Transcript Excitons, spin, and 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
EURISOL User Group, Florence, Jan. 2008
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Pre-equilibrium models are very successful to describe spectra and cross
sections of various types of nuclear reactions. For nucleon- and lightparticle-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.
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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.
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With inclusion of spin variables, the set of master equations is
dP(i, n, E , J , t )

dt
P(i, n  2, E , J , t ) (i, n  2, E , J )
 P(i, n  2, E , J , t ) (i, n  2, E , J )
 P(i, n, E , J , t )[ (i, n, E , J )   (i, n, E , J )  L(i, n, E , J )]

c
P
(
i
'
,
n
'
,
E
'
,
J
'
,
t
)


x ([i ' , n' , E ' , J ' ]  [i , n, E , J ])d
i ', J ', n ', x
and the spin-dependent intranuclear transition rates are
 ( E , J , n) 
2
| M |2 Yn X nJ

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 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.
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The spin function x+ for the |Dn|=2 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
 U n0 D ,
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.
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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
E V
E
 6.8  0.54 cm C
n0
Aproj
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
'
Ecm
(l )  Ecm  Erot (l )
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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.
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The greatest difference between the spin-independent formulation and that
including the angular momentum couplings is at the very beginning of the
reaction
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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 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 (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.
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