Microcanonical studies on isoscaling

The exponential scaling of isotopic yields is investigated for sources of different sizes over a broad range of excitation energies and freeze-out volumes, in both primary and asymptotic stages of the decay in the framework of a microcanonical multifragmentation model. It was found that the scaling parameters have a strong dependence on the considered pair of equilibrated sources and excitation energy and are affected by the secondary particle emission of the break-up fragments. No significant influence of the freeze-out volume on the considered isotopic ratios has been observed. Deviations of microcanonical results from grandcanonical expectations are discussed.     

Systematics of fragment observables

Multifragmentation is observed in many reaction types: light-ion-induced reactions at large incident energies (in the GeV region), central heavy-ion collisions from 30 to 100MeV/u, and peripheral heavy-ion collisions between 30 and 1000MeV/u or above. When nucleus-nucleus collisions are considered, another entrance channel parameter is the corresponding mass asymmetry. The first question which is addressed in this contribution is: do we observe similar reactions in each case? Multifragmentation may be related to a phase transition of nuclear matter. Some others features indicate that dynamical features are dominant. It is a priori possible that the underlying mechanisms are different in proton- and nucleus-induced reactions, in central and in peripheral collisions, at limited and at large bombarding energies. In order to see to what extent they can reflect similar behaviour, it is useful to compare the results of various reactions. The observables can be the fragment multiplicity, the mass distributions or the kinematical properties. In this contribution, we are looking for such general features. We will limit the discussion to the observations themselves, rather than the interpretation, which is the subject of numerous entries in this volume. The experimental results indicate that multifragmentation exhibits at the same time universal and entrance-channel-dependent properties.     

Comparisons of statistical multifragmentation and evaporation models for heavy-ion collisions

The results from ten statistical multifragmentation models have been compared with each other using selected experimental observables. Even though details in any single observable may differ, the general trends among models are similar. Thus, these models and similar ones are very good in providing important physics insights especially for general properties of the primary fragments and the multifragmentation process. Mean values and ratios of observables are also less sensitive to individual differences in the models. In addition to multifragmentation models, we have compared results from five commonly used evaporation codes. The fluctuations in isotope yield ratios are found to be a good indicator to evaluate the sequential decay implementation in the code. The systems and the observables studied here can be used as benchmarks for the development of statistical multifragmentation models and evaporation codes. An erratum to this article is available at .     

Fluctuations of fragment observables

This contribution presents a review of our present theoretical as well as experimental knowledge of different fluctuation observables relevant to nuclear multifragmentation. The possible connection between the presence of a fluctuation peak and the occurrence of a phase transition or a critical phenomenon is critically analyzed. Many different phenomena can lead both to the creation and to the suppression of a fluctuation peak. In particular, the role of constraints due to conservation laws and to data sorting is shown to be essential. From the experimental point of view, a comparison of the available fragmentation data reveals that there is a good agreement between different data sets of basic fluctuation observables, if the fragmenting source is of comparable size. This compatibility suggests that the fragmentation process is largely independent of the reaction mechanism (central vs. peripheral collisions, symmetric vs. asymmetric systems, light ions vs. heavy-ion-induced reactions). Configurational energy fluctuations, that may give important information on the heat capacity of the fragmenting system at the freeze-out stage, are not fully compatible among different data sets and require further analysis to properly account for Coulomb effects and secondary decays. Some basic theoretical questions, concerning the interplay between the dynamics of the collision and the fragmentation process, and the cluster definition in dense and hot media, are still open and are addressed at the end of the paper. A comparison with realistic models and/or a quantitative analysis of the fluctuation properties will be needed to clarify in the next future the nature of the transition observed from compound nucleus evaporation to multi-fragment production.     

Zpif's law in the liquid gas phase transition of nuclei

Zpif's law in the field of linguistics is tested in the nuclear disassembly within the framework of isospin dependent lattice gas model. It is found that the average cluster charge (or mass) of rank n in the charge (or mass) list shows exactly inversely to its rank, i.e., there exists Zpif's law, at the phase transition temperature. This novel criterion shall be helpful to search the nuclear liquid gas phase transition experimentally and theoretically. In addition, the finite size scaling of the effective phase transition temperature at which the Zpif's law appears is studied for several systems with different mass and the critical exponents of ν and β are tentatively extracted. Received: 2 June 1999 / Revised version: 8 October 1999     

Freeze-out volume in multifragmentation --Dynamical simulations

Stochastic mean-field simulations for multifragmenting sources at the same excitation energy per nucleon have been performed. The freeze-out volume, a concept which needs to be precisely defined in this dynamical approach, was shown to increase as a function of three parameters: freeze-out instant, fragment multiplicity and system size.     

