Multifragmentation of non-spherical nuclei

In studying the multifragmentation process, a large range of incident energies, changing by about four orders in magnitude, was covered and various types of projectiles, from proton till heaviest available ions, were probed. The reaction mechanism is often considered in terms of two-step scenario where the first, off-equilibrium and dynamical step results in the formation of thermalized source which then, in the second step, decays statistically into light particles and intermediate-mass fragments (IMF’s). Assuming that the thermal equilibrium is attained, various statistical multifragmentation models were employed for the second step (see [1–3] and references quoted therein). These models were so successful in providing an understanding of basic aspects of the multifragmentation process that the deviations between their predictions and the experimental data have been often taken as an indication for dynamical effects in the multifragmentation. This kind of simplistic ’cause effect’ interpretation may however be misleading due to several oversimplifying assumptions in the statistical calculations, such as , e.g., the spherical shape of the thermalized source. Indeed, one expects that the spherical shape can be perturbed during the dynamical phase and the density evolution has both compressed and rarefied zones which can give rise to a rather complicated source forms [4,5]. Perhaps more important are the angular momentum induced shape instabilities [6,7] which may cause large fluctuations of both the Coulomb barrier and the surface energy even for moderately high angular momenta (L ≤ 50h̄). Moreover, at high excitations, not only the quadrupole stiffness becomes small but also the fission saddle point moves towards larger elongations and smaller neck cross-sections [7], giving rise to some ’neck effects’ [6,7]. Hence, before discussing dynamical effects in the multifragmentation decay, one should study the effects of different shapes in the freeze-out configuration. In this paper, the non-spherical fragmenting source is considered within the statistical model and the observables sensitive to the shape of this source are searched for. Our statistical consideration is based on the MMMC method of the Berlin group [1]. In the MMMC method, one calculates all accessible states equally populated in the decay of thermalized system into N fragments. The microscopic thermodynamics used here describes the dependence of the volume of 6N-dimensional phase space on globally conserved quantities (energy, mass, charge, ...) and external constraints (like the spatial volume) to be defined by the first reaction stage. Within the microcanonical ensemble method, an explicit treatment of the fragment positions in the occupied spatial volume allows for a direct extension of the MMMC method [1] to the case of non-spherical shapes. Here, the source deformation is considered as an additional external constraint. Main results of our paper will be given for the source described as an axial ellipsoid : (x/Rx) + (y/Ry) + (z/Rz) = 1, with Rx = Ry 6= Rz. We assume that the freeze-out density of deformed system is the same as that of a spherical system with the radius Rsys = (RxRyRz), i.e., the volume of deformed system is conserved. This condition does not change the weight wr due to the accessible volume of the fragments in the Metropolis scheme of calculations [1]. On the other hand, it means that the ellipsoidal source shape depends on one additional parameter : the ratio of ellipsoid axes : R = Rx/Rz. The ratio R < 1 corresponds to the prolate form, while for the oblate form one has R > 1. An essential feature of non-spherical systems is that the deformation ’costs’ some extra energy Edef which is proportional to a change of nuclear surface with respect to the spherical shape. Since we do not consider shape evolution of the system but rather the influence of source shape on its thermodynamics, this energy Edef will be inaccessible for thermal motion and may be subtracted from the total excitation energy E∗. This subtraction have been done in all our calculations, however this point should be kept in mind if one tries to refer to the real values of energy pumped into the system. The source deformation will noticeably affect the moment of inertia and, together with the Coulomb energy which is calculated exactly for every multifragment configuration of non-spherical nucleus, becomes very important for describing rotating systems. As to the general scheme to account for the total angular momentum and the calculation of the statistical weight wpl of the configuration in the rotating frame, we closely follow Ref.