Similarities between the lattice gas model and some other models of nuclear multifragmentation

We continue our development of the nuclear lattice gas model by exploring links and similarities with other theoretical approaches to nuclear multifragmentation: the percolation model and the statistical multifragmentation model. It is shown that there exists a limit where the lattice gas model reduces to the percolation model. The similarity between the lattice gas model and the statistical multifragmentation model is more indirect and we utilize the equations of state in the two models. By using the law of partial pressures we obtain P-ϱ diagrams for the statistical multifragmentation model and find that these are remarkably similar to those obtained in the lattice gas model via an exact evaluation of the nuclear partition function on the lattice. For completeness, we also compute the P-ϱ diagram for a system obeying pure classical molecular dynamics with a simple two-body force.     

Nuclear multifragmentation: Basic concepts

We present a brief overview of nuclear multifragmentation reaction. Basic formalism of canonical thermodynamical model based on equilibrium statistical mechanics is described. This model is used to calculate basic observables of nuclear multifragmentation like mass distribution, fragment multiplicity, isotopic distribution and isoscaling. Extension of canonical thermodynamical model to a projectile fragmentation model is outlined. Application of the projectile fragmentation model for calculating average number of intermediate mass fragments and the average size of the largest cluster at different Z _bound, differential charge distribution and cross-section of neutron-rich nuclei of different projectile fragmentation reactions at different energies are described. Application of nuclear multifragmentation reaction in basic research as well as in other domains is outlined.     

Statistical description of nuclear break-up

We present an overview of concepts and results obtained with statistical models in the study of nuclear multifragmentation. Conceptual differences between statistical and dynamical approaches and the selection of experimental observables for identification of these processes are outlined. New and perspective developments, like inclusion of in-medium modifications of the properties of hot primary fragments, are discussed. We list important applications of statistical multifragmentation in other fields of research.     

Multifragmentation of Gold nuclei in the interactions with photoemulsion nuclei at 10.7 GeV/nucleon

Recent results from the EMU-01/12 collaboration are presented for 10.7 GeV/nucleon gold nuclei interactions in emulsion. The distributions of “bound” charge (Z _bound , Z _b3 ), multiplicity distributions, fragment correlations and fluctuations are discussed. The data are compared to similar results obtained on the ALADIN setup at 600 MeV/nucleon. It is shown that multifragmentation of gold nuclei at high and intermediate energies has common features. It is also obtained that the IMFs have reduced multiplicity at high energies. The data are analyzed within the scope of the statistical model of multifragmentation. This model requires the following predetermined model ingredients: mass, charge and excitation energy of nuclear residuals. The simple estimation method of these characteristics is proposed in the framework of the Glauber approach. It is shown that the multifragmentation model reproduces qualitatively the present data. A dramatic discrepancy between the predicted and experimental yield of two charged fragments is found. The evolution of transverse momentum of fragments as a function of Z _bound is presented. It is shown that the model greatly underpredicts the transverse momentum of fragments. It is interpreted as evidence of a strong radial flow of spectator fragments.     

Mechanisms of fragment production in heavy-ion reactions at intermediate energies

Comparative analysis of three typical models of nuclear disintegration, the statistical multifragmentation model (SMM), the quantum statistical model (QSM) and a generalized evaporation model (GEM), is carried out. The thermodynamical properties of a decaying system as well as observable characteristics in heavy ion collisions predicted by the different models are discussed. It is shown that these models yield quite similar results for low charge yields at higher excitation energies (E/A>6 MeV per nucleon) and it is suggested that the coincidence measurements of the intermediate mass fragment multiplicity and the neutron and proton multiplicity (or alternatively, the total bound charge) may be very useful for deducing the decay mechanism. The GEM is shown to differ from the other models in predicting a high Z residue peak.     

On the charge dispersion in high-energy proton-xenon collisions

The mass yield and the charge dispersion of secondary fragments produced in high-energetic proton-xenon bombardment are analysed in the frame of our statistical multifragmentation model. The critical mass distribution as well as the charge dispersion, which have led to the discussion of a nuclear liquid-gas phase transition, are easily reproduced within our model. A clear signal of a “phase transition” at T=5 MeV is found and is analysed in terms of various multifragment correlations.     

