Fission Product Yield Data for the Transmutation of Minor Actinide ...

a similar way, and we will concentrate below on the
analysis of super long mode fission. Contributions
of super long mode fission to the observed fission
cross-section are defined by the following equation:
s E s E s E
28
X
nFSL n nfSL n  n,xnffSL n
x = 1
( )= ( )+ ( )
(3.1.2)
coming from (n,xnf), x = 1, 2, 3...X fission reactions
of relevant equilibrated uranium nuclei, and can be
calculated using fission probability estimates
J P E :
s
p ( )
fSL
n
n,xnfSL (
J U max
n)= Â Ú
Jp
x+
1
Jp
fSL( x+
1)
Jp
0
E W ( U) P ( U) dU
(3.1.3)
where is the population of the (x + 1)th
nucleus at excitation energy U after emission of
x neutrons. The excitation energy Umax is defined by
the incident neutron energy (En) and energy
removed from the composite system by 238 Jp
Wx+1( U)
U(n,xnf)
reaction neutrons. The fission probability
Jp
P is related to the symmetric scission of
fSL( x+1)
( U)
the xth fissioning nucleus, and can be approximated
as:
Jp
fSL( x+ 1)
U
p p
Jp
fSLx
Jp
fSLx fSIx fSIIx
P
( )=
( U)+ ( U)+ ( U )
JpJp ( U ) + T ( U)+ T ( U)
(3.1.4)
T
J T J T T
(3.1.4)
where the super long mode fission probability
Jp
Jp Jp
( PfSL( x+1)
( U)
) depends on TfSLx ( U ) , TfSIx ( U ) ,
Jp Jp Jp
TfSIIx ( U ) , Tnx ( U ) and Tg x , the transmission
coefficients of the super long (SL), S1, S2 mode
fission, neutron emission and radiative decay
channels, respectively. The index ‘x’ refers to the
nuclides fissioning in the (n,xnf) reaction, and will
be omitted from the equations developed below for
simplicity.
Consider a double humped fission barrier
model [3.1.27] in which the neutron induced super
long mode fission process can be viewed as a two
step process (i.e. successive crossing over the inner
hump A and outer hump BSL), while separate outer
barrier humps BS1 and BS2 are assumed for S1 and
U ( )
nx
g x
S2 modes. Hence, the transmission coefficient of the
super long mode fission channel Jp T can be
fSL ( U )
approximated by the equation [3.1.28, 3.1.29]:
T
Jp
fSL
Jp
Jp
TfA U TfBU
( U )=
Jp
Jp
T U T U
fA
(3.1.5)
The fission transmission coefficients Jp Tfi U
are defined by the discrete transition states and
level density rfi(e, J, p) of the fissioning nucleus at
the inner and outer saddles (i = A, BSL, BS1, BS2):
Jp
fi
J
Â
K=-J JKp
fi
. (3.1.6)
The outer fission barrier height (E fBSL ) and
width (ћw BSL) are correlated with the axial
asymmetry of the outer saddle point, which
increases the level density r fBSL (e, J, p) at outer
saddle point deformations while the width should
also influence the energy dependence of the super
long mode yield.
3.1.2.2. Level density
( ) ( )
( )+ ( ) .
fB
( )= ( )
T U T U
+
( )
r e, J, p de
U
fi
Ú 0 1 + exp 2p
Efi + e -U
/ hwi
( ( ) )
( )
The total nuclear level density r(U, J, p) is
represented as the factorised contribution of quasiparticle
and collective states [3.1.30]. Quasi-particle
level densities r qp(U, J, p) were calculated with a
phenomenological generalized super fluid model by
Ignatyuk et al. [3.1.31], taking into account shell,
pairing and collective effects:
ρ(U, J, π) = K rot (U, J)K vib (U)ρ qp (U, J, π) (3.1.7)
where Krot (U, J) and Kvib (U, J) are the factors of
rotational and vibrational enhancement. At low
intrinsic excitation energies of a few MeV, fewquasi-particle
effects are essential, and their
treatment is described in Ref. [3.1.9]. At saddle and
ground state deformations, sym K is defined
rot ( U, J)
by the deformation order of symmetry adopted
from shell correction model calculations by Howard
and Möller [3.1.32]. Consider axially symmetric
deformations (U nuclei at equilibrium deformation,
neutron deficient U nuclei (N ≤ 144) at inner saddle
deformations and S1(S2) modes at outer saddle
deformations):