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

3.1.3.1. Fission barrier parameters
Shell correction model calculations by Howard
and Möller [3.1.32] for actinide nuclei predicted axial
asymmetry of the inner fission barrier A for neutron
numbers N > 144 and axial symmetry for N ≤ 144,
whereas the outer barrier B was predicted to be mass
asymmetric. Uranium inner and outer fission barrier
parameters are relevant for these asymmetries, and
were defined in Refs [3.1.36, 3.1.37]. These fission
barrier parameters fit neutron induced fission crosssection
data for actinide target nuclides up to
emissive fission thresholds, and reproduce available
neutron induced fission cross-sections up to E n of
20 MeV [3.1.38]. Inner E fA, ћω fA and outer E fB,
ћω fB barrier heights and widths correspond to
lumped (S1 + S2) mode asymmetric fission, that will
be employed to calculate the combined yield of
lumped (S1 + S2) mode fission. Fission barrier
parameters for light U nuclei (A < 230) were defined
by maintaining the difference (E fA(B) – B n), where B n
is the neutron binding energy (depends on the odd/
even type of the number of neutrons). Axial
symmetry is assumed at the inner saddle A for
neutron deficient U nuclei (E fAB < E fB ).
Outer barrier parameters for the super long
mode (E fBSL) and width (ћω BSL) depend on the
saddle symmetry. However, symmetry of the outer
saddle shape for the super long mode fission of U
nuclei has not been defined unambiguously. A
rather strong symmetric fission yield was observed
below the emissive fission threshold in the fission
reactions of lower mass nuclides 227 Ac, 228 Ac and
228 Ra formed in reactions 226 Ra(³He,p) 228 Ac,
226 Ra(³He,d) 227 Ac [3.1.7] and 226 Ra(t,p) 228 Ra [3.1.8],
respectively. The double humped fission barrier
model was used to fit the data, assuming arbitrarily
that symmetric fission “involves a separate outer
barrier that is axially asymmetric” [3.1.7, 3.1.8]. The
potential energy surface was investigated with the
shell correction model as a function of axial and
mass degrees of freedom in the vicinity of the
second saddle point for 228 Ra and 238 U [3.1.39]. Two
distinct saddle points were separated by ~1 MeV,
and were distinguishable for 228 Ra: the normal mass
asymmetric saddle (i.e. S1 or S2), which is stable
with respect to triaxial deformation; and the mass
symmetric saddle point with axially asymmetric
deformations (i.e. super long). As regards the
threshold energy and excitation energy dependence,
these conclusions are consistent with measured
fission probability data [3.1.7, 3.1.8]. However, the
situation is less certain in the case of the 238 U
30
fissioning nuclide. The mass asymmetric saddle and
mass symmetric outer saddle were found to be
separated by ~700 keV, but the ridge that separates
the two saddles of 238 U was lower than in the case of
228 Ra. The lower ridge separating the symmetric and
asymmetric mode valleys was discovered recently
by Möller et al. [3.1.40] through extensive shell
correction model calculations in the case of 234 U.
Möller et al. [3.1.40] have shown that mass
symmetric saddle and mass asymmetric saddle
modes are well separated in the case of 228 Ra until
scission, while for U nuclei the situation is less
certain. Thus, the symmetric outer saddle point for
234 U is ~1 to 2 MeV higher than the asymmetric saddle
point, but splitting of the asymmetric valley into S1
and S2 modes was not predicted. Since S1 and S2
saddle point asymmetries would be similar, we have
assumed one symmetric and one asymmetric mode
for fission probability calculations. This assumption
provides a means of interpreting the super long
mode fission yield in the 238 U(n,f) and 235 U(n,f)
reactions up to E n ~ 6 MeV. Partial S1 and S2 mode
fission yields could be easily fitted with adopted
values of the E fA(B) and ћω fA(B) parameters by
varying E fBS1(S2) and ћω BS1(S2) .
Statistical Hauser–Feshbach model calculations
have shown that level density modelling at the
equilibrium ground state of 238 U and saddle deformations
of 239 U represent a key tool in fitting
neutron induced super long mode fission crosssection
data [3.1.9] for 238 U(n,f) up to E n ~ 6 MeV.
Super long mode fission barrier parameters for the
239 U and 236 U fissioning nuclei were found to be
higher than those of asymmetric modes by
~3.5 MeV, and a barrier width ћω BSL = 2.25 MeV
was obtained. The contributions of lower mass U
nuclides via (n,xnf) reaction to the observed 239 U
symmetric fission might be obtained by assuming
the same difference of the outer barriers for the
symmetric super long and asymmetric S1(S2) fission
modes (E fBSL – E fBS1(S2) ) ~ 3.5 MeV. Shell correction
values are defined as (dW fBSL – dW fBS1(S2)) ~ 3.5
MeV, assuming dW fS1(S2) ~ 0.6 MeV [3.1.28] (i.e. for
the higher height of the outer saddle, the higher
shell correction value is assumed). Asymmetric
fission of 239 U will dominate for excitation energies
up to the emissive fission threshold, because a
fission barrier leading to the asymmetric valley is
much lower than one leading to the symmetric
valley. We assumed that 239 U and 236 U nuclei at the
outer mass symmetric saddles might be unstable
with respect to triaxial deformations. The same
instability was assumed for the inner saddle of U