SpinDependent PreEquilibrium Exciton Model Calculations for Heavy Ions

1
Spin-Dependent Pre-Equilibrium Exciton Model
Calculations for Heavy Ions

• E. Beták
• Institute of Physics SAS, 84511 Bratislava,
  Slovakia
• and
• Silesian University, 74601 Opava, Czech Republic

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 (nph). 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.
3
Originally, the exciton model did not consider
spin variables. They have been included
(Obloinský, Phys. Rev. C35 (1987), 407
Obloinský 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 (Beták, Fizika B12 (2003), 11), but as
the set of master equations is really huge in
this case, only some basic features have been
reported.
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
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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 Tls, and is of the same form as for
the compound nucleus. Pre-equilibrium g emission
(or absorption) is associated with two processes,
Dn0 and Dn2. If we assume E1 transitions
only, there are 3 values of spin starting from
the same initial value.
6
The spin function x for the Dn2 does not
depend on the exciton number, whereas the x0
does.
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• 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 n03 for reactions induced by
  nucleons and 4 for those
• by alphas. As the state of n1 transforms
  completely to n3 before
• the particle emission starts, n03 is the same
  as n01 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.

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
9
This procedure has been applied to 40Ca40Ca
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.
10
The greatest difference between the
spin-independent formulation and that including
the angular momentum couplings is at the very
beginning of the reaction
11
Comparison of calculated g energy spectra from
40Ca40Ca at 1000 MeV compared to the data
(Cardella et al., 9th Int. Conf. React. Mech.,
Varenna 2000, p. 427). Figure shows
spin-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.
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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 (Obloinský 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.