Critical exponents for nuclear multifragmentation: dynamical lattice model

We present a dynamical and dissipative lattice model, designed to mimic nuclear multifragmentation. Monte Carlo simulations with this model show a clear signature of critical behaviour and reproduce experimentally observed correlations. In particular, using techniques devised for finite systems, we could obtain two of its critical exponents, whose values are in agreement with those of the universality class to which nuclear multifragmentation is supposed to belong.     

Thermal multifragmentation of hot nuclei and liquid-fog phase transition

Multiple emission of intermediate-mass fragments (IMF) in the collisions of protons (up to 8.1 GeV), ⁴He (4 and 14.6 GeV), and ¹²C (22.4 GeV) on Au has been studied with the 4π setup FASA. In all the cases, thermal multifragmentation of the hot and diluted target spectator takes place. The fragment multiplicity and charge distributions are well described by the combined model including the modified intranuclear cascade followed by the statistical multibody decay of the hot system. IMF-IMF-correlation study supports this picture, giving a very short time scale of the process (≤70 fm/c). This decay process can be interpreted as the first-order nuclear “liquid-fog” phase transition inside the spinodal region. The evolution of the mechanism of thermal multifragmentation with increasing projectile mass was investigated. The onset of the radial collective flow was observed for heavier projectiles. The analysis reveals information on the fragment space distribution inside the breakup volume: heavier IMFs are formed predominantly in the interior of the fragmenting nucleus possibly due to the density gradient.     

The dynamical ingredients governing the multifragmentation process and collision dynamics in heavy ion collisions

By simulating numerically the reaction dynamics of heavy ion collisions within the modified quantum molecular dynamics (MQMD), we have studied the influences of the nucleon-nucleon (n-n) collision cross section with and without medium effect, momentum dependent interactions (MDI), equation of state (EOS) and the aggregating method of fragments on the multifragmentation process of heavy ion collisions with different beam energies. It is found that multifragmentation patterns of the final fragment distributions, the collective flows of fragments and single particles, collision number and nuclear matter density depend strongly on then-n cross section and momentum dependent interactions and the nuclear equation of state, especially these dependences are associated with beam energies. But the fragment multiplicity distribution patterns depend very weakly on the equation of state.     

Ternary versus binary fragmentation processes

Using a statistical multifragmentation model, the ratio of ternary to binary fragmentation processes is calculated for the reaction ¹⁰⁰Mo + ¹⁰⁰Mo as a function of the excitation energy and compared to experimental data. The results suggest that for excitation energies above 2 MeV/A ternary multifragmentation of the compound nucleus is not the dominant channel. It is shown that the data can be explained by considering the binary fragmentation of one of the colliding Mo nuclei.     

Is bimodality a sufficient condition for a first-order phase transition existence?

Here we present two explicit counterexamples to the widely spread beliefs about an exclusive role of bimodality as the first-order phase transition signal. On the basis of an exactly solvable statistical model generalizing the statistical multifragmentation model of nuclei, we demonstrate that the bimodal distributions can naturally appear both in infinite and in finite systems without a phase transition. In the first counterexample a bimodal distribution appears in an infinite system at the supercritical temperatures due to the negative values of the surface tension coefficient. In the second counterexample we explicitly demonstrate that a bimodal fragment distribution appears in a finite volume analog of a gaseous phase. In contrast to the statistical multifragmentation model, the developed statistical model corresponds to the compressible nuclear liquid with the tricritical endpoint located at one third of the normal nuclear density. The suggested parameterization of the liquid phase equation of state is consistent with the L. Van Hove axioms of statistical mechanics and it does not lead to an appearance of the nonmonotonic isotherms in the macroscopic mixed phase region which are typical for the classical models of the Van der Waals type. Peculiarly, such a way to account for the nuclear liquid compressibility automatically leads to an appearance of an additional state that in many respects resembles the physical antinuclear matter.     

Isospin and symmetry energy effects on nuclear fragment production in liquid-gas-type phase transition region

We have shown that the isospin of nuclei influences the fragment production during the nuclear liquid-gas phase transition. Calculations for Au¹⁹⁷, Sn¹²⁴, La¹²⁴ and Kr⁷⁸ at various excitation energies were carried out on the basis of the statistical multifragmentation model (SMM). We analyzed the behavior of the critical exponent τ with the excitation energy and its dependence on the critical temperature. Relative yields of fragments were classified with respect to the mass number of the fragments in the transition region. In this way, we have shown that nuclear multifragmentation exhibits a “bimodality” behavior. We have also shown that the symmetry energy has a small influence on fragment mass distribution; however, its effect is more pronounced in the isotope distributions of produced fragments.     

