A Grand Tour of Nuclear Fission Physics
267
to construct self-consistent configurations of the parent nucleus that are relevant
to fission, under the assumption of a solution consisting of independent quasiparticles, by constraining collective parameters of the nucleus (e.g., quadrupole and
octupole moments) [61, 78]. This type of approach falls under the larger category
of density functional theory (DFT) [79]. There are various ways to extract fission
observables using the microscopic method. For example, the set of HFB energies
as a function of collective parameters can be used within a scission-point model
[73], or to construct a PES as the basic framework for the methods discussed
above. Alternately, the time evolution of the system to scission can be treated
within the same microscopic framework. The time-dependent Hartree-Fock (TDHF)
method is a powerful approach that can describe the evolution of the nucleus to
scission [22, 80]. The TDHF approach is limited to the assumption of a single Slater
determinant solution at all times, however, it does not require collective-parameter
constraints and elementary single-particle degrees of freedom are automatically
included [61]. The TDHF approach was applied to the fission problem early on [22],
and later extend to include pairing [80, 81]. Recently, Bulgac et al. have developed
the time-dependent superfluid local density approximation (TDSLDA) [24, 82, 83],
a fully microscopic extension of DFT that includes pairing. In the TDSLDA, all
degrees of freedom are treated on an equal footing and all symmetries are correctly
implemented. This approach was recently applied to the study of fission dynamics
in 240 Pu [24].
Other approaches have attempted to go beyond the inherent restriction of the
TDHF method to a single determinant through configuration mixing [84]. The
mixing can be between microscopic states labeled by continuous parameters, as in
the generator coordinate method (GCM) [85, 86], or between microscopic states
in a discrete basis (e.g., configuration interaction in the shell model [87]). In
the GCM, the wave function of the nucleus is written as a linear superposition
of HFB solutions labeled by the relevant collective parameters (see, e.g., section
9.5 in [78]). Because the collective parameters can take on a continuous set of
values, the linear superposition takes the form of an integral over these parameters
and the unknown weights of the linear expansion are determined by a variational
principle [61, 78]. The application of the variational principle with the GCM wave
function leads to the so-called Hill-Wheeler equation [85], which is an integrodifferential non-local non-linear equation that is very difficult to solve for more
than one collective parameter [88]. A more tractable approach is to expand the
Hill-Wheeler equation to second order in the non-locality (i.e., the difference in
value between the collective parameters). This approximation is often accompanied
by the assumption of a Gaussian shape for the overlap between GCM states,
known as the Gaussian overlap approximation (GOA). With these simplifications,
the Hill-Wheeler equation reduces to a Schrödinger-like equation in the collective
coordinates, where the collective inertia tensor and potential are constructed from
the underlying single-particle degrees of freedom. The GCM is readily extended to
include a time dependence and, as in the static case, can be approximated by a timedependent collective Schrödinger equation. This time-dependent equation can then
267
to construct self-consistent configurations of the parent nucleus that are relevant
to fission, under the assumption of a solution consisting of independent quasiparticles, by constraining collective parameters of the nucleus (e.g., quadrupole and
octupole moments) [61, 78]. This type of approach falls under the larger category
of density functional theory (DFT) [79]. There are various ways to extract fission
observables using the microscopic method. For example, the set of HFB energies
as a function of collective parameters can be used within a scission-point model
[73], or to construct a PES as the basic framework for the methods discussed
above. Alternately, the time evolution of the system to scission can be treated
within the same microscopic framework. The time-dependent Hartree-Fock (TDHF)
method is a powerful approach that can describe the evolution of the nucleus to
scission [22, 80]. The TDHF approach is limited to the assumption of a single Slater
determinant solution at all times, however, it does not require collective-parameter
constraints and elementary single-particle degrees of freedom are automatically
included [61]. The TDHF approach was applied to the fission problem early on [22],
and later extend to include pairing [80, 81]. Recently, Bulgac et al. have developed
the time-dependent superfluid local density approximation (TDSLDA) [24, 82, 83],
a fully microscopic extension of DFT that includes pairing. In the TDSLDA, all
degrees of freedom are treated on an equal footing and all symmetries are correctly
implemented. This approach was recently applied to the study of fission dynamics
in 240 Pu [24].
Other approaches have attempted to go beyond the inherent restriction of the
TDHF method to a single determinant through configuration mixing [84]. The
mixing can be between microscopic states labeled by continuous parameters, as in
the generator coordinate method (GCM) [85, 86], or between microscopic states
in a discrete basis (e.g., configuration interaction in the shell model [87]). In
the GCM, the wave function of the nucleus is written as a linear superposition
of HFB solutions labeled by the relevant collective parameters (see, e.g., section
9.5 in [78]). Because the collective parameters can take on a continuous set of
values, the linear superposition takes the form of an integral over these parameters
and the unknown weights of the linear expansion are determined by a variational
principle [61, 78]. The application of the variational principle with the GCM wave
function leads to the so-called Hill-Wheeler equation [85], which is an integrodifferential non-local non-linear equation that is very difficult to solve for more
than one collective parameter [88]. A more tractable approach is to expand the
Hill-Wheeler equation to second order in the non-locality (i.e., the difference in
value between the collective parameters). This approximation is often accompanied
by the assumption of a Gaussian shape for the overlap between GCM states,
known as the Gaussian overlap approximation (GOA). With these simplifications,
the Hill-Wheeler equation reduces to a Schrödinger-like equation in the collective
coordinates, where the collective inertia tensor and potential are constructed from
the underlying single-particle degrees of freedom. The GCM is readily extended to
include a time dependence and, as in the static case, can be approximated by a timedependent collective Schrödinger equation. This time-dependent equation can then