Multifractal critical thermodynamics in nucleus-nucleus collisions

Critical points approach in the frames of multifractal thermodynamics is suggested to interpret the experimental data on nuclear multifragmentation which come from interactions in nuclear emulsion (in which ¹⁹⁷ ₇₉Au₁₁₈ nuclei of energy ∼1 GeV/nucleon break up into fragments) and from the charge distributions of projectile fragments in sulphur (³²S) fragmentation at 200 GeV/nucleon. It is also shown that multifragmentation after macro-solids collisions exhibits properties analogous to those observed in the nuclear multifragmentation experiments. Received: 19 October 1999     

Formation and deexcitation of the composite nucleus in the Ni + Al reaction at 28 A·MeV

The formation and the deexcitation of the composite nucleus formed during the Ni + Al reaction at 28 A·MeV has been studied with the 4 π multidetector AMPHORA. A rigourous selection of the experimental data is described in order to extract a central collision sample. Then different models are compared to the data. The incomplete fusion process is in agreement with the data. The azimuthal angle correlations of He-Li and Li-Li pairs have been used to discriminate sequential or instantaneous emission. The sequential deexcitation is more consistent with all the data. The different analyses allow to describe all the characteristics of the compound nucleus and finally a fusion cross section of 300 ± 100 mbarn has been measured. Received: 11 February 1998 / Revised version: 1 August 1998     

Two-proton small-angle correlations in central heavy-ion collisions: A beam-energy- and system-size-dependent study

Small-angle correlations of pairs of protons emitted in central collisions of Ca + Ca, Ru + Ru and Au + Au at beam energies from 400 to 1500MeV per nucleon are investigated with the FOPI detector system at SIS/GSI Darmstadt. Dependences on system size and beam energy are presented which extend the experimental data basis of pp correlations in the SIS energy range substantially. The size of the proton-emitting source is estimated by comparing the experimental data with the output of a final-state interaction model which utilizes either static Gaussian sources or the one-body phase-space distribution of protons provided by the BUU transport approach. The trends in the experimental data, i.e. system size and beam energy dependences, are well reproduced by this hybrid model. However, the pp correlation function is found rather insensitive to the stiffness of the equation of state entering the transport model calculations.     

Challenges in nuclear dynamics and thermodynamics

The purpose and contents of this topical issue, Dynamics and Thermodynamics with Nuclear Degrees of Freedom, which grew out of a series of workshops in the years 2004 and 2005, are introduced. The central topics are the nuclear density functional, nuclear multi-fragmentation, and nuclear phase transitions.     

Many-fragment correlations and possible signature of spinodal fragmentation

Abnormal production of events with almost equal-sized fragments was theoretically proposed as a signature of spinodal instabilities responsible for nuclear multifragmentation. Many-fragment correlations can be used to enlighten any extra production of events with specific fragment partitions. The high sensitivity of such correlation methods makes it particularly appropriate to look for small numbers of events as those expected to have kept a memory of spinodal decomposition properties and to reveal the dynamics of a first-order phase transition for nuclear matter and nuclei. This paper summarizes results obtained so far for both experimental and dynamical simulations data.     

Electromagnetic form factors of the nucleon

Dynamical and thermal characterizations of excited nuclear systems produced during the collisions between two heavy ions at intermediate incident energies are presented by means of a review of experimental and theoretical work performed in the last two decades. Intensity interferometry, applied to both charged particles (light particles and intermediate mass fragments) and to uncharged radiation (gamma rays and neutrons) has provided relevant information about the space-time properties of nuclear reactions. The volume, lifetime, density and relative chronology of particle emission from decaying nuclear sources have been extensively explored and have provided valuable information about the dynamics of heavy-ion collisions. Similar correlation techniques applied to coincidences between light particles and complex fragments are also presented as a tool to determine the internal excitation energy of excited primary fragments as it appears in secondary-decay phenomena.     

Influence of momentum-dependent interactions on balance energy and mass dependence⋆

We aim to understand the role of momentum-dependent interactions in transverse flow as well as in its disappearance. For the present study, central collisions involving masses between 24 and 394 are considered. We find that the momentum-dependent interactions have different impact in lighter colliding nuclei compared to heavier colliding nuclei. In lighter nuclei, the contribution of the mean field towards flow is smaller compared to heavier nuclei where binary nucleon-nucleon collisions dominate the scene. The inclusion of momentum-dependent interactions also explains the energy of the vanishing flow in the ¹²C + ¹²C reaction which otherwise was not possible with the static hard equation of state. An excellent agreement of our theoretical attempt is found for balance energy with experimental data throughout the periodic table.     

Instabilities in nuclear matter and finite nuclei

Spinodal instability in nuclear matter and finite nuclei is investigated. This instability occurs in the low-density region of the phase diagram. The thermodynamical and dynamical analysis is based on Landau theory of Fermi liquids. It is shown that asymmetric nuclear matter can be characterized by a unique spinodal region, defined by the instability against isoscalar-like fluctuation, as in symmetric nuclear matter. Everywhere in this density region the system is stable against isovector-like fluctuations related to the species separation tendency. Nevertheless, this instability in asymmetric nuclear matter induces isospin distillation leading to a more symmetric liquid phase and a more neutron-rich gas phase.     

