# Theory of Nuclear Fission

Nicolas Schunck  Nuclear and Chemical Sciences Division, Lawrence Livermore
National Laboratory, Livermore, California 94551, USA<EMAIL_ADDRESS>David
Regnier  CEA, DAM, DIF, 91297 Arpajon, France  Université Paris-Saclay, CEA,
LMCE, 91680 Bruyères-le-Châtel, France<EMAIL_ADDRESS>
###### Abstract

Atomic nuclei are quantum many-body systems of protons and neutrons held
together by strong nuclear forces. Under the proper conditions, nuclei can
break into two (sometimes three) fragments which will subsequently decay by
emitting particles. This phenomenon is called nuclear fission. Since different
fission events may produce different fragmentations, the end-products of all
fissions that occurred in a small chemical sample of matter comprise hundreds
of different isotopes, including $\alpha$ particles, together with a large
number of emitted neutrons, photons, electrons and antineutrinos. The
extraordinary complexity of this process, which happens at length scales of
the order of a femtometer, mostly takes less than a femtosecond but is not
completely over until all the lingering $\beta$ decays have completed – which
can take years – is a fascinating window into the physics of atomic nuclei.
While fission may be more naturally known in the context of its technological
applications, it also plays a pivotal role in the synthesis of heavy elements
in astrophysical environments. In both cases, experimental measurements are
not sufficient to provide complete data. Simulations are needed, yet at levels
of accuracy and precision that pose formidable challenges to nuclear theory.
The goal of this article is to provide a comprehensive overview of the
theoretical methods employed in the description of nuclear fission.

###### keywords:

Fission, Fission fragment yields , Cross sections , Prompt fission spectrum ,
Delayed spectrum , Large-amplitude collective motion , Energy density
functional theory , Time-dependent density functional theory

###### MSC:

[2010] 00-01, 99-00

††journal: Journal of LaTeX Templates

###### Contents

  1. 1 Introduction
  2. 2 Nuclear Deformation
    1. 2.1 Phenomenological Nuclear Mean Field
    2. 2.2 Energy Density Functional Theory
      1. 2.2.1 Single-Reference Energy Density Functional Theory: Self-Consistent Mean Field
      2. 2.2.2 Multi-Reference Energy Density Functional Theory: Configuration Mixing
    3. 2.3 Scission Configurations and Fission Fragments
  3. 3 Probabilities, Rates and Cross Sections
    1. 3.1 Spontaneous Fission
    2. 3.2 Neutron-Induced Fission
      1. 3.2.1 Fission Rates at High Energy
      2. 3.2.2 Fission Cross Sections for Fast Neutrons
      3. 3.2.3 Resolved Resonance Region: the R-matrix Formalism
  4. 4 Formation of Primary Fission Fragments
    1. 4.1 Statistical Models for Fission Fragment Distributions
    2. 4.2 Classical Dynamics
    3. 4.3 Time-Dependent Density Functional Theory
    4. 4.4 Time-Dependent Generator Coordinate Method
    5. 4.5 Comments on Adiabaticity and Dissipation
    6. 4.6 Ternary Fission
  5. 5 Emission of Light Particles
    1. 5.1 Distribution of Primary Fragments Properties
    2. 5.2 Prompt Particle Emission
    3. 5.3 Pre-fission Particle Emission
    4. 5.4 Applications of Fragments Deexcitations Models
      1. 5.4.1 Benchmarks of Prompt Observables
      2. 5.4.2 Probes into the Scission Mechanism
      3. 5.4.3 Evaluated Data for Nuclear Technology
      4. 5.4.4 Fission Event Generators in Neutron Transport Codes
      5. 5.4.5 Experimental Data Analysis
    5. 5.5 Delayed Emission
  6. 6 Toward a Consistent Description of Fission
    1. 6.1 Number of Particles
    2. 6.2 Deformations of the Fragments
    3. 6.3 Spin Distributions
    4. 6.4 Excitation Energy
  7. 7 Conclusion

##  1 Introduction

The first experimental evidence of nuclear fission was obtained in late 1938
by German scientists Hahn and Strassman [1, 2], who observed that “isotopes of
Barium ($Z=56$) are formed as a consequence of the bombardment of Uranium
($Z=92$) with neutrons” [excerpt from [3]]. The first theoretical
interpretation of this phenomenon was given shortly thereafter by Lise Meitner
and was based on the liquid drop model of the atomic nucleus [3]. Only a few
months later, N. Bohr and J.A. Wheeler published the first comprehensive
theoretical study of the process [4]. These initial discoveries pertained to
what we now call neutron-induced fission; G.N. Flerov and K.A. Petrzhak
separately discovered the spontaneous fission of Uranium isotopes in 1941 [5,
6].

In simple terms, nuclear fission is the process by which a heavy atomic
nucleus divides into two (binary) or three (ternary) fragments. Although there
is no formal definition for it, the usual consensus is that any object bigger
than an $\alpha$ particle $(Z=2,N=2$) formed by the breaking up of a heavy
nucleus is considered a fission fragment; anything smaller would be considered
as particle emission or radioactivity. For a given sample of fissioning
material, not all nuclei fission identically: the relative proportion of each
isotope is called the primary fission fragments distribution. Fission
fragments are always in an excited state. Therefore, as soon as they are
formed, they undergo a sequence of decays. Initially, this decay proceeds
mainly through neutron emission. When the energy of the fragment becomes
smaller than the separation energy $S_{n}$ of a single neutron, photon
emission becomes the dominant mode of deexcitation. This initial phase of the
decay process is very rapid, of the order of $\tau\lesssim 10^{-13}$s and
defines what is called the prompt particle emission. Since neutron emission
changes the isotopic composition of the sample, it is customary to refer to
the distribution of fission fragments after prompt emission as the independent
fission yields. After neutron and photon emission, many fragments may be
unstable against $\beta$ decay – the conversion of a neutron into a proton
with the emission of an electron and antineutrino. Beta decay takes place over
time scales ranging from a few picoseconds up to years. Since the daughter
nuclei produced in the decay may also be excited, $\beta$ decay is often
followed by photon and sometimes neutron emission. This phase of the decay of
the fission fragments is called delayed emission. At the end of all emissions,
both the isotopic and elemental composition of the sample have changed. The
relative proportion of each isotope is encoded in what is called the
cumulative fission product yields [7]. Figure 1 gives a qualitative
representation of the entire process.

Figure 1: Schematic representation of the fission process. A heavy atomic
nucleus, possibly formed in a nuclear reaction, deforms until it divides into
two (sometimes three) fragments, thereby releasing a large amount of energy of
the order of about 200 MeV for actinide nuclei. The excited fission fragments
rapidly emit neutrons followed by photons (prompt fission spectrum). About a
few picoseconds after the scission of the original nucleus, many of the
resulting fission fragments undergo a sequence of $\beta$ decays in
competition with additional photon and or neutron emission (delayed emission).

From a theoretical perspective, nuclear fission is a striking example of
large-amplitude collective motion for a quantum many-body system. In other
words, it results, or emerges, from the quantum many-body effects induced by
in-medium nuclear forces. The characteristic features of atomic nuclei, e.g.,
the competition between individual and collective behavior, their self-bound
nature and the fact that they are open quantum systems, all play a role in our
understanding of the various phases of the fission process. Finally, fission
is intrinsically a time-dependent process that eventually populates a huge
number of output channels. This short enumeration should already suggest to
the informed reader that the theoretical description of nuclear fission poses
enormous challenges. In addition, fission plays a major role in understanding
the limits of nuclear stability, especially concerning superheavy elements [8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Partly for this reason, fission is
one of the main mechanisms that terminates the cycle of nuclear reactions
responsible for the formation of elements in astrophysical environments,
especially in the rapid-neutron capture process (r process) [20, 21, 22, 23].
Last but not least, the fission mechanism is at the heart of nuclear
technology. Apart from the standard example of nuclear energy [24, 25],
details of the fission process are also important for nuclear forensics [26]
and international non-proliferation efforts and safeguards by the
international community. Knowledge about the physics of the fission process is
summarized in evaluated nuclear data libraries such as JEFF [27] and ENDF
[28], which list the cross sections of a large range of nuclear reactions
along with the fission yields, or ENSDF [29], which is more oriented toward
nuclear structure data.

Whether one is interested in the fundamental description of fission or its
applications in other scientific or technological disciplines, the recent
years have brought many interesting developments. The rapid and sustained
increase in computational capabilities has rejuvenated fission theory by
enabling simultaneously realistic large-scale simulations of fission
properties [22, 30] and more fundamental descriptions of the process [31, 32].
Many of these most recent results and methods have yet to find their way in
the nuclear data libraries, which – as far as fission is concerned – often
rely on very phenomenological models and concepts from the 1950ies or 1960ies.
At the same time, the quality and robustness of these libraries keeps
improving partly thanks to the introduction of novel techniques from
artificial intelligence and machine learning to evaluate and verify nuclear
data [33, 34, 35].

Surprisingly there are only a few textbooks providing a comprehensive
introduction to nuclear fission, especially fission theory, and they are just
a few years old [36, 37, 38]. In the past five years, three review articles
have been published covering either microscopic methods [31], experimental
results [39] or phenomenology [40]. While these papers have provided a welcome
update of recent results, their focus is mostly on the chain of events leading
to the divide of the nucleus and not so much on the probability that this
process takes place nor on the decay of the fragments, even though nearly all
experimental information on fission is collected, directly or indirectly,
during this decay phase. We also feel that there is a disconnect between the
basic nuclear science community, the goal of which is understand the basic
mechanisms at play in fission, and the specialized user communities were
fission data is employed. For example, the producers and users of global
nuclear data libraries may not always be aware of the most recent developments
in nuclear theory that could impact the process of evaluation; conversely,
nuclear scientists may not always be aware of the possible impact of their
work nor of the constraints (especially in terms of precision) imposed by
applications. The goal of this review is to try to provide an overview of
fission theory that could be beneficial to both types of communities.

Our presentation is organized to follow the chronology of the fission process.
Following Bohr [4, 41, 42], one still thinks of fission as a deformation
process. In Section 2, we will therefore discuss in details how the
qualitative concept of deformation emerges from the nuclear mean field theory,
and how fission fragments are mapped to specific, deformed configurations of
the fissioning nucleus at the point of scission. This section has some overlap
with [31], but it is necessary to introduce some very basic concepts that will
be used throughout this review. Section 3 deals with the probability that
fission occurs. In the case of spontaneous fission, this probability is in
fact the main quantity that must be computed to determine the fission half-
life; in induced fission, it is related to the very challenging problem of
computing the fission cross section, namely the probability that fission takes
place relatively to other possible mode of decay such as neutron scattering,
evaporation, photoemission, etc. The mechanisms by which the nucleus will
change deformation and reach the point of scission are described in Section 4.
In the past two or three decades, this is perhaps the area where the most
progress has been made. The static models of fission en vogue from the 1950ies
to the 1990ies have been replaced by explicit time-dependent description,
either semi-classical or full quantum-mechanical. As mentioned earlier, the
decay of fission fragments has been seldom addressed in a review. It will be
presented in Section 5. This decay leads to the emission of neutrons and
photons and to a series of $\beta$ decays. Some fundamental problems in basic
science such as the role of fission in nucleosynthesis mechanisms [43, 44, 45,
23, 46, 47] or what has been dubbed the antineutrino reactor anomaly [48, 49,
50, 51, 52, 53, 54, 55] cannot be properly understood without invoking fission
fragment deexcitation. Finally, Section 6 will review some of the most recent
attempts at predicting the properties of the primary fission fragments after
the point of scission with microscopic methods. By connecting the physics of
large amplitude collective motion with the statistical decay of the fragments,
these efforts offer the prospect of building a truly consistent theory of
fission in the near future.

