266
W. Younes
particle states. From there, the “shell energy” can be calculated as the sum of
single-particle energies over the occupied states. The difference between this shell
energy and one calculated using a continuous smoothed density of states gives the
shell correction energy [54, 61]. A similar approach gives the pairing correction as
the difference in pairing correlation energy (the difference in ground-state energies
of the nucleus with and without pairing) using the actual single-particle states and a
smooth density of states [54, 61]. There are several methods to generate the potential
function needed for the shell and pairing corrections, but the method of folded
potentials [53, 54] is especially well adapted to the wide range of shapes encountered
in fission.
The macroscopic-microscopic model gives the energy of the nucleus as a
function of its shape, and this can be used to generate a potential energy surface
(PES) as a function of shape parameters. The PES can then be used to locate fission
barriers [62] and extract their properties, which in turn can provide estimates of
fission rates [56, 57]. More importantly, the PES is a fundamental ingredient in
dynamical (i.e., time-dependent) calculations of the fission process. One way to
model fission dynamics is by performing a random walk across the PES. Ward et al.
[63] adopted this approach, using a Metropolis algorithm with transition rates across
the PES determined by level densities obtained with a combinatorial technique.
Another approach is to solve the classical Langevin equation for the system
[64]. The Langevin approach essentially models the evolution of the system as
that of a Brownian particle coupled to a heat bath. The Langevin approach has
been coupled to statistical particle emission models at each time step to predict
properties of particles emitted before scission [65], and has been successfully used
to calculate fission-fragment mass, charge, and angular distributions, as well as
fission probabilities and cross sections [66–68]. A special case of the Langevin
method in the strong-damping limit, where inertia can be ignored, has also been
successfully applied to the calculation of fragment mass distributions for a wide
range of parent nuclei [69].
Scission-point models are another type of approach to fission developed starting
in the 1950s [70, 71] to predict fission-fragment properties, and have continued
to evolve through the present day [72, 73]. These models use purely statistical
arguments at scission, without invoking the dynamical evolution of the parent
nucleus up to that point. Scission-point models rely on two critical assumptions: (1)
the fission-fragment properties can be determined entirely from an analysis of the
system at scission, and (2) a thermodynamic equilibrium exists at scission between
the fragments. A probability distribution for the configuration at scission can then
be constructed based on the energy of the system in this configuration. Using this
probability distribution, average properties of the fragments can be obtained. For
example, in recent work by Lemaître et al. [73], charge yields were calculated for a
wide range of actinides using this approach.
Microscopic descriptions of the fission process start from protons, neutrons,
and an effective (i.e., in-medium) interaction between them [74, 75]. The HartreeFock [76, 77] and Hartree-Fock-Bogoliubov (HFB) approximation can be used
W. Younes
particle states. From there, the “shell energy” can be calculated as the sum of
single-particle energies over the occupied states. The difference between this shell
energy and one calculated using a continuous smoothed density of states gives the
shell correction energy [54, 61]. A similar approach gives the pairing correction as
the difference in pairing correlation energy (the difference in ground-state energies
of the nucleus with and without pairing) using the actual single-particle states and a
smooth density of states [54, 61]. There are several methods to generate the potential
function needed for the shell and pairing corrections, but the method of folded
potentials [53, 54] is especially well adapted to the wide range of shapes encountered
in fission.
The macroscopic-microscopic model gives the energy of the nucleus as a
function of its shape, and this can be used to generate a potential energy surface
(PES) as a function of shape parameters. The PES can then be used to locate fission
barriers [62] and extract their properties, which in turn can provide estimates of
fission rates [56, 57]. More importantly, the PES is a fundamental ingredient in
dynamical (i.e., time-dependent) calculations of the fission process. One way to
model fission dynamics is by performing a random walk across the PES. Ward et al.
[63] adopted this approach, using a Metropolis algorithm with transition rates across
the PES determined by level densities obtained with a combinatorial technique.
Another approach is to solve the classical Langevin equation for the system
[64]. The Langevin approach essentially models the evolution of the system as
that of a Brownian particle coupled to a heat bath. The Langevin approach has
been coupled to statistical particle emission models at each time step to predict
properties of particles emitted before scission [65], and has been successfully used
to calculate fission-fragment mass, charge, and angular distributions, as well as
fission probabilities and cross sections [66–68]. A special case of the Langevin
method in the strong-damping limit, where inertia can be ignored, has also been
successfully applied to the calculation of fragment mass distributions for a wide
range of parent nuclei [69].
Scission-point models are another type of approach to fission developed starting
in the 1950s [70, 71] to predict fission-fragment properties, and have continued
to evolve through the present day [72, 73]. These models use purely statistical
arguments at scission, without invoking the dynamical evolution of the parent
nucleus up to that point. Scission-point models rely on two critical assumptions: (1)
the fission-fragment properties can be determined entirely from an analysis of the
system at scission, and (2) a thermodynamic equilibrium exists at scission between
the fragments. A probability distribution for the configuration at scission can then
be constructed based on the energy of the system in this configuration. Using this
probability distribution, average properties of the fragments can be obtained. For
example, in recent work by Lemaître et al. [73], charge yields were calculated for a
wide range of actinides using this approach.
Microscopic descriptions of the fission process start from protons, neutrons,
and an effective (i.e., in-medium) interaction between them [74, 75]. The HartreeFock [76, 77] and Hartree-Fock-Bogoliubov (HFB) approximation can be used