Prompt light particle emission in the 36Ar + 58Ni reaction at 95 MeV/nucleon

The prompt component at intermediate velocity of light charged particles is investigated. An improved coalescence model coupled to the intra-nuclear cascade code ISABEL is used to obtain light complex particle energy spectra and multiplicities as a function of impact parameter. The results are compared with experimental data from the ³⁶Ar + ⁵⁸Ni experiment at 95 MeV/nucleon, performed with the INDRA 4π detection system. The calculated prompt component is found to rather well reproduce proton spectra. For complex light charged particles the calculated components well populate the high energy part of spectra. Prompt emission can therefore explain the large transverse energies experimentally observed at mid-rapidity. Received: 27 July 2000 / Accepted: 20 November 2000     

Energy and angular momentum sharing in dissipative collisions

Primary and secondary masses of heavy reaction products have been deduced from kinematics and E-ToF measurements, respectively, for the direct and reverse collisions of ⁹³Nb and ¹¹⁶Sn at 25 AMeV. Light charged particles have also been measured in coincidence with the heavy fragments. Direct experimental evidence of the correlation of energy-sharing with net mass transfer has been found using information from both the heavy fragments and the light charged particles. The ratio of hydrogen and helium multiplicities points to a further correlation of angular momentum sharing with net mass transfer. Received: 18 September 2000 / Accepted: 2 December 2000     

Dynamical models for fragment formation

The various dynamical models for fragment formation in nuclear collisions are discussed in order to bring out their relative advantages and shortcomings. After discussing the general requirements for dynamical models that aim to describe fragment formation, we consider the various mean-field models that incorporate fluctuations and then turn to models based on molecular dynamics.     

Precompound charged particle emission (PCE) — a mechanism beyond element production by complete fusion

Complete fusion reactions (xn-channels) using actinide targets are observed for values of the effective fissilities x _eff ∼ 0.80 in the sub-pb range of production cross sections. The elements produced at this limit are Z = 108–112. Beyond complete fusion, heavier elements might still be produced by reaction mechanisms releasing part of the nuclear charge before an equilibrated compound system might have been reached. Precompound Charged particle Emission (PCE) is proposed as a possible mechanism following complete fusion. A scheme delivering isotopes of elements Z = 110–115 is discussed, and experimental evidence for such a process is presented. Compound systems, the atomic numbers of which are smaller than in complete fusion reactions, might be produced in ⁴⁸Ca induced reactions on actinides with larger cross sections than those at the limits of complete fusion. Besides complete fusion, the PCE-mechanism should be considered as an alternative to interpret the ⁴⁸Ca-induced reactions on actinides.     

Emission of unbound 8Be and 12C * (0+ 2) clusters in compound nucleus reactions

We have studied the emission of light unbound clusters, ⁸Be and ¹²C ^* (0⁺₂), in the reactions ¹⁸O + ¹³C ^{31} Si ^{23} Ne + ⁸Be and ²⁸Si + ²⁴Mg ^{52} Fe ^{40} Ca + ¹²C ^* (0⁺₂). The -ray spectra obtained in coincidence with ⁸Be and ¹²C ^* (0⁺₂) emission have been studied relative to the statistical emission of two or three -particles. The angular-momentum-to-energy balance of the cluster emission is compared with that of multiple- emission. The properties of the energy spectra of the binary process and the population of the residual nuclei by cluster emission are discussed. It is observed that cluster emission carries away less excitation energy on average than the sequential emission of the individual components.     

Nuclear multifragmentation, its relation to general physics

Heat can flow from cold to hot at any phase separation even in macroscopic systems. Therefore also Lynden-Bell's famous gravo-thermal catastrophe must be reconsidered. In contrast to traditional canonical Boltzmann-Gibbs statistics this is correctly described only by microcanonical statistics. Systems studied in chemical thermodynamics (ChTh) by using canonical statistics consist of several homogeneous macroscopic phases. Evidently, macroscopic statistics as in chemistry cannot and should not be applied to non-extensive or inhomogeneous systems like nuclei or galaxies. Nuclei are small and inhomogeneous. Multifragmented nuclei are even more inhomogeneous and the fragments even smaller. Phase transitions of first order and especially phase separations therefore cannot be described by a (homogeneous) canonical ensemble. Taking this serious, fascinating perspectives open for statistical nuclear fragmentation as test ground for the basic principles of statistical mechanics, especially of phase transitions, without the use of the thermodynamic limit. Moreover, there is also a lot of similarity between the accessible phase space of fragmenting nuclei and inhomogeneous multistellar systems. This underlines the fundamental significance for statistical physics in general.