12
S. Hilaire and S. Goriely
Given an initial compound nucleus state, fission occurs by tunnelling through all
accessible fission barriers. Therefore, for a single barrier, the fission transmission
coefficient is given by
T f (E, J, π) =
d(J,π)
T hw (E − ε d ) +
E+B n
E c
ρ (ε, J, π) T hw (E − ε) dε
in which ε corresponds to the transition states’ energies. These transition states are
discrete up to a given arbitrary threshold E c , and, as for the compound nucleus at
normal deformation, are then described by a NLD, ρ(ε, J, π ), beyond E c .
The potential energy surface often displays several barriers and the fission transmission coefficient used in the Hauser–Feshbach model takes more complicated
forms [23–25]. For multiple humped fission barriers, one also accounts for the fact
that there exist potential wells between the barriers in which quantum states can be
located, usually called class-II or class-III states depending upon whether they are
located between the first and second barrier or between the second and the third. If
these class-II/III states have a spin and parity corresponding to that of the compound
nucleus from which fission occurs, they induced a resonance effect in the fission
transmission coefficient for which more or less refined treatments are possible [23–
25].
When one uses traditional (i.e. based on analytical expressions) methods to
compute fission cross section, a large number of parameters have to be adjusted
to reproduce at best experimental data. One can adjust the fission barrier heights
and widths, the transition states and their corresponding NLD parameters as well
as the eventual class-II/III states. With increasing kinetic energy of the projectile,
several residual nuclei come into play. For a 10 MeV neutron incident on 238 U, for
instance, both 239 U (first-chance fission) and 238 U (second-chance fission) fission
barrier parameters have to be simultaneously fine-tuned and the higher the incident
energy, the larger the number of fissioning nuclei. If this makes the fine-tuning more
complicated, it also provides a way to get more constraints than the single 238 U
neutron-induced fission cross section, provided one wants to coherently model the
several fission chances.
To be more precise, the fact that the fourth chance of 238 U fission is governed
by the fission barriers of 236 U implies that the latter should also provide a good
description of the first chance of 235 U since in both cases the nucleus which
undergoes fission is the same. Another constraint can also be obtained by noticing
that the fission barriers parameters enabling a proper description of photo-induced
fission on 238 U should also provide a good second-chance fission of neutron-induced
fission of 238 U. Therefore, a coherent modelling of fission means that the same set
of input parameters should provide simultaneously the various fission chances of the
various fissioning systems encountered within a given isotopic chain. An illustration
of the results obtained within such a modelling framework is given in Fig. 5. If the
price to pay by considering all these constraints is an important amount of work,
S. Hilaire and S. Goriely
Given an initial compound nucleus state, fission occurs by tunnelling through all
accessible fission barriers. Therefore, for a single barrier, the fission transmission
coefficient is given by
T f (E, J, π) =
d(J,π)
T hw (E − ε d ) +
E+B n
E c
ρ (ε, J, π) T hw (E − ε) dε
in which ε corresponds to the transition states’ energies. These transition states are
discrete up to a given arbitrary threshold E c , and, as for the compound nucleus at
normal deformation, are then described by a NLD, ρ(ε, J, π ), beyond E c .
The potential energy surface often displays several barriers and the fission transmission coefficient used in the Hauser–Feshbach model takes more complicated
forms [23–25]. For multiple humped fission barriers, one also accounts for the fact
that there exist potential wells between the barriers in which quantum states can be
located, usually called class-II or class-III states depending upon whether they are
located between the first and second barrier or between the second and the third. If
these class-II/III states have a spin and parity corresponding to that of the compound
nucleus from which fission occurs, they induced a resonance effect in the fission
transmission coefficient for which more or less refined treatments are possible [23–
25].
When one uses traditional (i.e. based on analytical expressions) methods to
compute fission cross section, a large number of parameters have to be adjusted
to reproduce at best experimental data. One can adjust the fission barrier heights
and widths, the transition states and their corresponding NLD parameters as well
as the eventual class-II/III states. With increasing kinetic energy of the projectile,
several residual nuclei come into play. For a 10 MeV neutron incident on 238 U, for
instance, both 239 U (first-chance fission) and 238 U (second-chance fission) fission
barrier parameters have to be simultaneously fine-tuned and the higher the incident
energy, the larger the number of fissioning nuclei. If this makes the fine-tuning more
complicated, it also provides a way to get more constraints than the single 238 U
neutron-induced fission cross section, provided one wants to coherently model the
several fission chances.
To be more precise, the fact that the fourth chance of 238 U fission is governed
by the fission barriers of 236 U implies that the latter should also provide a good
description of the first chance of 235 U since in both cases the nucleus which
undergoes fission is the same. Another constraint can also be obtained by noticing
that the fission barriers parameters enabling a proper description of photo-induced
fission on 238 U should also provide a good second-chance fission of neutron-induced
fission of 238 U. Therefore, a coherent modelling of fission means that the same set
of input parameters should provide simultaneously the various fission chances of the
various fissioning systems encountered within a given isotopic chain. An illustration
of the results obtained within such a modelling framework is given in Fig. 5. If the
price to pay by considering all these constraints is an important amount of work,