##  2 Nuclear Deformation

The concept of nuclear deformation is a phenomenological one. The nuclear
Hamiltonian $\hat{H}$ that determines, in principle, all nuclear properties,
is rotationally invariant [56, 57]. Theories that attempt to solve directly
the many-body Schrödinger equation, such as the no-core shell model [58] or
the nuclear shell model [59] never need consider deformation: the eigenvectors
of $\hat{H}$ are always rotationally-invariant. This observation also applies
to the other types of many-body methods that give approximations to the true
eigenvectors of $\hat{H}$ [60, 61, 62].

However, it was discovered long ago that one could describe, qualitatively and
quantitatively, many important nuclear properties such as, e.g., the value of
nuclear quadrupole moments, the strength of electromagnetic transitions
between excited states or the particular distribution of discrete energy
levels, by invoking collective behaviors associated with a deformed average
nuclear potential in which nucleons move independently [63, 64, 65]. While
originally introduced in a phenomenological manner, the concept of deformation
can be formalized as a spontaneous symmetry breaking of the nuclear mean
field: the average potential is not invariant under rotation [66]. Deformation
is a very pervasive concept in fission theory. Potential energy surfaces,
i.e., the potential energy as a function of nuclear deformation, determine
fission paths and the large-amplitude dynamics of the nucleus. The number of
particles in fission fragments is typically mapped onto the deformation of the
fissioning nucleus at the point of scission. The overlap, in the quantum
mechanical sense, between the nuclear wave functions at different deformations
is directly related to the calculation of fission probabilities. For these
reasons, models to compute deformed nuclei are the cornerstones of any fission
theory.

Historically, such models were based on phenomenological mean-field
approaches, where the average nuclear potential is parametrized directly with
an appropriate mathematical function. Examples of such phenomenological
potentials include the Nilsson, Woods-Saxon, and Folded-Yukawa potentials.
Within this framework, some of the most comprehensive studies of fission were
reported already in the 1970ies [67, 68]. In recent years, this phenomenology
has also been used to provide large-scale, systematic surveys of nuclear
properties [69, 70, 71, 72, 73]. We briefly review this approach in Section
2.1. Nuclear energy density functional theory provides a more fundamental
description of nuclear deformation, since it basically relates the average
potential to an energy density functional or an effective pseudo-potential
that describe effective interactions between nucleons [66]. The theory can be
viewed as a (considerable) extension of the self-consistent Hartree-Fock
theory [74, 75, 76] and is inspired in part by the density functional theory
of electrons in atoms [77]. As mentioned above, nuclear deformation emerges
when the density of particles breaks some of the symmetries of the nuclear
Hamiltonian [78]. While computationally more involved, the advantage of EDF
methods is that they provide (effective) Hamiltonians and an arsenal of many-
body techniques to compute quantum numbers. While there were pioneering
applications of EDF to fission in the 1970ies [79, 80], it is fair to say that
self-consistent calculations of fission properties only became relevant in the
late 1980ies and 1990ies. In the past two decades, these methods have now
become competitive with more phenomenological ones thanks to advances in
computing capabilities [31]. We will summarize the most relevant notions of
the EDF approach in Section 2.2. Finally, we will discuss in Section 2.3 how
to infer from potential energy surfaces the points at which the nucleus breaks
into two fragments.

###  2.1 Phenomenological Nuclear Mean Field

What is known as the macroscopic-microscopic approach to nuclear structure is
at the heart of nuclear phenomenology. It relies on modeling the atomic
nucleus as a quantum, charged liquid drop with additional corrective terms
that account for quantum many-body effects. In the language of modern many-
body theory, one would call such an approach a 0-body theory with corrective
terms of 1- and 2-body origin. In this picture, fission is the extreme
distortion of the droplet that leads to its breakup. In more details, the
total energy of the atomic nucleus is written as the sum of several
contributions: (i) the macroscopic energy $E_{\rm mac}(\boldsymbol{q})$,
typically represented by the liquid drop or droplet model, (ii) a shell
correction $\delta E_{\rm shell}(\boldsymbol{q})$ originating from the single-
particle degrees of freedom and (iii) a pairing correction $\delta E_{\rm
pair}(\boldsymbol{q})$ to account for nuclear superfluidity. Each of these
three terms parametrically depends on the underlying deformation
$\boldsymbol{q}=(q_{1},\cdots,q_{N})$ of the nuclear shape,

$E(\boldsymbol{q})=E_{\rm mac}(\boldsymbol{q})+\delta E_{\rm
shell}(\boldsymbol{q})+\delta E_{\rm pair}(\boldsymbol{q})$ (1)

The leading term in this expression is the macroscopic energy which is
computed as the energy of a charge drop of nuclear matter in the liquid drop
model (LDM). This finite piece of nuclear matter can also be expressed as a
series of powers of $Z$ and $A$ – the lepdotermous expansion [81, 82] – and be
related to predictions from more microscopic theories [83, 84, 85]. Initial
parameterizations of the liquid drop were relatively simple [86]; more recent
models have included additional terms to mock up various effects (charge
asymmetry, odd-even effect of pairing, proton form factors, etc.) that vary
smoothly with $Z$ and $N$ [69, 70, 73]. Historically, the LDM provided the
theoretical basis for the first explanation of fission by Meitner, Bohr and
Wheeler [3, 4]. The stability of a liquid drop against fission is captured in
the dimensionless fissility parameter $x$, which is the ratio of the Coulomb
to the surface energy of the drop [4, 41]. Empirically, the fissility
parameter can be approximated as

$x\approx\frac{Z^{2}}{47A(1-\eta I^{2})},$ (2)

with $I=(N-Z)/A$ and $\eta=-a_{\rm ssym}/a_{\rm surf}$ is the ratio of the
surface-symmetry to surface coefficients of the leptodermous expansion of the
nuclear energy. These two terms are proportional to $I^{2}A^{2/3}$ and
$A^{2/3}$, respectively. They can be either fitted to reproduce average
properties of nuclei [81], or computed from a model of nuclear matter.

Even if the macroscopic energy represents nearly 99% of the total binding of a
heavy atomic nucleus such as an actinide, corrective terms must be added to
obtain a more precise estimate of the total energy. The shell correction
effectively simulates the fact that neutrons and protons are quantum-
mechanical particles occupying single-particle levels, the distribution of
which impacts the stability of the nucleus, hence its binding energy [87, 88].
Generically, it reads

$\delta
E=\sum_{k\in\mathcal{S}}e_{k}-\left\langle\sum_{k\in\mathcal{S}}e_{k}\right\rangle$
(3)

where $\mathcal{E}=\\{e_{k}\\}_{k=1,\dots,p}$ refers to a full set of single-
particle (s.p.) energies and $\mathcal{S}$ is a subset of $\mathcal{E}$. To
extract the energy for the lowest configuration at deformation
$\boldsymbol{q}$, $\mathcal{S}$ would simply contain the $Z$ and $N$ lowest
energies; any other combination of s.p. levels would yield the energy of an
excited state. The symbol $\langle\cdots\rangle$ refers to the Strutinsky
averaging procedure to produce the “smooth” energy. Full details about how to
compute this term are given in [68]; variants of the standard Strutinsky
procedure to remove the spurious effect of continuum or pairing effects are
discussed in [89, 90, 91, 92]. The introduction of the shell correction was
key to explain two major features of the fission process: (i) the energy
threshold at which fission takes place is nearly constant for actinides,
contrary to LDM predictions; (ii) the LDM cannot account for the existence of
fission isomers. These meta-stable states are more deformed than the ground
state thanks to the stabilizing effect of the shell correction.

The shell correction can be interpreted as having a one-body origin: it is
associated with an independent-particle model of the nucleus, where each
nucleon moves, independently of one another, in the average mean field. In the
language of second quantization, the resulting Hamiltonian operator only
contains one-body operators. To describe the different energy spectra between
even-even and odd nuclei, pairing correlations must be taken into account
explicitly; see [93] for a comprehensive introduction to nuclear pairing.
Historically, the effect of pairing on the total energy was quantified by
introducing a pairing correction $\delta E_{\rm pair}$ akin to the shell
correction. In most applications of the macroscopic-microscopic model, this
pairing correction is computed from the BCS solution and can thus be
interpreted as a corrective term of two-body origin. All three terms of (1)
can be modified to describe nuclei at finite angular momentum or finite
temperature [94, 95, 96, 97, 98, 99, 100, 101]; see also the textbook [102].

Figure 2: Left: Funny Hills parameterization [67] of the nuclear shape for
$(h,\alpha)=(0.2,0.3)$ with $c$ varying from 1.0 to 2.2 by steps of 0.3.
Right: Axial shapes with the Cassini oval parameterization [103] for
$u=0.0,0.4,0.8,1.0,1.2$. The value $u=0$ corresponds to the sphere, the value
$u=1.0$ to the point of scission and the value $u=1.2$ to separated fragments.
Figures reproduced with permission from [31] courtesy of Schunck; copyright
2016 by Institute of Physics.

All terms in the decomposition (1) of the total energy depend on the
deformation of the nuclear shape which, in this approach, is an input to the
calculation. The determination of a suitable parameterization of the nuclear
shape is in fact a challenging problem. Given a parameterization $\mathcal{P}$
and a set of shape parameters $\boldsymbol{q}=(q_{1},\cdots,q_{N})$, the
following criteria should be met: (i) there should be a one-to-one
correspondence between any vector $\boldsymbol{q}$ and the actual shape; (ii)
since the shape parameterization is an input to any calculation, the number
$N$ of components of $\boldsymbol{q}$, i.e., the number of shape parameters,
should be as small as possible (iii) the parameterization must be capable of
describing weakly-deformed, compact shapes, extremely elongated shapes and
even disjoint shapes representing the separated fragments. The standard
multipole expansion of the nuclear radius [104], for example, does not meet
any of these criteria: it is not bijective [105, 106], the number of
parameters needed to describe well-deformed shapes increases with deformation,
and it cannot characterize separated fragments. Parametrizations more adapted
to describing fission include the Funny-Hills parameterization [67], the Los
Alamos parameterization [42], Cassini ovals [103] or Legendre polynomials
[107]; see also [31] for an explicit summary of some of these
parameterizations. Figure 2 shows an example of the Funny Hills and Cassini
ovals parameterization.

Figure 3: One-dimensional representation of fission barriers in actinide
nuclei. Note that the scission point is not represented at scale: in practice,
it is “far” from the outer barrier. The distance in the deformation space
between the location of the scission point and that of the fission isomer may
be four to five times larger than the distance between the fission isomer and
the ground state.

This nuclear phenomenology based on the macroscopic-microscopic model has been
extremely successful in predicting many facets of nuclear properties [69, 70,
71, 73, 100, 108]. In the context of fission, this approach provides a
computationally efficient, realistic method to perform large-scale
calculations of deformation properties, namely potential energy surfaces
(PES). In many actinides, there is empirical evidence that the PES require
several deformation degrees of freedom to adequately capture the full dynamics
of fission: elongation of the whole nucleus, reflection asymmetry of the
shape, triaxiality parameters, but also deformation parameters characterizing
the fragments themselves [109]. Recent investigations proved that fragments
can also be reflection asymmetric, which would require at least another two
parameters to describe the shape asymmetry for each fragment [110, 111]. Such
multi-dimensional PES cannot be visualized, but a lot of important information
can be extracted by projecting them onto a one-dimensional subspace of lowest
energy: the fission path. In actinide nuclei, this fission path is typically
characterized by a deformed potential minimum (i.e. the classical ground
state) and an additional, meta-stable minimum at larger deformations: the
fission isomer. Both minima are separated by potential energy barriers: the
inner barrier, between the ground state and the fission isomer, and the outer
barrier, between the fission isomer and regions corresponding to ever larger
deformations. Precise calculations of these fission barriers are especially
important for estimating spontaneous fission half-lives and fission cross
sections, as will be presented in Section 3. Figure 3 shows a typical,
schematic representation of the one-dimensional fission path in actinide
nuclei. In such one-dimensional representations, the saddle point is simply
the point on the fission path that has the highest energy (here: the top of
the inner barrier).

