Mass distributions for nuclear disintegration from fission to evaporation
By a proper choice of the excitation energy per nucleon we analyze
the mass distributions of the nuclear fragmentation at various
excitation energies. Starting from low energies (between and
MeV/nucleon) up to higher energies about MeV/n, we
classified the mass yield characteristics for heavy nuclei
() on the basis of Statistical Multifragmentation Model.
The evaluation of fragment distribution with the excitation energy
show that the present results exhibit the same trend as the
Keywords: mass distribution, fission, nuclear multifragmentation.
Properties of nuclear reactions have been under investigation for several decays. Experimental and theoretical studies on nuclear reactions are very important not only for the context of nuclear physics but also for our understanding of astrophysical events, such as supernova explosion and formation of neutron stars . At low excitation energies up to MeV per nucleon, the fission of compound nucleus or its evaporation to small particles have been observed both theoretically and experimentally. At the excitation energies in between and MeV per nucleon one may observe a U-shape distribution corresponding to partitions with a few small fragments and one big residual fragment, which looks like an evaporative emission. At excitation energies greater than MeV per nucleon one may observe a mass distribution of intermediate mass fragments (nuclear multifragmentation). During this process, the hot and compressed nuclei tend to expand in thermodynamical equilibrium. Then they enter the region of subsaturation densities, where they become unstable to density fluctuations and break up into the fragments. In this case, it is believed that a liquid-gas phase transition is manifested [2, 3, 4, 5]. At higher energies MeV/n, big fragments disappear, and an exponential fall of the mass distribution with mass number A may be seen.
By a proper choice of the excitation energy per nucleon we analyze the mass distributions of the nuclear fission of heavy elements such as U and Ra at various excitation energies on the basis of Statistical Multifragmentation Model (SMM). We have also discussed the mass distribution of the nuclear fragmentation at higher energies up to MeV/n.
2 Calculations and results
Nuclear fission is one of the most interesting channel of de-excitation of heavy nuclei at low excitation energies up to MeV/nucleon, at which nuclear density exhibits small fluctuations around the equilibrium density fm. In order to study compound nucleus fission one should consider a correct physical description in the light of liquid drop model with shell effects at various deformation modes, where the determination of angular momentum and dynamical variables becomes very important. In our calculations, we consider the Bohr-Wheeler approach  in SMM, where the partial width of the compound nucleus is given by
where is the level density of compound nucleus, the level density at the saddle point, the excitation energy for the nucleus, and the height of the fission barrier. Here, the shell effect on the level densities of the nucleus can be neglected at high energies . The fission barrier is determined by Myers and Swiatecki , and we use the results in Ref.  for the level density at saddle point.
For orientation, we have shown the results of our calculations at low excitation energies for the fission of U in Fig. 1. In this figure, one may see a double peaked distribution at an excitation energy of MeV/n on the upper-left panel, which can be interpreted as asymmetric fission, where the mass ratio of the most probable fragments is about to . In the upper-right panel one may see a triple peaked mass distribution, which is usually interpreted in terms of a single symmetric peak and a double-peaked distributions. In the lower panels we show the characteristic of symmetric fission, associated with higher excitation energies. The evaluation of mass yield distribution with the excitation energy show that the present results are in agreement with those obtained experimentally in the reactions induced by protons and deuterons on Ra .
It is also observed that the proportion of the symmetric component increases with increasing excitation energy, in agreement with experimental results. In order to make a comparison with , we have chosen the nucleus Ra, and showed the results of our calculations in Fig. 2, from which one may see a triple peaked mass distribution at MeV/n, and a single symmetric distribution at MeV/n. The triple peaked distribution can be interpreted in terms of a symmetric fission associated with higher excitation energies, and an asymmetric component associated with lower excitation energies. In the right top and bottom panels, we also show the results in percentage of the variation of the mass yield with mass number A. These results are very similar to the obtained experimentally in Ref. . We also review the fragmentation phenomena at higher excitation energies on the basis of SMM. According to SMM, one assumes a micro-canonical ensemble of breakup channels, and the system should obey the laws of conservation of energy E*, mass number A and charge number Z. The probability of generating any breakup channels is assumed to be proportional to its statistical weight as
where denotes the entropy of a multifragment state of the breakup channel j. The breakup channels are generated by Monte Carlo method according to their weights. Light fragments with mass number and charge number are considered as stable particles (nuclear gas) with masses and spins taken from the nuclear tables. Only translational degrees of freedom of these particles contribute to the entropy of the system. Fragments are treated as heated nuclear liquid drops, and their individual free energies are parameterized as a sum of the bulk, surface, Coulomb and symmetry energy contributions
The bulk contribution is given by , where T is the temperature, the parameter is related to the level density, and MeV is the binding energy of infinite nuclear matter. Contribution of the surface energy is , where 18 MeV is the surface coefficient, and MeV the critical temperature of the infinite nuclear matter. Coulomb energy contribution is , where c denotes the Coulomb parameter obtained in the Wigner-Seitz approximation, , with the charge unit e, fm, and is the normal nuclear matter density (0.15 fm). And finally, the symmetry term is , where MeV is the symmetry energy parameter. All the parameters given above are taken from the Bethe-Weizsäcker formula and correspond to the assumption of isolated fragments with normal density in the freeze-out configuration.
In Fig. 3 we show the results of our calculation for typical mass distributions for the fragmentation of Ra nucleus, at excitation energies of E* = , and MeV/n. One may see from this figure that at an excitation energy of E* = MeV/n (corresponding temperature T MeV), there is a U-shape distribution corresponding to partitions with few small fragments and one big residual fragment. In the so called transition region (T MeV), however, variation of T with E* exhibits a plateau-like behavior that can be interpreted as a sign of liquid-gas phase transition in the system [11, 12, 13], and one observes a smooth transformation for E* = MeV/n in the same figure. At high temperatures (T MeV) for E* = MeV/n, the big fragments disappear and an exponential-like fall-off is observed. For higher excitation energies (E* MeV/n) the nuclei tend to evaporate to nucleons and small fragments. All these results are in good agreement with experimental data [5, 14, 15, 16, 17]. In view of these theoretical results it is instructive to demonstrate the possibility of the application of such approaches for the analysis of experimental data for nuclear reactions and astrophysical studies [1, 18].
Many helpful discussions with A.S. Botvina are gratefully acknowledged. The authors thank the Board of Scientific Research Project of Selçuk University(BAP). R.O. thanks GSI for hospitality, where part of this work was carried out.
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