Critical temperature for the nuclear liquid-gas phase transition (from multifragmentation and fission)

Critical temperature T _c for the nuclear liquid-gas phase transition is estimated from both the multifragmentation 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 T _c as a free parameter. It is concluded for both cases that the critical temperature is higher than 15 MeV. The text was submitted by the authors in English.     

Liquid-fog and liquid-gas phase transitions in hot nuclei

Thermal multifragmentation of hot nuclei is interpreted as the nuclear liquid-fog phase transition inside the spinodal region. The exclusive data for p(8.1 GeV) + Au collisions are analyzed within the framework of the statistical model SMM. It is found that the partition of hot nuclei is specified after expansion to a volume equal to V _t = (2.6 ± 0.3)V ₀. The freeze-out volume is found to be twice as large: V _f = (5 ± 1)V ₀. The similarity between multifragmentation and ordinary fission is discussed. The text was submitted by the authors in English.     

Breakup temperature of target spectators in 197Au + 197Au collisions at E/A = 1000 MeV

Breakup temperatures were deduced from double ratios of isotope yields for target spectators produced in the reaction ¹⁹⁷Au + ¹⁹⁷Au at 1000 MeV per nucleon. Pairs of ^3,4He and ^6,7Li isotopes and pairs of ^3,4He and H isotopes (p, d and d, t) yield consistent temperatures after feeding corrections, based on the quantum statistical model, are applied. The temperatures rise with decreasing impact parameter from 4 MeV for peripheral to about 10 MeV for the most central collisions. The good agreement with the breakup temperatures measured previously for projectile spectators at an incident energy of 600 MeV per nucleon confirms the universality established for the spectator decayat relativistic bombarding energies. The measured temperatures also agree with the breakup temperatures predicted by the statistical multifragmentation model. For these calculations a relation between the initial excitation energy and mass was derived which gives good simultaneous agreement for the fragment charge correlations. The energy spectra of light charged particles, measured at τ_lab = 150°, exhibit Maxwellian shapes with inverse slope parameters much higher than the breakup temperatures. The statistical multifragmentation model, because Coulomb repulsion and sequential decay processes are included, yields light-particle spectra with inverse slope parameters higher than the breakup temperatures but considerably below the measured values. The systematic behavior of the differences suggests that they are caused by light-charged-particle emission prior to the final breakup stage.     

Thermal source parameters in Au+Au central collisions at 35 A MeV

Central Au+Au collisions at 35 A MeV are analyzed to find the characteristics of a thermal source, in the framework of the statistical multifragmentation model SMM. A recently introduced backtracing protocol provides an effective comparison of theory and experiment. For the first time the distributions of the central source parameters (density, mass number, excitation energy) are found. The collective energy of primary fragments is also deduced. It is shown that the backtracing procedure allows an estimation of the pre-equilibrium emission.     

Multifragmentation of gold nuclei by light relativistic ions: Thermal breakup versus dynamical disintegration

The multiple emission of intermediate-mass fragments (IMF) is studied for collisions of p, ⁴He, and ¹²C on Au with the 4π FASA setup. The mean multiplicities of IMF saturate at a value of around 2 for incident energies above 6 GeV. An attempt at describing the observed IMF multiplicities in the two-stage scenario, a fast cascade followed by a statistical multifragmentation, fails. Agreement with the measured IMF multiplicities is obtained by introducing an intermediate expansion phase and modifying empirically the excitation energies and masses of remnants. The angular distributions and energy spectra from p-induced collisions are in agreement with the scenario of “thermal” multifragmentation of a hot and expanded target spectator. In the case of ¹²C + Au(22.4 GeV) and ⁴He (14.6 GeV) + Au collisions, deviations from a pure thermal breakup are seen in the fragment energy spectra, which are harder than those both from model calculations and from the measured ones for p-induced collisions. This difference is attributed to a collective flow with the expansion velocity at the surface of about 0.1c (for ¹²C + Au collisions).     

Onset of multifragmentation dominance at 1 GeV proton-induced nuclear reaction for target nuclei with A 160

The onset of the multifragmentation regime is studied by means of 1 GeV proton-induced nuclear reactions at various target nuclei. Analysing the measured longitudinal momentum transferred to the fragmenting nucleus it is found that the multifragmentation process seems to dominate over the induced fission in the mass region A 160. The qualitative behaviour of the experimental findings is reproduced by considering a simple model.