Historically, the predictions of fission isomeric states and “double-humped”
fission barriers was a major success of the macroscopic-microscopic approach
[67, 68, 112, 113], as was the explanation of asymmetric fission by
reflection-asymmetric shapes stabilized by the shell correction [103].
Refinements in the models combined with modern computational capabilities have
led to global predictions of static fission barriers [114, 115, 116, 117, 118,
119, 120]. Such calculations are necessary, since as we go away from the
valley of stability, the topology of the potential energy surfaces may change
quite dramatically and the traditional features of fission barriers in
actinides may not apply anymore. Figure 4 shows a systematics of fission
barrier heights (more precisely, the highest fission barrier) in heavy and
superheavy nuclei taken from [22]. It is clear from these results that
uncertainties from nuclear models alone can reach a few MeV, which corresponds
to dozens of orders of magnitude in spontaneous fission half-lives; see
Section 3.1.

Figure 4: Highest fission barrier heights for heavy and superheavy elements
with $84\geq Z\geq 120$ and $118\geq N\geq 250$ for three different models of
nuclear deformation: FRLDM corresponds to the macroscopic-microscopic
calculation of [115] while the BPCM [121] and HFB14 [122] are two different
version of energy density functionals; see discussion in Section 2.2. Figures
reproduced with permission from [22] courtesy of Giuliani; copyright 2018 by
The American Physical Society.

###  2.2 Energy Density Functional Theory

The EDF approach is the modern name for what was known for a long time as the
self-consistent mean field theory. Its root can be found both in the
generalized Hartree-Fock theory, which was formalized in a few seminal papers
in the early 1960ies [123, 124], as well as in the density functional theory
for electrons [77, 125]. In a nutshell, the basic idea of the EDF approach is
to compute the nuclear mean field from a suitable model of nuclear forces
among nucleons rather than parameterizing it as in the macroscopic-microscopic
approach. In practice, this calculation is often based on a model for in-
medium, two-body effective nuclear interactions (or pseudo-potentials) such as
the Skyrme [126] or Gogny [127] force. When computing the expectation value of
such pseudo-potentials on a many-body product state of single-particle wave
functions (Slater determinant), both the resulting energy and the nuclear mean
field become functionals of the density of particles [128]. Direct
parameterizations of the energy density have also been proposed, with varying
success [129, 130, 131, 121, 132]. Note that the effective potentials need not
be restricted to two-body terms only: three- and higher-order many-body
interactions could be included as well [133, 134]. The EDF approach comprises
a full hierarchy of approximations and methods yielding increasing accuracy
and precision in the determination of nuclear properties. One may consider two
main categories: single-reference energy density functional (SR-EDF) and
multi-reference density functional (MR-EDF). They are most of the time used to
compute static properties of the nucleus but could also be implemented in a
time-dependent flavor. The EDF approach has a decade-old rich history and it
would go far beyond the scope of this review to describe it in too many
details. We refer the interested reader to [66, 128, 135, 136, 137, 138, 139].
Below, we will only summarize the essential concepts and formulas most
relevant to fission theory.

####  2.2.1 Single-Reference Energy Density Functional Theory: Self-Consistent
Mean Field

Today, the single-reference EDF approach is largely built on the Hartree-Fock-
Bogoliubov (HFB) theory, where the nuclear many-body wave function is
approximated by a generalized product of quasiparticle wave functions [128,
57, 66]. In the formalism of second quantization, these quasiparticles (q.p.)
are associated with annihilation and creation operators
$(\beta_{\mu},\beta_{\mu}^{\dagger})$ and the many-body wave function reads

$\ket{\Phi}=\prod_{\mu}\beta_{\mu}\ket{0},$ (4)

where $\ket{0}$ is the particle vacuum. The q.p. operators are themselves
related to the annihilation and creation operators $(c_{k},c_{k}^{\dagger})$
of an arbitrary single-particle basis by the Bogoliubov transformation [124]

$\displaystyle\beta_{\mu}$ $\displaystyle=\sum_{m}\left[U^{\dagger}_{\mu
m}\,c_{m}+V^{\dagger}_{\mu m}\,c_{m}^{\dagger}\right],\vskip 6.0pt plus 2.0pt
minus 2.0pt$ (5a) $\displaystyle\beta_{\mu}^{\dagger}$
$\displaystyle=\sum_{m}\left[V^{T}_{\mu m}\,c_{m}+U^{T}_{\mu
m}\,c_{m}^{\dagger}\right].$ (5b)

Let us consider an effective two-body pseudo-potential $\hat{V}$ characterized
by the matrix elements $v_{ijkl}\equiv\braket{ij}{\hat{V}}{kl}$ in the single-
particle basis, with $\bar{v}_{ijkl}$ the anti-symmetrized version of these
matrix elements ($\bar{v}_{ijkl}=v_{ijkl}-v_{ijlk}$). The HFB energy reads

$E[\rho,\kappa,\kappa^{*}]=\sum_{ij}t_{ij}\rho_{ji}+\frac{1}{2}\sum_{ijkl}\bar{v}_{ijkl}\rho_{lj}\rho_{ki}+\frac{1}{4}\sum_{ijkl}\bar{v}_{ijkl}\kappa^{*}_{ij}\kappa_{kl}.$
(6)

where $t_{ij}$ are the matrix elements of the kinetic energy operator, $\rho$
is the one-body density matrix and $\kappa$ the pairing tensor. These two
quantities encode the basic degrees of freedom of the HFB theory. They can be
expressed in terms of the matrices of the Bogoliubov transformation as
$\rho=V^{*}V^{T}$ and $\kappa=V^{*}U^{T}$. The main idea of the HFB theory is
to minimize the energy (6) with respect to the matrix elements of $U$ and $V$.
This results in the HFB equation

$\big{[}\mathcal{H},\mathcal{R}\big{]}=0,$ (7)

where $\mathcal{H}$ is the HFB matrix and $\mathcal{R}$ the generalized
density, respectively given by

$\mathcal{H}=\left(\begin{array}[]{cc}h&\Delta\\\
-\Delta^{*}&-h^{*}\end{array}\right),\qquad\mathcal{R}=\left(\begin{array}[]{cc}\rho&\kappa\\\
-\kappa^{*}&1-\rho^{*}\end{array}\right).$ (8)

In these expressions, $h=t+\Gamma$ is the mean field, $t$ is the single-
particle kinetic energy potential and
$\Gamma\equiv\Gamma_{ik}=\sum_{jl}\bar{v}_{ijkl}\rho_{lj}$ the mean field
potential, and
$\Delta\equiv\Delta_{ij}=\tfrac{1}{2}\sum_{kl}\bar{v}_{ijkl}\kappa_{kl}$ is
the pairing field. Both objects are functionals of $\rho$ and $\kappa$.

Applications of the formalism rely on effective two-body potentials $\hat{V}$
such as the Skyrme or Gogny interactions; see [128, 140, 141] for recent
reviews. There is also a relativistic version of the EDF approach based on
solving a self-consistent Dirac equation [142, 143, 66]. Both non-relativistic
and relativistic EDFs depend on a small set of adjustable parameters that must
be calibrated on data. The most commonly used parameterizations in fission
studies are the SkM* [144] and UNEDF1 [145] Skyrme EDFs, the D1S
parameterization of the Gogny force [146], and the DD-PC1 [147] and DD-ME2
[148] parameterizations of relativistic Lagrangians. Pairing correlations are
central to the HFB theory. As can be seen from (6), the same effective pseudo-
potential $\hat{V}$ should be at the origin of all components of the energy:
the particle-hole part proportional to $\rho$ and the particle-particle one
proportional to $\kappa$. This requirement is enforced only in the case of the
Gogny force. In practice, users of the Skyrme EDF adopt a different effective
potential in the pairing channel based, e.g., on simple density-dependent
forces [149], separable finite-range interactions [150], or even non-empirical
pairing forces built out of chiral effective field theory [151, 152, 153].
Such choices are largely motivated by pragmatic reasons such as the
divergences induced by zero-range effective potentials, which require a
regularization procedure [154, 155] or/and the difficulty in probing the
pairing channel when fitting the parameters of the EDF [66].

The HFB approach works best when the symmetries of the nuclear Hamiltonian are
broken. For example, the minimization of the HFB energy may lead to solutions
where both densities $\rho$ and $\kappa$ are not rotationally-invariant. These
symmetries are not restricted to spatial ones such as translational and
rotational invariance, axial symmetry or reflection symmetry: they also
include time-reversal or isospin symmetry [156, 157, 158]. In fact the ansatz
(4) by itself breaks a fundamental symmetry of the nuclear Hamiltonian:
particle number. Therefore, HFB solutions, strictly speaking, correspond to
wave packets of Slater determinants with different particle numbers [57].

The SR-EDF framework based on deformed HFB solutions is thus the direct,
“microscopic” equivalent of the macroscopic-microscopic approach and can in
fact be viewed as its theoretical justification [159]. Qualitatively, the main
difference between the two is how the average nuclear potential is determined:
it is approximated by a suitable mathematical function in the macroscopic-
microscopic approach and computed self-consistently from a model of nuclear
forces in the HFB theory. Potential energy surfaces can be constructed by
imposing various constraints on the HFB solutions. For example, the
expectation value of the multipole moment operators
$\hat{Q}_{\lambda\mu}(\boldsymbol{r})=r^{2}Y_{\lambda\mu}(\theta,\varphi)$,
where $Y_{\lambda\mu}(\theta,\varphi)$ are the spherical harmonics [160], is
given by

$\braket{\hat{Q}_{\lambda\mu}}=\int
d^{3}\boldsymbol{r}\,\hat{Q}_{\lambda\mu}(\boldsymbol{r})\rho(\boldsymbol{r}),$
(9)

where $\rho(\boldsymbol{r})$ is the total density of particles. These moments
have a straightforward physics interpretation: $\hat{Q}_{20}$ quantifies the
degree of elongation of the shape, $\hat{Q}_{30}$ the degree of left-right
asymmetry, $\hat{Q}_{40}$ the bulging around the mid-point of the shape, etc..
By solving the HFB equation (7) with constraints on the values
$\braket{\hat{Q}_{\lambda\mu}}$ as given by (9), we can generate a manifold of
HFB solutions representing many different shapes. Figure 5 shows a simple
example of such a potential energy surface in the case where only two
constraints were imposed on $\braket{\hat{Q}_{20}}$ and
$\braket{\hat{Q}_{30}}$.

Figure 5: Potential energy surface of 240Pu in the $(Q_{20},Q_{30})$
collective space. The four volume renderings give a visual representation of
the nuclear shape at various locations on the PES. The calculation was
performed for the SkM* parameterization of the Skyrme functional and
corresponds to Fig. 3 of [161].

####  2.2.2 Multi-Reference Energy Density Functional Theory: Configuration
Mixing

While the SR-EDF approach introduces symmetry-breaking as a means to build
more correlations in what would be otherwise a very simplified nuclear wave
function, the multi-reference EDF ansatz takes these deformed solutions and
combines them [162]. Schematically, this mixing takes the form

$\ket{\Psi}=\int dq\,f(q)\ket{\Phi(q)},$ (10)

where $q$ is a generic label for the HFB solutions $\ket{\Phi(q)}$ given by
(4) and $f(q)$ are some weight functions to be determined [163]. This quantum
configuration mixing can be used, in particular, to restore all the broken
symmetries and therefore recover the quantum numbers that were lost during the
symmetry breaking [164]. In this case, the labels $q$ correspond to the
parameters of the symmetry group, the basis states $\ket{\Phi(q)}$ are
typically given by $\ket{\Phi(q)}\equiv\hat{R}\ket{\Phi}$ where $\hat{R}$ is
the generator of the relevant symmetry group, and $f(q)$ is an appropriate
weight function. In the simplest case of particle-number projection, which is
associated with the $U(1)$ symmetry group, the ansatz (10) thus becomes

$\ket{\Psi}=\int d\varphi\,\left(\frac{1}{2\pi}e^{-i\varphi
N}\right)e^{i\varphi\hat{N}}\ket{\Phi}=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\,e^{i\varphi(\hat{N}-N)}\ket{\Phi}.$
(11)

The most relevant symmetries for fission include parity [165], particle number
[166, 167] and angular momentum [168]. MR-EDF techniques also provide a
powerful way to improve the ansatz for the nuclear wave function by adding yet
another layer of correlations that could not be captured at the SR-EDF level
[169]. One of the most standard techniques of MR-EDF is the generator
coordinate method (GCM) [170, 171]. The GCM is especially well adapted to
describing collective correlations that involve all nucleonic degrees of
freedom and play such a central role in understanding fission. In such a case,
the labels $q$ correspond to different values of some constraint operators
$\hat{Q}$ and the basis states are the constrained HFB solutions. In fact,
GCM-based techniques can be used to extract a purely collective Hamiltonian
called the Bohr Hamiltonian that is somewhat reminiscent of a quantized
version of the deformed liquid drop, in the sense that it depends on global,
collective coordinates that typically represent nuclear deformation [172, 173,
174].

