.8 ± 0.2, and 8.1 ± 0.2 MeV, respectively. Note that Fisher's prescription is reasonable when the temperature of the system is close to the critical one. In fact, it is not the case. Application of this model may give a spurious value for Tc. The shortcomings of Fisher's prescription were already discussed in the literature [18, 19].
It should be noted that in some papers the term "critical temperature" is not used in the strict thermodynamic sense given above. In [20] multifragmen-tation in Au + Au collisions at 35 MeV per nucleon was analyzed with the so-called Campi plot [21] 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. The more appropriate term here is the "breakup temperature". This temperature corresponds to the onset of the fragmentation of the nucleus entering the phase-coexistence region. Sometimes the term "limiting temperature" is also used for it. Analysis of the experimental data on the "limiting temperatures" in [22] resulted in estimation of Tc for the symmetric nuclear matter, which was found to be equal to 16.60 ± 0.86 MeV.
Having in mind the shortcomings of Fisher's prescription we have estimated the nuclear critical temperature within the framework of the SMM [11]. This model describes well different properties of thermal disintegration of target spectators produced in collisions of relativistic light ions. The yield of IMF, Y(Z), depends on the contribution of the surface free
energy to the entropy of a given final state, therefore it is sensitive to the value of the critical temperature. This is well demonstrated in paper [23]. Experimental data for p (8.1 GeV) + Au collisions are compared with model predictions. The combined INC + SMM model is used with the intranuclear cascade prescription (INC) to describe the first stage of the reaction. The comparison of the measured and calculated IMF charge yields provides a way to estimate Tc. In [24] it was done by the analysis of the fragment charge distributions for the p (8.1 GeV) + Au reaction with Tc as the only free parameter. It was found that Tc = = 20 ± 3 MeV.
In the next paper by the FASA Collaboration [25] the value Tc = 17 ± 2 MeV was obtained from the analysis of the same data using a slightly different separation of the events. In contrast to [24], the analysis was done for the fragments in the range Z = = 4-11 to exclude the influence of nonequilibrium emission of Li. It was found that the fragment yield depends not only on the critical temperature but also on the breakup volume Vt. The optimal values of Vt and Tc were obtained by the least-square fitting of the data [23]. It was concluded that Tc ^ 18 MeV.
Thus, different experimental estimations of the critical temperature from fragmentation data are very contradictory. This is a reason to look for other observables that are sensitive to the critical temperature for the liquid—gas phase transition. It was suggested in [26] to analyze the temperature dependence of the fission probability to estimate Tc. The following two sections are based on [27].
2. TEMPERATURE DEPENDENCE OF FISSION BARRIER
The fissility of heavy nuclei is determined by the ratio of the Coulomb and surface free energies: the larger the ratio, the smaller the fission barrier. As the temperature approaches the critical one from below, the surface tension (and surface energy) gradually decreases, and the fission barrier becomes lower. Thus, the measurement of fission probabilities for different excitation energies allows an estimate of how far the system is from the critical point. Temperature effects in the fission barrier were considered in a number of theoretical studies based on different models (see, e.g., [3, 28-33]). The effect is so large for hot nuclei that the barrier vanishes, in fact, at temperatures of 4-6 MeV for the critical temperature Tc in the range 15-18 MeV.
In terms of the standard liquid-drop conventions [34], the fission barrier can be represented as a function of temperature by the following relation:
Bf (T, Ts) = ES(TS) - E0(T) + Ec(Ts) - (4)
- E0(T) = E0(T) [(Ba - 1) + 2x(T)(Bc - 1)].
Here, Bs is the surface (free) energy at the saddle point in units of surface energy E0(T) of a spherical drop; Be is the Coulo
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