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Spin-Dependent Pre-Equilibrium Exciton Model Calculations for Heavy Ions

Spin-Dependent Pre-Equilibrium Exciton Model Calculations for Heavy Ions. E. Běták Institute of Physics SAS, 84511 Bratislava, Slovakia and Silesian University, 74601 Opava, Czech Republic.

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Spin-Dependent Pre-Equilibrium Exciton Model Calculations for Heavy Ions

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  1. Spin-Dependent Pre-Equilibrium Exciton Model Calculations for Heavy Ions E. Běták Institute of Physics SAS, 84511 Bratislava, Slovakia and Silesian University, 74601 Opava, Czech Republic EURISOL User Group, Florence, Jan. 2008

  2. Pre-equilibrium models are very successful to describe spectra and cross sections of various types of nuclear reactions. For nucleon- and light-particle-induced reactions at lower energies, the exciton model is probably the most transparent one. Therein, the states of the reactions are classified according to the number of particles p above and holes h below the Fermi level, together called excitons n (n=p+h). The time development of the system is governed by the set of master equations describing the equilibration process and competing emission during nuclear reaction. (Rather often, the lengthy solution of the master equations is replaced by some closed expressions in practice.) There are two essential quantities of the exciton model, which determine basic behaviour of the reaction and consequently the calculated spectra and cross sections, namely the initial exciton number n0 and the intensity of the intranuclear transitions. The latter one is expressed using some (effective) potential or the interaction matrix element. Either of these approaches – when applied in reality – leads to some parameterization. EURISOL User Group, Florence, Jan. 2008

  3. Originally, the exciton model did not consider spin variables. They have been included (Obložinský, Phys. Rev. C35 (1987), 407; Obložinský and Chadwick, Phys. Rev. C41 (1990), 1652) 20 years ago. In order to apply the exciton model to the heavy ion collisions, the initial configuration needs to be solved. Initial analyses indicated some general scaling with projectile and the incident energy. This dependence has been justified using the overlap of the partner nuclei (target and projectile) in the momentum space (Cindro et al., Phys. Rev. Lett. 66 (1991), 868; Cervesato et al., Phys. Rev. C45 (1992), 2369, and others). Combination of these two basic ingredients is the main premise for the use the pre-equilibrium exciton model also for heavy-ion reactions. Some intentions in this direction have been marked few years ago (Běták, Fizika B12 (2003), 11), but as the set of master equations is really huge in this case, only some basic features have been reported. EURISOL User Group, Florence, Jan. 2008

  4. With inclusion of spin variables, the set of master equations is and the spin-dependent intranuclear transition rates are Here, Y is the energy part (exactly the same as in the spin-independent case), and X is the angular momentum part EURISOL User Group, Florence, Jan. 2008

  5. The angular momentum part X for intranuclear transitions (dumping) is strongly dependent on spin for low exciton numbers, and nearly constant for high ones The spin dependence of the nucleon emission is given by Tl’s, and is of the same form as for the compound nucleus. Pre-equilibrium g emission (or absorption) is associated with two processes, Dn=0 and |Dn|=2. If we assume E1 transitions only, there are 3 values of spin starting from the same initial value. EURISOL User Group, Florence, Jan. 2008

  6. The spin function x+for the |Dn|=2 does not depend on the exciton number, whereas the x0does. EURISOL User Group, Florence, Jan. 2008

  7. The initial exciton number is the main variable, which determines • - How hard the spectra of outgoing particles and/or gammas are. • The most energetic part of the spectra is • where D is related to the type of emitted particles (=2 for nucleons). • Thus, the most energetic part may be used for slope analysis to • determine n0. In early years of the exciton model, the slope analysis • gave typically n0=3 for reactions induced by nucleons and =4 for those • by alphas. As the state of n=1 transforms completely to n=3 before • the particle emission starts, n0=3 is the same as n0=1 here. For heavy • ions, nice systematics emerged, which scaled all types of HI collisions • into the same curve. • What is the neutron-to-proton ratio of the emission in the hard part • of the spectrum. EURISOL User Group, Florence, Jan. 2008

  8. The initial exciton number calculated (in the spin-independent formulation) from the overlaps of the colliding nuclei in phase space reproduces the empirical data and can be well approximated by simple formulae (Korolija et al., Phys. Rev. Lett. 60 (1980), 193; Cindro et al., Phys. Rev. Lett. 66 (1991), 868 and Fizika B1 (1992), 51; Ma et al., Phys. Rev. Lett. 70 (1993) and Phys. Rev. C48 (1993), 448), which e.g. read Spin can be introduced by simple subtracting the rotational energy of the double-nuclear system. Thus, the effective energy which is responsible for approaching of the nuclei in their radial coordinate is now EURISOL User Group, Florence, Jan. 2008

  9. This procedure has been applied to 40Ca+40Ca collisions at 1000 MeV (25 A MeV). The distribution of the initial exciton numbers and corresponding excitation energies and also the corresponding contribution to the cross section (of a creation of a composite system with specified initial excitation energy and spin) are shown below. EURISOL User Group, Florence, Jan. 2008

  10. The greatest difference between the spin-independent formulation and that including the angular momentum couplings is at the very beginning of the reaction EURISOL User Group, Florence, Jan. 2008

  11. Comparison of calculated g energy spectra from 40Ca+40Ca at 1000 MeV compared to the data (Cardella et al., 9th Int. Conf. React. Mech., Varenna 2000, p. 427). Figure showsspin-independent formulation, that with angular momentum couplings, and also the possibility of using the Generalized Lorentzian (in the non-spin calculations). There was no adjustment of the parameterers for the g emission, and similarly no tuning of the details of pre-equilibrium calculations to get them closer to the data. EURISOL User Group, Florence, Jan. 2008

  12. CONCLUSIONS: The formulation of the pre-equilibrium exciton model has been enhanced by the possibility to apply it to the heavy-ion collisions with consideration of spin variables. This approach includes 3 essential ingredients: - Angular momentum couplings (Obložinský et al.) - Initial configuration determined by the overlap in the momentum space (Cindro et al.) - Consideration of the angular momentum of colliding nuclei by the reduction of energy available for other degrees of freedom. Thus, the exciton model becomes a suitable tool to study the charge equilibration in heavy ion collisions. EURISOL User Group, Florence, Jan. 2008

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