###  2.3 Scission Configurations and Fission Fragments

Qualitatively, the moment where the two fragments just separate and the whole
nucleus breaks apart is called the scission point. This picture results from
the analogy of the fission process with the deformation of a charged liquid
drop that overcomes surface tension [3]. Within the macroscopic-microscopic
model of Section 2.1, such a naive concept of scission can still make sense:
in all the parameterizations of the nuclear shape specifically designed to
describe very elongated shapes, there always exists a set of parameter values
for which the neck between the two prefragments reduces to a single point –
the scission point [42, 103, 175].

One may thus be tempted to simply define scission as the locations on the
potential energy surfaces where the two fragments are fully formed and well
separated. However, when determining the solutions to the nuclear mean field,
whether semi-phenomenologically as in Section 2.1 or self-consistently as in
Section 2.2, the obtained wave functions are for the whole nucleus: if only
because of the Pauli principle, some of the single-particle wave functions
spread over each of the two prefragments. In other words, the system is
(heavily) entangled [176]. In fact, the system remains so even when the two
fragments are far apart from one another. These difficulties are compounded by
the use of a variational principle (in the EDF approach) or by the naive
filling of orbits (in the macroscopic-microscopic approach). For nuclear
shapes corresponding to well-separated fragments – such configurations are
sometimes called the asymptotic conditions – the total energy computed in both
frameworks is, in practice, the one that minimizes the energy of each
fragment. As a result, calculations of fission fragments properties for such
configurations are highly non-physical: against all experimental evidence,
fragments have no excitation energy.

To mitigate these problems, one is thus forced to resort to phenomenological
criteria to define scission before the nuclear shape truly corresponds to two
fragments. The simplest of these criteria involve setting a critical value for
the density of particles in the neck region between the prefragments [177] or
for the number of particles in that same region [161]. Random neck rupture
models provide an additional layer of refinement to such approaches [178, 179,
180]. Since fission in general, and scission in particular, is the result of a
competition between attractive nuclear forces and repulsive Coulomb forces,
one may define scission as the point where the nuclear interaction energy
between the two prefragments is smaller than the Coulomb repulsion between
them [181]. However, as mentioned above, the two prefragments are entangled
and the very notion of a nuclear and Coulomb energy for a prefragment is
ambiguous at best. In fact, it was shown that such quantities are not
invariant under unitary transformations of the whole fissioning nucleus [176].
It is possible to design specific transformations that maximize the
localizations of quasiparticles in each prefragment and minimize their
interaction energy and, as a consequence, change the definition of scission
[161]. More recently, there have been a few attempts to better understand the
physics of scission by borrowing from configuration-interaction and reaction
theory techniques, the goal being to establish a continuous pathway between
pre- and post-scission configurations [182, 183, 184].

From these remarks, it should be clear that there is no unique definition of
scission and that predictions of fission fragment properties, including their
initial conditions in terms of numbers of particle, excitation energy, spin
and parity, are highly model-dependent. This has a significant impact on the
modeling of the deexcitation of the fission fragments, which is largely
determined by these initial conditions as will be discussed in Section 5.
Furthermore, the class of fission models called the scission-point models are
based on using the hypothesis of statistical equilibrium at scission to
determine fission fragment properties [185, 186]. The predictions of such
models are contingent on what definitions of scission they adopt.

##  3 Probabilities, Rates and Cross Sections

The previous section summarized the various theoretical methods used to
compute the properties of deformed atomic nuclei – including the extreme
deformations characteristic of the point of scission. This prologue was
necessary to introduce the basic concepts of fission theory: deformation (or
symmetry breaking), collective space, potential energy surfaces, fission
barriers, scission point, etc. The next three sections will build upon these
notions to address the three most fundamental problems in fission theory:
computing the probability that fission takes place – possibly in competition
with other decay modes, characterizing the evolution of the nucleus from a
near ground-state configuration to the point of scission and predicting the
outcome of the overall process, namely all the particles emitted and the
fission products. This section is focused on fission probabilities.

In nuclear physics jargon, fission probabilities are related to what is called
the entrance channel, which we can define here as the mechanism that produces
the proper conditions for an atomic nucleus $(Z,N)$ to undergo fission. Many
of the elements heavier than Lead have in fact a significant probability of
undergoing fission simply as a consequence of the large number of protons they
contain. The spontaneous fission half-life $\tau_{\rm sf}$ is the key quantity
that theorists must compute and compare with experimental measurements. This
is in fact very challenging since the experimental half-lives cover about 30
orders of magnitude, from about $\tau_{\rm sf}\approx 4.1\mu$s for 250No to
$\tau_{\rm sf}>$ 14 billions years for 232Th [36].

The case of induced fission is even more complex. By definition, the
fissioning nucleus is formed in a nuclear reaction involving a projectile, a
target and a set of initial conditions: type of incoming particle, beam
energy, impact parameter. While neutron-induced and photofission (fission
induced by photons) are the most common and most important cases, other
scenarii such as $\beta$-delayed fission (where fission follows the $\beta$
decay of an isotope), proton-induced fission or fission in heavy-ion
collisions are also important, especially in nucleosynthesis or superheavy
science. It is often assumed that fission takes place after the equilibration
of all degrees of freedom that follows the reaction, resulting in what is
called the compound nucleus [4, 187]. The validity of this approximation
depends on how typical fission times compare with the relaxation time of the
various degrees of freedom including single-particle, collective vibrations
such as the Giant Dipole Resonance (GDR) or neutron emission [188]. Estimates
of these various timescales suggest that the notion of a compound nucleus is
valid for up to several dozens of MeV of excitation energy [188, 189, 190]. In
such cases, the main challenge for nuclear theory becomes describing the exact
structure of this compound nucleus. There are also instances where fission
takes place before full equilibration. This is the quasifission, that is
especially important in heavy ion collisions [191].

###  3.1 Spontaneous Fission

As discussed earlier in Section 2.1, fission results from the competition
between Coulomb repulsion and surface tension. This mechanism is encoded
either directly, in the values of the liquid drop parameters, or indirectly in
the parameters of the energy functional, and is best understood as a
deformation process. This process itself depends on the variations of the
potential energy as a function of the deformation of the nuclear shape – the
potential energy surface (PES). If first-year students of quantum mechanics
were to look at an example of a one-dimensional PES such as Fig. 3, they would
be immediately reminisced of quantum tunneling. Indeed, spontaneous fission is
most conveniently described as quantum tunnelling through a (multi-
dimensional) potential energy barrier. Qualitatively, the problem thus posed
is quite simple: initially, the nucleus is in its ground state. The
probability of fission is simply related to the probability of tunneling from
this initial condition to a deformed shape outside the barrier. Such a
probability is often computed semi-classically with the WKB approximation
[192] and it is given by the following formula

$\tau_{1/2}^{\rm
SF}=\frac{1}{\nu}\exp\left(\frac{2}{\hbar}\int_{a}^{b}ds\sqrt{2B(s)(V(s)-E_{0})}\right),$
(12)

where $s$ is the curvilinear abscissa along the most likely fission path,
i.e., the trajectory in the $N$-dimensional collective space spanned by the
collective variables $\boldsymbol{q}=(q_{1},\dots,q_{N})$ that minimizes the
classical action [36]. The nuclear potential energy along this path is noted
$V(s)$ and $B(s)$ is the collective inertia. $\nu$ is the number of assaults
to the barrier per unit time [67]. The ground-state energy is noted $E_{0}$
and the points $a$ and $b$ are referred to as the inner and outer turning
points, respectively.

While qualitatively simple, the devil hides in the details. Firstly, in
addition to involving the introduction of a set of collective variables
$\boldsymbol{q}$, the derivation leading to (12) also relies on an explicit
decoupling between these collective variables and the other nuclear degrees of
freedom, which expresses the fact that the collective motion is confined
within the space spanned by the variables $\boldsymbol{q}$ [193, 194, 195,
196, 174, 139]. As a result, the potential energy cannot be taken directly as
either the macroscopic-microscopic potential energy (1) or the HFB energy (6).
Instead, these values are modified by zero-point energy corrections, the
specifics of which depend on the nature of the collective variables and of the
theoretical framework used to express the decoupling. Most of the time, these
zero-point energy corrections are computed with the GCM [171]. Importantly,
their values do not only affect the overall potential energy surface
$V(\boldsymbol{q})$, but also the initial energy of the nucleus $E_{0}$: in
principle, the latter should correspond to the lowest eigenstate of the
collective Hamiltonian. In practice, most calculations treat the value of
$E_{0}$ as an adjustable parameter that can be used to quantify uncertainties
of the predictions.

Secondly, in contrast to textbook examples of tunnelling, the potential energy
surface is defined in the abstract space that represents deformation. In
practice the set of collective coordinates $\boldsymbol{q}$ can be either the
deformations of the nuclear surface, see Section 2.1, or the expectation value
of relevant observables, see Section 2.2. This means that the ter $B$ in the
WKB formula is not related to the mass of the nucleus, but is a more abstract
object called the collective inertia tensor. It is in fact a rank-2 tensor
$\mathsf{B}(\boldsymbol{q})\equiv B_{ij}(\boldsymbol{q})$ with $i,j=1,\dots,N$
where $N$ is the number of collective variables. The calculation of this
object is far from simple. Traditional formulas rely on the GCM or the
adiabatic time-dependent HFB (ATDHFB) approximation; see [31] for a summary of
the relevant formulas. Most practical implementations of these methods neglect
the effect of velocity, time-odd terms induced by the generalized momenta
$\boldsymbol{p}$ associated with the collective variables $\boldsymbol{q}$,
both in the GCM where these momenta should be introduced explicitly, which
would double the number of collective variables [197], or in the ATDHFB
formulas where their effect is simply neglected [198, 199]. In addition, both
the GCM and ATDHFB collective inertia involve the derivative operator
$\partial/\partial q_{i}$. This operator is often reduced to a local
approximation: this perturbative approximation has been shown to have a large
impact on half-lives [200, 201, 202]. Because of these limitations, other
methods have been proposed to compute the collective inertia, including
“dynamic” path calculations [203] and the adiabatic self-consistent collective
coordinate method [204, 205]. Recently, there have been very promising
attempts to perform “exact” ATDHFB calculation of the collective inertia
tensor by inverting the full QRPA inertia [206]. Results suggest rather large
differences with both local and non-local cranking approximations as seen in
Fig. 6.

Figure 6: Top panel: Comparison between the perturbative cranking
approximation to the collective mass tensor $\mathsf{M}=\mathsf{B}^{-1}$ and
the exact calculation along the symmetric fission path of 240Pu. Bottom panel:
ratio between the exact and perturbative cranking inertia. Figures reproduced
with permission from [206] courtesy of Washiyama; copyright 2021 by The
American Physical Society.  Figure 7: Summary of spontaneous fission
calculations with the SkM* EDF for even-even SH nuclei. (a) Inner fission
barrier heights $E_{A}$ (in MeV); (b) SF half-lives log10 $\tau_{\rm SF}$ (in
seconds); (c) $\alpha$-decay half-lives log10 $\tau_{\alpha}$ (in seconds);
(d) Dominant decay modes. If two modes compete, this is marked by coexisting
triangles. Figures reproduced with permission from [16] courtesy of Staszczak;
copyright 2013 by The American Physical Society.

