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W. Younes
be solved using standard techniques and used to predict fission-fragment properties
such as mass distributions, TKE, TXE, etc. [89–92]. The GCM can also be extended
to include single-particle degrees of freedom [93] while also eliminating the need
for a GOA [94]. The standing challenge for these GCM approaches is to include
all relevant collective and single-particle degrees of freedom in a computationally
tractable way [94, 95]. Another difficulty in this approach is that the GCM states
for different values of the collective parameters are not guaranteed to be orthogonal,
which complicates the interpretation of the results [88].
The discrete-basis approach to fission provides an alternative to the GCM by
mixing states that are orthogonal from the start, and where single-particle degrees
of freedom can be built in explicitly [96–98]. In this method, a set of configurations
must first be identified that can be used as a discrete basis to adequately model
fission dynamics near scission. Next, the continuum wave functions of post-scission
states have to be described, and their coupling to pre-scission states has to be
calculated. In this way, the discrete-basis method will be able to describe the latter
stages of the fission process, which are especially challenging to model within the
GCM framework, because of non-adiabatic behavior [94].
In [96, 97] the construction of a discrete basis for the description of fission
dynamics is illustrated by a toy model for the fictitious fission of 32 S into two 16 O
fragments. The ground state of 32 S can be constructed simply by filling the lowest
shell-model orbitals for the protons and neutrons: 0s 1/2 , 0p 3/2 , 0p 1/2 , 0d 5/2 , and
1s 1/2 . Identifying the magnetic substate quantum numbers with K, the projection
of the angular momentum on the symmetry axis of the nucleus, we define a K π
partition as the number of pairs of nucleons of a given type occupying states with
given K quantum number and parity π . For the 16 O + 16 O system the K π partition
corresponds to the ground state of a single 16 O nucleus and its parity partner, so
that their ± combination has good parity. By comparing K π partitions of the initial
32 S and final 16 O + 16 O configurations, the “fission” process can be seen to proceed
through a set of intermediate 2p-2h states.
The same principles can be applied to more realistic Hartree-Fock calculations
of heavy nuclei [98]. Configurations with a given K partition (here parity is no
longer necessarily a good quantum number) can be followed as a function of the
quadrupole constraint (Q 20 ) to scission. In some cases, these configurations will
display a minimum in energy as a function of Q 20 just before scission and are
labeled as “cliff” states, while others will have no minimum to hold the nucleus
back from separating into fragments and are labeled “glider” states. When following
the lowest-energy configurations along the fission path for 236 U [98], the transitions
between different K partitions before scission involve single or double pair jumps,
while the transition between glider states at scission can involve significantly more
pairs. The major rearrangement of the K partition at scission is not easily described
in approaches that rely on shape parameters alone (e.g., the GCM without singleparticle extensions), and is more naturally described in a discrete-basis approach
[99].
W. Younes
be solved using standard techniques and used to predict fission-fragment properties
such as mass distributions, TKE, TXE, etc. [89–92]. The GCM can also be extended
to include single-particle degrees of freedom [93] while also eliminating the need
for a GOA [94]. The standing challenge for these GCM approaches is to include
all relevant collective and single-particle degrees of freedom in a computationally
tractable way [94, 95]. Another difficulty in this approach is that the GCM states
for different values of the collective parameters are not guaranteed to be orthogonal,
which complicates the interpretation of the results [88].
The discrete-basis approach to fission provides an alternative to the GCM by
mixing states that are orthogonal from the start, and where single-particle degrees
of freedom can be built in explicitly [96–98]. In this method, a set of configurations
must first be identified that can be used as a discrete basis to adequately model
fission dynamics near scission. Next, the continuum wave functions of post-scission
states have to be described, and their coupling to pre-scission states has to be
calculated. In this way, the discrete-basis method will be able to describe the latter
stages of the fission process, which are especially challenging to model within the
GCM framework, because of non-adiabatic behavior [94].
In [96, 97] the construction of a discrete basis for the description of fission
dynamics is illustrated by a toy model for the fictitious fission of 32 S into two 16 O
fragments. The ground state of 32 S can be constructed simply by filling the lowest
shell-model orbitals for the protons and neutrons: 0s 1/2 , 0p 3/2 , 0p 1/2 , 0d 5/2 , and
1s 1/2 . Identifying the magnetic substate quantum numbers with K, the projection
of the angular momentum on the symmetry axis of the nucleus, we define a K π
partition as the number of pairs of nucleons of a given type occupying states with
given K quantum number and parity π . For the 16 O + 16 O system the K π partition
corresponds to the ground state of a single 16 O nucleus and its parity partner, so
that their ± combination has good parity. By comparing K π partitions of the initial
32 S and final 16 O + 16 O configurations, the “fission” process can be seen to proceed
through a set of intermediate 2p-2h states.
The same principles can be applied to more realistic Hartree-Fock calculations
of heavy nuclei [98]. Configurations with a given K partition (here parity is no
longer necessarily a good quantum number) can be followed as a function of the
quadrupole constraint (Q 20 ) to scission. In some cases, these configurations will
display a minimum in energy as a function of Q 20 just before scission and are
labeled as “cliff” states, while others will have no minimum to hold the nucleus
back from separating into fragments and are labeled “glider” states. When following
the lowest-energy configurations along the fission path for 236 U [98], the transitions
between different K partitions before scission involve single or double pair jumps,
while the transition between glider states at scission can involve significantly more
pairs. The major rearrangement of the K partition at scission is not easily described
in approaches that rely on shape parameters alone (e.g., the GCM without singleparticle extensions), and is more naturally described in a discrete-basis approach
[99].