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Multifragmentation and the liquid-gas phase transition of nuclear matter

One of the major axes of research in present day nuclear physics is the study of the decay of hot and compressed nuclear matter. Interestingly, in between the lower excitation energy regime of light particle evaporation and the high energy limit of complete vaporization, copious production of fragments is observed. This passage from a nuclear liquid to a nucleon gas, via the intermediate phenomenon of multifragmentation, has suggested numerous interpretations which extend from a thermodynamical approach to one that emphasises the dynamics of the process. The achievement of thermal and chemical equilibrium, the measurement of temperatures, densities and pressures, and the observation of a liquid-gas phase transition are only some amongst the most studied subjects.
Figure: Temperature as a function of excitation energy per nucleon from the study of projectile fragments in peripheral collisions at relativistic energies (top; recent data from GSI combined with older measurements from CERN SC, and JINR Dubna) and in dissipative collisions at lower energies (bottom; data from GANIL).

One of the most striking results obtained in this domain are the recent publications related to the so-called nuclear caloric curve. These curves result from an independent measurement of both temperature T (via measurement of ratios of isotopic yields) and excitation energy E* (via calorimetry) for a nuclear system. The upper part of Figure [*] shows the first of such measurements. One may notice the following features: i) at low energy a square root dependence of T on E* as predicted for a Fermi gas, ii) a indication of a plateau possibly corresponding to a saturation of this temperature above an excitation energy of 3 MeV per nucleon, followed by iii) a linear rise whose slope is close to the one associated to an ideal gas. These features and a possible interpretation of the plateau in terms of a latent heat could represent a signal of the long sought liquid-gas phase transition. In a similar measurement (lower part of Figure [*]) temperatures deduced from a series of isotopic ratios differ among each other. These differences can be resolved by taking into account effects such as the population of excited states, secondary decays, and excluded volume.

Figure: Yields of fragments as a function of charge number for central collisions in intermediate energy dissipative collisions in comparison with predictions from a statistical model. Shown is the distribution of all fragments (left) and of the largest (Z1), 3rd largest (Z3) and 5th largest (Z5) fragment. Data from the Multics Coll. taken at MSU.

The full lines in this figure show the predictions of a quantum statistical model where these effects are taken into account and where one assumes, for each excitation energy, an equilibrated state of given density and temperature. These results do not exhibit the simple features as shown in the upper part of Figure [*] and it is presently investigated to what extent the differences can be attributed to expansion effects, to the time dependence of the temperature of the system, and to the fact that different isotopes freeze out at different times.

Multifragmentation: Is equilibrium achieved? ...

    One of the most significant and recent breakthroughs is the understanding that copious multifragmentation can indeed be achieved if low nuclear densities are reached as indicated in the phase diagram (Figure [*]). Dynamical models show, e.g. for central collisions, that a first phase of compression is followed by expansion leading to the low density region where multifragmentation occurs. Using statistical models one finds significant fragmentation only if the densities decrease well below normal nuclear densities. Therefore, multifragmentation may be the experimental indicator that the critical/spinodal region in the phase diagram has been reached. It is, however, a subject of intense theoretical debate to what extent the fragmenting systems is really equilibrated or still has a 'memory' of the earlier phase. The present hypothesis is that the velocity of the expansion which follows the primary compression depends on the symmetry of the colliding nuclei, on the incident velocity and on the impact parameter. If it is too small in order to reach sufficiently low densities the system reaches a turning point and shrinks back to its initial density (see second arrow in Figure [*]). If the initial expansion velocity is sufficiently large, low densities will be achieved and the medium will become unstable against volume fragmentation producing a large number of nucleons and fragments. This is, for such a scenario, the freeze-out configuration, comparable to the saddle point in nuclear fission. Before this freeze-out is reached, the dynamics of the expansion allows for the emission of nucleons and fragments (in small numbers) from the surface, thus cooling the system. The duration of the expansion, its velocity, the final densities reached before break-up and the amount of cooling are questions that are addressed and will be answered in the coming years. A question of special interest arising from this scenario is if and when thermodynamical equilibrium is reached and to what extent the observed fragment yields (Z,A) and spectra are consistent with the statistical models applicable to such a situation. A violent decompression phase or a strong thermal pressure will result in collective radial flow. Decoupling this flow from the thermal motion, the analysis of the production of intermediate mass fragments (IMF) by statistical codes give generally an accurate description as long as the size, the density and the excitation energy of the system at freeze-out are adjusted freely. This is shown in Figure [*] and can also be seen in Figure [*]. Whether the values obtained for this configuration are also consistent with the dynamics has to be verified for the various circumstances where multifragmentation is observed. The evolution of the nuclear system formed depends on the initial condition of the system: impact parameter, relative velocity, mass asymmetry, total size of the system, etc.. This has led experimental studies to look at the evolution of systems formed under very different conditions: central collisions for symmetric systems where compression effects are important, peripheral collisions (or central collisions of asymmetric systems) for a more gentle heating of the nucleus. In most of these cases, copious multifragmentation is found, extending up to the vaporisation of the system. The large variety of beams accessible in Europe has clearly been a remarkable asset in these studies.

...Not necessarily

    The thermodynamical picture for the evolution of hot nuclear matter is not the only one prevailing: such alleys as dynamical effects, mechanical instability and percolative phenomena are suggested and it needs to be studied how they relate to the equilibrium picture. It is, for example, of importance to point out that the freeze-out configurations are rather systematically located within the spinodal region (see Figure [*]). In this region, any mechanical instabilities will rapidly grow and give rise to multifragmentation. It is therefore a challenge for future studies to disentangle which effect, mechanical instability or phase space, is predominant for the production of the IMFs. The observation of a power law behaviour for the size distribution of the fragments has triggered a number of studies that have looked for evidence of critical behaviour. These analyses consider nuclear multifragmentation as one example of a critical phenomenon and attempts are made to extract from the data the related critical exponents. Comparison with percolative and liquid-gas systems show remarkable similarities. The existence of power law behaviour in fragment size distributions still has to be fully understood and progress in this type of critical analysis should eventually relate this nuclear phenomenon to the various universal classes of fragmentation, including thermodynamical processes. Not all experimental situations populating a similar excitation energy range lead to copious multifragmentation: an example is the heating of nuclear matter with high energy anti-protons. In this case a strong reduction of IMF production is observed when comparing to ion beams. Assuming that anti-proton heating leads to a well defined equilibrated source, the production rates of IMFs observed are reproduced with statistical codes using as input parameter normal nuclear density. It is still necessary to understand the particular dynamical path followed by a system heated with anti-protons, i.e. what initial excitation energy is reached and how the system evolves to apparent equilibrium and statistical decay. 

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