Spontaneous fission half-lives depend very sensitively on each of these
ingredients. Because of the exponential factor, small variations in the value
of the potential energy, zero-point energy or collective inertia can change
the half life by orders of magnitude [12, 207, 201]. Among the quantities that
can impact the final result are the number and type of collective variables
$\boldsymbol{q}$ that define the PES. It has been known for a long time, for
example, that pairing correlations can have a dramatic impact on $\tau_{\rm
sf}$, largely by reducing the collective inertia which, in a simple solvable
one-dimensional model scales like $B\propto 1/\Delta^{2}$ where $\Delta$ is
the pairing gap [208, 209, 210, 211, 212]. This observation was an incentive
to take collective variables associated with pairing correlations, such as the
constraint on the number fluctuations
$\braket{\Delta\hat{N}^{2}}=\braket{\hat{N}^{2}}-\braket{\hat{N}}^{2}$ or the
pairing gap $\Delta$ [213, 201, 214, 215, 216, 217].

Figure 8: Sensitivity of the spontaneous fission lifetimes to different values
of the ground-state energy log${}_{10}[\tau_{\rm sf}(E_{0}=0.5\,{\rm
MeV})/\tau_{\rm sf}(E_{0}=1.5\,{\rm MeV})]$ computed with different collective
inertias: ATDHFB (top panels), GCM (middle panels), and semi-empirical inertia
formula (bottom panels). Figures reproduced with permission from [22] courtesy
of Giuliani; copyright 2018 by The American Physical Society.

A global analysis of spontaneous fission half-lives in heavy nuclei, based on
quantum tunnelling through one-dimensional potentials including triaxiality
and reflection-asymmetry, can be found in [17]. Earlier results were largely
focused on superheavy elements with either macroscopic-microscopic methods
[218, 219, 8], Gogny HFB [15, 220], Skyrme Hartree-Fock+BCS [12] or Skyrme HFB
[16]. Figure 7 shows an example of self-consistent calculation of fission
barriers, spontaneous fission and $\alpha$-decay half-lives in superheavy
elements. Results suggest that in superheavy neutron-rich nuclei, spontaneous
fission is the dominant channel of decay. This is especially important to
properly model the r process [21]. Recent large-scale surveys of fission modes
in heavy nuclei have attempted to probe the influence of various model
parameters such as the method to compute the collective inertia or various
corrections to the ground-state energy [22]. For example, Fig. 8 shows how the
spontaneous fission half-life varies depending on the total value $E_{0}$ of
the ground-state energy. As expected because of the exponential factor in
(12), a variation of the ground-state energy of only 1 MeV can change half-
lives by up to 5 order of magnitude.

Over the years, there have been several attempts to go beyond the semi-
classical WKB approximation for spontaneous fission rates. Functional-integral
methods are, formally, the most developed alternatives [221, 222, 223, 224].
They lead to an imaginary-time evolution equation for the nucleus under the
barrier, the solutions of which are called instantons. This theory has been
tested in a few semi-realistic cases [225, 226]. Other more recent approaches
include solving directly the time-dependent Schrödinger with a complex
absorbing potential [227], and using a configuration-interaction approach
(similar to the nuclear shell model) to test the validity of the adiabatic
hypothesis, namely the fact that the potential energy surface is precomputed
[228, 229]. Although most of these attempts have been tested with toy model
Hamiltonians and sometimes involved important numerical simplifications, they
clearly suggest that the adiabatic approximation may overestimate the
spontaneous fission half-life significantly.

To finish this section, we should mention a particularly exotic spontaneous
fission mode called “cluster” radioactivity, where one of the fragments is a
small nucleus of up to mass $A\approx 20$ that is much smaller than its
partner [230]. While initial descriptions of this decay mode were based on
extrapolating the $\alpha$-decay model of Gamow [231], it was shown early on
that standard fission model techniques based on computing tunneling
probabilities across multi-dimensional potential energy surfaces could be very
successful [232, 233]. The main difference is that these PES must probe
extremely asymmetric shapes with, e.g., a much larger ratio
$\braket{\hat{Q}_{30}}/\braket{\hat{Q}_{20}}$ than for traditional fission
modes. This phenomenon was studied extensively in the framework of the SR-EDF
approach [234, 18] and was suggested as the primary decay mode of Oganesson
($Z=118$) isotopes [19, 235].

###  3.2 Neutron-Induced Fission

Neutron-induced fission plays a special role both in basic science and in
technological applications of fission. As the name indicates, it is a special
case of fission that is triggered when a neutron interacts with a (heavy)
target $(Z,N)$ of binding energy $B(Z,N)$. Assuming for now that the neutron
is absorbed by the target, the energy of the resulting nucleus is then
$B(Z,N)+E_{n}$: the resulting nucleus is in an highly excited state of energy
$E=B(Z,N)+E_{n}-B(Z,N+1)$ (binding energies are negative). In the case of the
235U(n,f) reaction, for example, the binding energies of 235U and 236U are
$B(^{235}\mathrm{U})=-1783.102$ MeV and $B(^{236}\mathrm{U})=-1789.648$ MeV
leading to an excitation energy of $E=E_{n}+6.546$ MeV. In the standard view
of fission as a deformation process, neutron-induced fission occurs or, more
precisely, has a large probability to occur whenever $E$ (in the compound
nucleus) exceeds the height of the highest fission barrier. For some
naturally-occurring elements such as 235U, this happens regardless of the
energy $E_{n}$ of the incident neutron: these elements are called fissile.
Odd-mass nuclei are a lot more likely to be fissile than even-even ones:
pairing correlations induce an additional gain in binding energy after the
absorption of the neutron, which increases the chance for the compound nucleus
to be at an energy above the fission barrier. Elements where fission can be
triggered by higher-energy neutrons, with $E_{n}>1$ MeV, are called
fissionable and a typical case of interest for nuclear technology is 238U.

Figure 9: Overview of some of the most relevant reaction channels for the
compound nucleus. Depending on the energy of the incident neutron, other
reaction mechanisms may be competing. These include direct reactions, where
the incoming particle interacts with specific rotational/vibrational modes of
the target, and pre-equilibrium reactions, where several single particle
excitations/deexcitations take place without a complete thermalization of the
compound system. The total cross section is the sum of the cross sections of
each process.

Qualitatively, the fission cross section $\sigma_{nf}$ can be thought of as
the probability (expressed as a surface) that an incoming beam’s neutron is
captured by the target and that the resulting nucleus decays through fission.
It is obviously a function of the energy of the incident neutron. A very
pedagogical introduction to this topic can be found in [38]. In practice, the
theoretical treatment of neutron-induced fission is thus markedly more complex
than that of spontaneous fission. Because of the substantial amount of
excitation energy in the fissioning nucleus many decay channels such as
neutron emission $(n,xn)$, radiative capture $(n,\gamma)$, etc. can be open.
Crucially, the fission cross section also depends on the entrance channel,
i.e., the probability that the neutron is absorbed in the first place – a
process that competes with scattering, both elastic $(n,el)$ and inelastic
$(n,n^{\prime})$. Therefore, the variations of the fission cross section with
the nucleus $(Z,N)$ or the energy $E_{n}$ of the incident neutron depend not
only on how the characteristics of the fission process itself vary with these
quantities, but also on how every other channel is affected. For example,
neutron emission from an excited nucleus cannot occur if the energy is lower
than the neutron separation energy $S_{n}$. This channel is thus entirely
closed for all $E<S_{n}$ (the cross section is zero), a situation that can
happen for instance in low energy gamma-induced fission. This will in turn
impact in a non-trivial way how the fission cross section will change as a
function of $E$.

Figure 9 lists some of the various channels that should typically be
considered in a heavy nucleus such as actinides and transactinides. It is
important to mention that the various output channels may be populated by
processes with very different time scales. In short-time processes, the
neutron may simply scatter without interacting with the internal region of the
target nucleus. This corresponds to what is called direct reactions, which
constributes to the elastic scattering channel and possibly the inelastic one
when low energy vibrational/rotational modes of the target are excited. In
intermediate-time processes at higher energies, the neutron can lead to
successive excitations/deexcitations of single particles degrees of freedom
(DoFs) of the target with not enough time for the whole system to thermalize.
This is the pre-equilibrium part of the cross section [236], which populates
mostly the inelastic scattering channel. Finally, the longer-time processes
consist in the absorption of the neutron leading to a thermalized compound
nucleus. In this case, one assumes that the compound nucleus is in a strongly-
damped regime [237, 238, 239, 240, 241]: its wave function is a very complex,
incoherent linear superposition of many different types of intrinsic,
collective and coupled wave functions. The compound nucleus itself emits
particles and populate a variety of output channels. Due to its typically
large time scale, neutron-induced fission falls into this category of compound
nucleus processes.

Figure 10: Fission cross sections in the reaction 239Pu(n,f) as a function of
the energy of the incident neutron. Data taken from the EXFOR database [242,
243]; measurements from [244]. Left: thermal and resonance region $E_{n}<0.1$
MeV; both axes are in log scale. Right: fast region $E_{n}>0.1$ MeV; the
x-axis is in log scale.

As an example, Fig. 10 shows the variations of the fission cross section as a
function of the incident neutron energy for the case of 239Pu(n,f). At very
low energies $E_{n}<0.1$ eV (the thermal regime), the cross section varies
smoothly with the energy and is roughly inversely proportional to the neutron
velocity. At such low-energies, the $(n,n^{\prime})$ and $(n,xn)$ channels are
closed and the fission cross section results from the competition between
elastic scattering, photodeexcitation and actual fission. In fact, the
smoothly decreasing trend mostly reflects the fact that the probability of
absorption decreases with neutron energy (or equivalently: the scattering
probability increases). The epithermal or resolved resonance region ranges
from 1 to about $10^{4}$ eV. In this range of neutron energies, the resonances
associated with the capture of the neutron by the target can be resolved
experimentally. In this maze of resonances, only a few actually corresponds to
the physics of fission while most of them are related with the probability of
capturing the neutron. Finally, the region of $E_{n}>100$ keV corresponds to
what is defined as fast neutrons. At such energies, most decay channels are
open: $(n,\gamma)$, $(n,n^{\prime})$ and $(n,xn)$. The rapid increase of
$\sigma_{nf}$ at around 5-6 MeV corresponds to second-chance fission, i.e., a
neutron is emitted from 240Pu but leaves enough energy in the 239Pu daughter
that this nucleus fissions. Note that several features of the resonance and
fast-neutron region are markedly different in fissile and fissionable
elements, as illustrated in Figure 13 page 13.

####  3.2.1 Fission Rates at High Energy

One of the earliest theories attempting to compute the probability of induced
fission was proposed by Kramers [245]. The main hypothesis is to assume that
the fission rate can be obtained by computing the flux of a current of
probability $\boldsymbol{j}(\boldsymbol{q},\boldsymbol{p},t)$ through the
saddle point. This semi-classical and statistical model depicts the nucleus as
a few shape collective coordinates $\boldsymbol{q}$ and their associated
momenta $\boldsymbol{p}$ interacting with a thermal bath of residual DoFs. The
current $\boldsymbol{j}(\boldsymbol{q},\boldsymbol{p},t)$ is computed from the
probability distribution function $f(\boldsymbol{q},\boldsymbol{p},t)$
representing the probability for the nucleus to be at point
$(\boldsymbol{q},\boldsymbol{p})$ at time $t$. The equation of motion for this
quantity is the Kramers equation [246],

$\displaystyle\frac{\partial f}{\partial
t}(\boldsymbol{q},\boldsymbol{p},t)=\sum_{\alpha\beta}\left[-B_{\alpha\beta}p_{\beta}\frac{\partial}{\partial
q_{\alpha}}+\left(\frac{\partial V}{\partial
q_{\alpha}}+\sum_{\delta}\frac{1}{2}\frac{\partial B_{\beta\gamma}}{\partial
q_{\alpha}}p_{\beta}p_{\delta}\right)\frac{\partial}{\partial
p_{\alpha}}\right.$
$\displaystyle\left.+\sum_{\gamma}\Gamma_{\alpha\beta}B_{\beta\gamma}\frac{\partial}{\partial
p_{\alpha}}p_{\delta}+D_{\alpha\beta}\frac{\partial^{2}}{\partial
p_{\alpha}\partial p_{\beta}}\right]f(\boldsymbol{q},\boldsymbol{p},t),$ (13)

where $D_{\alpha\beta}(\boldsymbol{q})=\Gamma_{\alpha\beta}(\boldsymbol{q})T$
is the dissipation tensor, $T$ the temperature of the bath,
$\Gamma_{\alpha\beta}$ the friction tensor and
$B_{\alpha\beta}(\boldsymbol{q})$ the collective inertia tensor. We define the
reduced friction tensor $\boldsymbol{\beta}=\mathsf{\Gamma}\mathsf{B}$, that
is,
$\boldsymbol{\beta}\equiv\beta_{\alpha\gamma}=\sum_{\gamma}\Gamma_{\alpha\beta}B_{\beta\gamma}$.
The Kramers equation is the extension of the Fokker-Planck equation in
presence of a potential $V(\boldsymbol{q})$ [247, 248]. In addition to not
assuming a stationary process, the advantage of such approaches based on
diffusion is that they include a mechanism to couple intrinsic excitations to
collective motion through the dissipation tensor $D_{\alpha\beta}$.

In its original formulation, Kramers used a one-dimensional version of (3.2.1)
where he further assumed that the potential well near the ground state was
approximately parabolic with a frequency $\omega^{\prime}$ and that the top of
the barrier was also well described by an inverted parabola of frequency
$\omega_{0}$. In such a case, he obtained the following fission rate
$\lambda_{f}$ in the limit of large dissipation,

$\displaystyle\lambda_{f}=\frac{\omega^{\prime}}{2\pi\omega_{0}}\left(\sqrt{\frac{\beta^{2}}{4}+\omega^{\prime
2}}-\frac{\beta}{2}\right)e^{-E_{f}/kT},$ (14)

where $T$ is the nuclear temperature, which is given (approximately) by
$T\approx\sqrt{E^{*}/a}$ with $a$ the level-density parameter, $E^{*}$ the
excitation energy, and $\beta$ is the one-dimensional version of the reduced
friction tensor introduced earlier. This approach was later generalized in
multi-dimensional cases [249, 250, 251, 252, 253]. In this case, assuming that
the local minimum of the potential $V(\boldsymbol{q})$ characterizing the
ground state is at $\boldsymbol{q}_{\rm gs}$ and the saddle point at
$\boldsymbol{q}_{\rm B}$, the formula for the fission rate becomes

$\lambda_{f}=\frac{1}{2\pi}e^{-V_{B}/kT}\left(\frac{\det W^{\rm(gs)}}{|\det
W^{\rm(B)}|}\right)^{1/2}\Lambda,$ (15)

with

$W^{\rm(gs)}_{\alpha\beta}=\left.\frac{\partial^{2}V}{\partial
q_{\alpha}q_{\beta}}\right|_{\boldsymbol{q}=\boldsymbol{q}_{\rm gs}},\qquad
W^{\rm(B)}_{\alpha\beta}=\left.\frac{\partial^{2}V}{\partial
q_{\alpha}q_{\beta}}\right|_{\boldsymbol{q}=\boldsymbol{q}_{\rm B}},$ (16)

and $\Lambda$ given by the only positive eigenvalue of the matrix equation
$\det|\mathsf{M}\Lambda^{2}+\boldsymbol{\beta}\Lambda+\mathsf{W}^{\rm(B)}|=0$,
where $\mathsf{M}$ is the collective mass tensor,
$\mathsf{M}=\mathsf{B}^{-1}$. Projecting such multi-dimensional calculations
on one-dimensional trajectories is equivalent to introducing a mass correction
and an additional friction term [254].

Figure 11: Fission rate calculated for the nucleus 248Cf in the case of two-
body dissipation for excitation energy $E^{*}=30$ MeV (a) and $E^{*}=1500$ MeV
(b). The solid, dashed, and dotted curves correspond to the Langevin
calculations in three-, two-, and one-dimensional collective potential energy
surfaces, respectively. Figures reproduced with permission from [255] courtesy
of Nadtochy; copyright 2007 by The American Physical Society.

Early studies of fission rates were limited since numerically solving the
Kramers equation (3.2.1) is notoriously difficult [256]. However, the
solutions to this equation, which are probability distribution functions, can
also be sampled by solving the stochastic Langevin equation (26). Given a
large enough number of full Langevin trajectories, the time-dependent fission
rate is given by $\lambda_{f}(t)=-1/N(t)dN/dt$ where $N(t)$ is the number of
trajectories that did not escape the saddle point at time $t$ [257, 255]. At
small $t$, this fission rate is nearly 0, while at larger $t$ calculations
show that the rate converges to a nearly constant, quasistationary value [258,
259, 255] that matches what one would obtain from the solution of the Kramers
equation [260], as illustrated in Fig. 11. The transient time needed for the
system to reach quasistationary flow plays an important role when other decay
channels such as neutron or $\gamma$ emission are taken into account: during
that time, fission plays a relatively smaller role compared with these other
processes, and this can affect, e.g., the number of pre-scission neutrons
[261]. As shown in Fig. 11, the fission rate is very sensitive to the size of
the collective space and the excitation energy, but it also depends strongly
on the value of the dissipation coefficient [262]. Note that because of its
classical character, this approach cannot describe lower-energy effects
related to resonances or even tunneling. Until now, it has therefore mostly
been applied to calculations of fission rates at very high energies such as
encountered in heavy-ion collisions.

####  3.2.2 Fission Cross Sections for Fast Neutrons

The approach presented above allows computing fission rates (or probabilities)
but has not been used to the calculation of fission cross sections yet.
Predicting a fission cross section is actually more demanding as it requires
the proper description of both the input channel and of the competition
mechanism between all the possible output channels (e.g. $\gamma$ and neutron
emission). In the fast-neutron range ($E_{n}\gtrsim 100$ keV), such
calculations are usually performed within the Hauser-Feshbach theory [263]
that relies on a statistical picture of nuclear structure.

##### General Theoretical Framework

In the high-energy range the density of resonances is so high that it is
impossible to resolve them experimentally. As a consequence, theoretical
approaches focus on estimating the average cross sections over a small sliding
window in energy. The standard theory to describe the average cross section
for compound-nucleus processes in this energy range is the Hauser-Feshbach
theory. A pedagogical review of this approach can be found in [187]. The
Hauser-Feschbach framework relies heavily on the Bohr hypothesis stating that
the probability for an output channel depends only on the energy, spin and
parity of the compound nucleus and is independent of the details of its
formation process [264, 265, 266, 267, 187, 188]. Its goal is to compute the
average cross section $\sigma_{\alpha\beta}(E,J,\pi)$ that describes the
formation through an entrance channel generically noted $\alpha$ of a compound
nucleus of energy $E$, spin $J$ and parity $\pi$ that subsequently decays
through a channel $\beta$. Channels refer both to a certain combination of
projectile and targets, e.g., $\alpha\equiv n+^{239}\mathrm{Pu}$, but also to
specific combinations of quantum numbers for each of the constituents. In this
context, fission is depicted by one macroscopic channel without details about
the exact fragmentations. For example both the 138Fe+102Ca and 140Cs+100K mass
splits will be part of what is considered the fission channel of 240Pu. Given
an entrance channel $\alpha=a+A$ corresponding to an incident particle $a$ and
a target $A$, and an exit channel $\beta$ (e.g. neutron or gamma emission),
the cross section for the process, assuming no width fluctuations, reads [187]

$\sigma_{\alpha\beta}(E,J,\pi)=\pi\lambdabar^{2}\sum_{J\pi}\frac{2J+1}{(2j_{a}+1)(2J_{A}+1)}\sum_{l_{\alpha}j_{\alpha}l_{\beta}j_{\beta}}\frac{T_{\alpha
l_{\alpha}j_{\alpha}}^{J\pi}(E)T_{\beta
l_{\beta}j_{\beta}}^{J\pi}(E)}{\displaystyle\sum_{\delta}T_{\delta}(E)},$ (17)

where $j_{a}$ and $J_{A}$ are the total angular momentum of the incident and
target nuclei, $J$ is the total compound-nucleus angular momentum,
$l_{\alpha}$, $j_{\alpha}$ and $l_{\beta}$, $j_{\beta}$ are the orbital and
angular momentum coupling in the channels $\alpha$ and $\beta$, and
$\lambdabar$ is the reduced wavelength of the incident particle $a$.

The coefficients $T_{\alpha}(E)\equiv T_{\alpha
l_{\alpha}j_{\alpha}}^{J\pi}(E)$ are the transmission coefficients for the
channel $\alpha$ at energy $E$. In neutron-induced fission, the typical decay
channels involve neutron emission, $\gamma$ emission, and actual fission,
while the entrance channel is neutron absorption. In the case of resolved
resonances, $T_{\alpha}(E)=2\pi\Gamma_{\alpha}/D$ where $\Gamma_{\alpha}$ is
called the width of the channel $\alpha$, which is inversely proportional to
the lifetime $\tau$ of the state through
$\Gamma_{\alpha}=\hbar/\tau_{\alpha}$, and $D$ is the level spacing at energy
$E$ [268]. In practice, the partial widths fluctuate rapidly with the energy.
The average cross section is obtained by averaging (17) over a small energy
window centered on $E$ by using the relation

$\left\langle\frac{\Gamma_{\alpha}(E)\Gamma_{\beta}(E)}{\displaystyle\sum_{\delta}\Gamma_{\delta}(E)}\right\rangle=W_{\alpha\beta}\frac{\braket{\Gamma_{\alpha}(E)}\braket{\Gamma_{\beta}(E)}}{\displaystyle\sum_{\delta}\braket{\Gamma_{\delta}(E)}},$
(18)

where $\braket{\dots}$ refers to the energy averaging. The actual transmission
coefficients used in (17) are thus defined as the average quantities
$T_{\alpha}(E)=2\pi\braket{\Gamma_{\alpha}}/D$ and $W_{\alpha\beta}$ is called
the width-fluctuation correction. If $\braket{\Gamma}/D\ll 1$, then this
correction can be calculated semi-analytically [269]. Let us emphasize that we
have discussed here only the simplest version of the Hauser-Feshbach theory
and have restricted ourselves to the formula most relevant for fission. In-
depth discussions and additional references can be found in several review
articles [270, 187]. One of the main limitations of Hauser-Feshbach theory is
its assumption that channels do not interfere, i.e., direct interactions
cannot take place. Several extensions of the theory were developed in the
1970ies to address these issues but they are not applied very frequently [271,
272, 273].

From a practical perspective, the application of Hauser-Feshbach theory to
fission implies that we “only” have to determine the transmission coefficients
of all the relevant processes. Their expression depends on the type of channel
involved, e.g., the neutron optical model gives the neutron transmission
$T_{n}(E)$. In this review, we are only going to discuss the calculation of
$T_{f}(E)$; see [274] and references therein for additional details about the
neutron-capture, neutron-emission and $\gamma$ transmission coefficients.

##### Calculation of Transmission Coefficients

The standard method to determine fission transmission coefficients $T_{f}(E)$
consists in computing the penetrability through a one-dimensional fission
barrier [275]. This involves first the construction of a collective potential
and inertia as a function of a single collective coordinate associated with
the elongation of the fissioning system. The probability that the compound
nucleus decay through fission derives from the transmission of a collective
plane wave from the low-deformations to the high-deformation domain. In other
words, the transmission coefficient is extracted from the solution of a
stationary version of an equation of the type (31) (see Section 4.4). It is
important to realize that such a wave equation is not the Schrödinger equation
for a particle, but the equation of motion for a collective wave packet
traveling through the collective space (=deformation space) characterizing the
fission process. Although this approach could in principle be rooted in first-
principle theories [275] it is still mostly guided by phenomenological
considerations. When the energy of the system lies above the fission barrier,
this approach is often combined with the transition-state model [276]: at the
top of each barrier are a sequence of discrete excited states of energies
$\epsilon_{k}$. For each such state, there is an associated transmission
coefficient (20) determined from a collective potential up-shifted by an
energy $\epsilon_{k}$. These states are typically modeled as members of
collective rotational bands built on top of vibrational bandheads. The total
fission transmission coefficient thus becomes

$T_{f}(E)=\sum_{k}f_{k}T_{f}(E-\epsilon_{k}),$ (19)

where $f_{k}=1$ if the spin and parity of the transition state equals that of
the compound nucleus, and is zero otherwise. Note that (19) can be extended by
additionally considering a continuum of excited states [277].

The main ingredients of this approach are the collective potential and the
associated inertia. For fissile nuclei such as 239Pu and 235U, the energy $E$
of the compound nucleus always exceeds the highest fission barrier
irrespective of the energy of the incident neutron. In this regime,
transmission coefficients are most of the time determined from the formula
given by Hill and Wheeler [278]. It is obtained by assuming that the
collective potential is an inverted parabola of frequency $\omega_{0}$ and
height $E_{f}$. If we work within a one-dimensional collective space and
further assume that the collective inertia does not depend on the collective
variable, then the wave equation can be solved analytically and yields the
following Hill-Wheeler formula for the transmission coefficient,

$T_{f}(E)=\frac{1}{1+\exp\big{[}2\pi(E_{f}-E)/\hbar\omega_{0}\big{]}}.$ (20)

Note that the resulting transmission coefficient is independent of the
collective inertia. In practice, (20) is often applied by using semi-realistic
estimates of the fission barrier height $E_{f}$ from either macroscopic-
microscopic [279, 280] or fully microscopic calculations [122] and by choosing
$\omega_{0}$ as an adjustable parameter. In the case of the double-humped
fission barrier typical of actinide nuclei, the full transmission coefficient
is computed by coupling the coefficients of the first ($T_{A}$) and second
($T_{B}$) barriers according to

$T_{f}=\frac{T_{A}T_{B}}{T_{A}+T_{B}},$ (21)

if we assume a statistical equilibrium in the class II states [281].

Equation (20) could in principle be generalized to a more realistic collective
space. Along this line, we can mention the early work of Cramer and Nix
solving the case of two inverted parabola on a finite support connected by a
parabolic shape [282]. The subsequent description of quasi-bound states in the
outer potential qualitatively explains the presence of resonances in the
fission cross section at energies below the barrier height; see Fig.12.
Similar studies are performed in Ref. [283] Going even further, one can add an
imaginary part to the collective potential, typically in the second well, to
account for the damping of collective vibrations. This idea leads to the
fission optical model and is for instance discussed in Ref. [284, 285]. As
shown in the right panel of Fig.12, such an approach reproduces nicely the
large (few 100 keV) resonances just below the fission threshold.

Figure 12: Left: Fission transmission coefficient as a function of the
excitation energy of the fissioning system. The one humped Hill-Wheeler
estimation (red curve) is compared to its counterpart when coupling two
barriers as in Eq. (21) (green dotted line) as well as the one obtained from a
Cramer-Nix potential (full black line). While the Hill-Wheeler formula gives a
smooth evolution with energy, the Cramer-Nix approach predicts resonances in
qualitative agreement with experiment. Courtesy of P. Tamagno [281]. Right:
234U cross section predicted by the extended optical model for fission (full
black line) compared to various experimental data. Figure reproduced with
permission from [284] courtesy of M. Sin; copyright 2007 by The American
Physical Society.

It could also be possible to solve Eq.(20) numerically for an arbitrary
potential and inertia functions. Doing so still poses a number of practical
challenges. Encoding the physics of fission requires collective spaces with
more than one deformation degree of freedom. Most importantly, the Hill-
wheeler equation was obtained by neglecting the deformation dependence of the
collective inertia: in this case, the final formula is independent of it. In a
more realistic calculation, the transmission coefficient would most certainly
depend on it – like the spontaneous fission rate of (12). The problem of which
boundary conditions to adopt is also not trivial. Finally, the extreme
sensitivity of the fission cross section to the barrier height as well as to
the fine structure of the resonant states in the secondary potential well
makes such approaches highly challenging.

####  3.2.3 Resolved Resonance Region: the R-matrix Formalism

When the energy of the compound nucleus becomes close to the top of the
fission barrier, or is even lower than it, the situation becomes more complex
as both the level structure of the two potential wells and their coupling
begin playing a role [286, 287]. In particular, the excited states associated
with the second potential well, which correspond to more elongated shapes and
have a higher probability to fission, play the role of doorway states for the
fission exit channel [288, 289, 290]. The presence of such intermediate
structures – states that are neither continuum states with a very broad width
nor are pure single-particle or collective states with a very narrow width –
in the cross sections partly explains the structure of the resolved resonance
region at $10\,\mathrm{eV}<E_{n}<10\,\mathrm{keV}$ in Fig. 10, together with
the resonances that come from the entrance channel and reflect an enhanced
probability that the neutron is absorbed.

While the Hauser-Feschbach theory captures well the smooth-energy trends of
the cross sections, it is, by design, not able to describe the narrow
resonances observed in the intermediate-energy range. The description of these
fine structures relies on another formalism: the R-matrix theory. The R-matrix
theory is a general, multi-channel approach especially designed to handle
resonances [291, 292]. More specifically, it is the reference method used for
evaluating the neutron cross sections in the resolved resonance region (RRR)
and is extensively used to fit experimental cross section data [293, 294].

We recall that in the standard R-matrix theory of a reaction between a
projectile and a target, space is divided into two regions, the interaction
(or internal) and asymptotic regions, and the system target+projectile is also
divided into two subsystems with different numbers of particles and quantum
numbers (=the channels $\alpha$) [291, 292]. The wave function in the external
region is the product of the intrinsic wave functions of each subsystem and an
asymptotic wave function for the relative motion of the center of mass; in the
internal region, it is a many-body wave function. The actual R matrix is a
mathematical object that connects the wave functions of all possible output
channels at the boundary between the internal and external regions.

Fission requires a special treatment in the context of the R-matrix due to the
fact that it is not by itself a well defined output channel but can actually
lead to a large number of different fragmentations. The extension of the
R-matrix formalism to the case of fission was first formalized by Lynn [295,
296, 297]; a comprehensive and pedagogical presentation of his model can also
be found in Ref. [298]. The basic idea is to identify a set of collective
variables $\boldsymbol{\eta}$ that defines the fission channel and separate it
from all the other degrees of freedom $\boldsymbol{\xi}$ of the nucleus. In
all applications of this formalism, the vector $\boldsymbol{\eta}$ has only
one component $\eta$ that typically coincides with the axial quadrupole
deformation $\beta$ of the nucleus – which is also one of the collective
variables used to define PES; see Section 2. The associated collective motion
will only include the corresponding $\beta$-vibrational modes, i.e., the
shapes vibrations of the axial quadrupole moment of the nucleus. In this
approach the many-body Hamiltonian is expanded as [298]

$\hat{H}=\hat{H}_{\rm vib}(\boldsymbol{\eta})+\hat{H}_{\rm
int}(\boldsymbol{\xi})+\hat{H}_{c}(\boldsymbol{\eta},\boldsymbol{\xi}),$ (22)

where the collective Hamiltonian $\hat{H}_{\rm vib}(\boldsymbol{\eta})$ acts
only on the fission mode variables $\boldsymbol{\eta}$. The intrinsic
Hamiltonian operator $\hat{H}_{\rm int}(\boldsymbol{\xi})$ acts only on the
intrinsic variables $\boldsymbol{\xi}$, which includes both single- or quasi-
particle degrees of freedom, rotational modes and any non-$\beta$-vibrational
mode. $\hat{H}_{c}(\boldsymbol{\eta},\boldsymbol{\xi})$ encodes the coupling
between the fission modes and all intrinsic excitations. In analogy to
standard R-matrix theory, the total space, here the collective space of
deformations $\boldsymbol{\eta}$, is divided into two regions, internal and
external. Traditionally, the fission channel “radius” that defines the
boundary is put near, or at the outer barrier denoted $\boldsymbol{\eta}_{0}$
[297].

Figure 13: Left: Fission cross section for the 240Pu(n,f) reaction in the
resonance region for $0\leq E_{n}\leq 4$ keV. The data is taken from [299].
Right: Same quantity for neutrons energies $10\,\text{keV}\leq E_{n}\leq
50\,\text{MeV}$. Data taken from [299, 300].

In the internal region, the eigenstates $X_{\lambda}$ of the complete
Hamiltonian (22) are expanded as

$X_{\lambda}=\sum_{n}\sum_{\nu}C^{\lambda}_{n\nu}(\boldsymbol{\eta}_{0})\Phi^{\boldsymbol{\eta}_{0}}_{n}(\boldsymbol{\eta})\chi_{\nu}(\boldsymbol{\xi}),$
(23)

where the basis wave functions of the vibrational modes are denoted by
$\Phi^{\boldsymbol{\eta}_{0}}_{n}(\boldsymbol{\eta})$. The notation indicates
that these are eigenfunctions of the collective, vibrational Hamiltonian that
only depend on $\boldsymbol{\eta}$ but with a specific boundary conditions set
at $\boldsymbol{\eta}=\boldsymbol{\eta}_{0}$. The basis wave functions for the
intrinsic excitations are $\chi_{\nu}(\boldsymbol{\xi})$ and correspond to
eigenstates of the intrinsic Hamiltonian. They define the Bohr fission
channels. For the double-humped barrier typical of actinides, collective
vibrational states can only be built inside the two potential wells
corresponding to the ground state and the fission isomer. Therefore, one can
adapt the expansion (23) as follows,

$X_{\lambda}=\sum_{n}\sum_{\nu}C^{\lambda;I}_{n\nu}(\boldsymbol{\eta}_{0})\Phi^{\boldsymbol{\eta}_{0};I}_{n}(\boldsymbol{\eta})\chi_{\nu}(\boldsymbol{\xi})+\sum_{n}\sum_{\nu}C^{\lambda;II}_{n\nu}(\boldsymbol{\eta}_{0})\Phi^{\boldsymbol{\eta}_{0};II}_{n}(\boldsymbol{\eta})\chi_{\nu}(\boldsymbol{\xi}),$
(24)

where the additional labels $I$ and $II$ refer to the first and second well
respectively. The computation of the matrix elements of the total Hamiltonian
with the ansatz (24) makes manifest the potential couplings between the two
wells. Different regimes of coupling (weak coupling, uniform mixing) lead
ultimately to different resonant structures in the fission cross section which
are presented in great details in [275]. It includes phenomena going from the
large (100 keV) sub-barrier vibrational resonances observed for instance in
230Th(n,f) to the clusters of low-energy (a few keV) fission resonances
illustrated in Fig. 13. In this formalism it is possible to interpret such
structures as the interplay between class I and class II states. Upon
formation, the nucleus has some probability (and associated width) to pass the
first fission barrier and to exit in the second potential well. In that second
well, the system has a (much higher) probability of fissioning. The amount of
couplings depends on the energy of the nucleus and on all other possible decay
modes inside the two wells.

The final product of the formal R-matrix theory is a link between the
scattering matrix, hence the cross section, and a set of parameters related to
the internal wave functions $X_{\lambda}$. The theory was initially successful
at explaining qualitatively the large number of narrow resonances
(intermediate structures) in the fission sub-barrier cross sections [295, 296,
297, 289, 301]. It is now used routinely as an advanced theoretical framework
in the context of cross section evaluation to fit existing measurements [302,
303]. To our knowledge, the R-matrix formalism has not yet been used to
predict (instead of fitting or analyzing) features of resonances starting from
a first-principle Hamiltonian. Although possible in principle, such
predictions would require the knowledge of the internal region at a level of
precision that is most likely beyond what is possible with state-of-the-art
many-body theories.

In recent years, the focus of such approaches has also shifted to the
prediction of the average fission cross section in the continuum energy range
[304, 305, 306]. Figure 14 shows an example of the fission cross sections in
several Plutonium isotopes obtained within this formalism. The idea is to
consider a stochastic ensemble of realistic R-matrices that account for a
statistical ensemble of resonances. One then determines the cross section as
an average over all the cross sections obtained with the R-matrix samples.
This kind of approach somehow bridges the R-matrix formalism with the Hauser-
Feshbach approach.

Figure 14: Fission cross-sections as a function of the energy of the incident
neutron for a chain of Plutonium isotopes. Calculations were performed using
R-matrix theory for sub-barrier fission and are compared to experimental
measurements and reference evaluations. Figures reproduced with permission
from [304] courtesy of Bouland; copyright 2013 by The American Physical
Society.

##  4 Formation of Primary Fission Fragments

In the previous two sections, we have described how the concept of nuclear
fission emerges naturally from a description of the atomic nucleus in a
symmetry-breaking mean field, where fission probabilities are related to
quantum tunnelling through a multi-dimensional potential energy surface and
fission cross sections to the presence and structure of deformed excited
states. In this section, we assume that fission is happening and we review the
methods capable of predicting the primary fragments properties. One may
distinguish two classes of theoretical approaches. The first one relies on
defining an ensemble of scission configurations and populating them according
to a statistical distribution. The so-called scission point models that we
discuss in Section 4.1 are prime examples of this class of methods. A second
option is to describe explicitly the dynamics of the process as the nucleus
evolves from its weakly-deformed initial state toward configurations where two
primary fragments can be identified. Such models of fission dynamics can be
further subdivided into two sub-categories:

  * •

Real-time, time-dependent calculations simulate a single fission event, that
is, the evolution in time of a single configuration of the fissioning nucleus.
A single solution of the Langevin equation or a single random walk (Section
4.2) or a fully self-consistent time-dependent Hartree-Fock (TDHF) or time-
dependent Hartree-Fock-Bogoliubov (TDHFB) trajectory (Section 4.3) are
examples of such techniques;

  * •

Time-dependent collective models simulate the time-dependent evolution of the
probability distribution of populating a specific configuration. The most
common example of such an approach is the quantum-mechanical time-dependent
generator coordinate method (TDGCM) presented in Section 4.4 [307]. Although
it has been applied very rarely in practical calculations, its classical
equivalent would be the Kramers equation [246].

The two classes of methods are obviously related. For example, sampling many
different single-configuration trajectories allows reconstructing probability
distributions: this is how fission fragment distributions can be extracted
from Langevin or random walks simulations and could, in principle, be compared
with the equivalent time-dependent collective model. As we will discuss in
more details in Section 6, the main goal of time-dependent models of fission
dynamics is to predict actual observables related to the primary fission
fragments such as their distribution in charge and mass, the total kinetic
energy (TKE) distribution, or the sharing of excitation energy between the
fragments. These initial conditions are important inputs to be used in fission
event generators to simulate the deexcitation of the fragments and compare
with experimental measurements.

###  4.1 Statistical Models for Fission Fragment Distributions

Historically, solving explicitly time-dependent differential equations to
directly simulate fission dynamics often exceeded the available computational
resources. At the same time, there was a consensus that the characteristics of
the potential energy surface of the fissioning nucleus at scission played a
key role in determining the fission fragment mass and charge distributions. In
particular, it had become clear that the nuclear shell structure and, more
generally, the deformation properties of the fission fragments were important
drivers in setting their relative abundance [185, 186]. What is known as the
scission-point model is an attempt to take advantage of these observations to
obtain realistic estimates of the primary fission fragment distributions
without simulating the dynamics explicitly. Scission-point models rely on the
hypothesis of quasi-statistical equilibrium at scission: it is assumed that
the dynamics of fission populates scission configurations in a statistical
manner. Scission-point models have been relatively successful and have the
clear advantage of being computationally very cheap since all that is needed
is a “good” potential energy surface: it entirely avoids simulating fission
dynamics.

In most implementations of scission-point models, the PES is not computed from
the potential energy of the fissioning nucleus, but instead defined as the sum
of the potential energy of each fragment $V_{i}(\boldsymbol{q}_{i})$, $i=1,2$
and of the interaction between them that depends mostly on their relative
distance $d_{12}$. The latter term contains both a Coulomb and nuclear
component,

$\displaystyle V(\boldsymbol{q})=$ $\displaystyle
V_{1}(Z_{1},N_{1},\boldsymbol{q}_{1})+V_{2}(Z_{2},N_{2},\boldsymbol{q}_{2})$
$\displaystyle+V_{\rm
Coul}(Z_{1},N_{1},\boldsymbol{q}_{1},Z_{2},N_{2},\boldsymbol{q}_{2},d_{12})+V_{\rm
nucl}(Z_{1},N_{1},\boldsymbol{q}_{1},Z_{2},N_{2},\boldsymbol{q}_{2},d_{12})$
(25)

After the total potential energy $V(\boldsymbol{q})$ of the system at scission
has been tabulated, the probability of populating a given fragment at a given
excitation energy $E^{*}$ can be computed in several ways. One option is to
relate it to a statistical Boltzman factor, i.e. $\propto
e^{-V(\boldsymbol{q})/T}$ where $T$ is a “collective temperature”, which can
be either a parameter of the model or related to excitation energy via the
usual $E^{*}\approx aT^{2}$ expression [186, 308]. Another option consists in
relating it to the level density in each fragment at the current excitation
energy $E_{i}^{*}$ of the fragment $i$ [309, 310]. This latter prescription
requires introducing a mechanism to share the total excitation energy (TXE)
available among the fragments; see discussion in Section 6.4. Obviously,
different assumptions can be made about the ingredients of the calculation,
such as the size of the deformation space for each fragment, the specific
model of the interaction between the fragments, and the method used to
populate each fragmentation. Overall, scission-point models reproduces fairly
well the mass yields, for example in the neutron-induced fission of major
actinides [308], or the transition between symmetric and asymmetric fission in
heavy actinides such as Fm isotopes [311, 312]. The models can be extended to
predict other important observables such as the charge, mass, the TKE of the
reaction and neutron emission in actinides [313].

The low computational cost of scission-point models enables performing
systematic studies over a large range of fissioning systems (e.g. in Ref.
[310]). A key ingredient of these approaches is the ensemble of configurations
considered in the neighborhood of scission. This ensemble is inevitably chosen
in a somewhat arbitrary way since the very concept of scission remains ill-
defined, as mentioned in Section 2.3. As a consequence, different scission-
point models may predict significantly different primary fragments
distributions as illustrated in the right panel of Fig. 15.

Figure 15:  Primary fragments mass distribution for the spontaneous fission of
252Cf. Left: Comparison of the scission point model of Pasca et al. (full
line) to experimental data (black dots). Figure reproduced with permission
from [314] courtesy of H. Pasca; copyright 2013 by The American Physical
Society. Right: Comparison of the scission point model SPY-2 with a smoothing
(full red line) to experimental data (black dotted line). The raw predictions
of SPY-2 (green full line) along with the predictions of SPY-1 (blue dotted
line) are also plotted for the record. Figures reproduced with permission from
[310] courtesy of J.-F. Lemaitre; copyright 2013 by The American Physical
Society.

###  4.2 Classical Dynamics

Fission is by nature a time-dependent process and one would be forgiven to
think that static approaches such as the scission-point model may not capture
all facets of the phenomena. Going beyond such models therefore requires
describing the time evolution of the nucleus starting from its nearly
spherical shape up to well-separated primary fragments. A fully microscopic
theory of this process involves a quantum and real-time evolution of more than
200 nucleons characterized by their position and spin/isospin quantum numbers.
The complexity of this task scales exponentially with the number of DoFs and
prevents solving exactly the associated Schrödinger equation on classical
computers. A long-term goal of fission theory is thus to find reliable
approximations, or reductions, of this quantum many-body dynamics that capture
the physics of fission while staying computationally tractable.

The first possible level of approximation is based on a classical and
statistical picture of the dynamics, which leads to the well-known Langevin
equations. These equations have a long history in physics and we refer to the
review articles of Abe et al. [246] and Fröbrich & Gontchar [315] for a
detailed discussion of their various forms in the context of nuclear physics.
In this section, we simply want to recall the basic form of the Langevin
equations and highlight some of the results that have been obtained from their
application to fission.

The Langevin equations originate from a transformation of the classical
nucleonic degrees of freedom into (i) a few collective coordinates
$\boldsymbol{q}(t)=(q_{1}(t),\dots,q_{N}(t))$ and their associated momenta
$\boldsymbol{p}(t)=(p_{1}(t),\dots,p_{N}(t))$ (ii) the set of all remaining
degrees of freedom usually dubbed intrinsic DoFs. The idea is to choose
properly the collective variables such that the formation of the primary
fragments is mostly independent of the detailed characteristics of the
intrinsic DoFs. With this in mind, the intrinsic DoFs are often interpreted as
a thermal bath interacting with the collective DoFs. The resulting Langevin
equations of the system describe the dynamics of the collective DoFs and read

$\displaystyle\dot{q}_{\alpha}=$
$\displaystyle\sum_{\beta}B_{\alpha\beta}p_{\beta},$ (26a)
$\displaystyle\dot{p}_{\alpha}=$
$\displaystyle-\frac{1}{2}\sum_{\beta\gamma}\frac{\partial
B_{\beta\gamma}}{\partial q_{\alpha}}p_{\beta}p_{\gamma}-\frac{\partial
V}{\partial
q_{\alpha}}-\sum_{\beta\gamma}\Gamma_{\alpha\beta}B_{\beta\gamma}p_{\gamma}+\sum_{\beta}\Theta_{\alpha\beta}\xi_{\beta}(t).$
(26b)

The scalar field $V(\boldsymbol{q})$ is the potential energy,
$\mathsf{B}(\boldsymbol{q})\equiv B_{\alpha\beta}$ is the collective inertia
tensor and
$\mathsf{\Gamma}(\boldsymbol{q})\equiv\Gamma_{\alpha\beta}(\boldsymbol{q})$
the coordinate-dependent friction tensor. A dissipation term is encoded in the
random force $\boldsymbol{\xi}(t)$. Its strength
$\mathsf{\Theta}\equiv\Theta_{\alpha\beta}$ is related to the friction tensor
through the fluctuation-dissipation theorem,
$\sum_{\beta}\Theta_{\alpha\beta}\Theta_{\beta\gamma}=\Gamma_{\alpha\gamma}T$,
with $T\equiv T(\boldsymbol{q})$ the local temperature of the bath of
intrinsic DoFs at point $\boldsymbol{q}$.

There are many different ways to compute the ingredients entering (26), that
is, the fields $V(\boldsymbol{q})$, $\mathsf{B}(\boldsymbol{q})$,
$\Gamma(\boldsymbol{q})$. These quantities must be pre-computed by a nuclear
model capable of giving realistic estimates of nuclear deformation properties.
Traditional applications of the Langevin equation often rely on the
macroscopic-microscopic model described in Section 2.1 to compute them.
However, there have been recently a few attempts to couple microscopic inputs
computed from the EDF approach with the semi-classical dynamics given by the
Langevin equation [316, 317, 235, 318]. In this case, $V(\boldsymbol{q})$ is
the HFB energy and $\mathsf{B}(\boldsymbol{q})$ the collective inertia tensor
briefly discussed in Section 3.1. Along the same lines, it is very important
to keep in mind that the formal and technical difficulties in determining the
scission configurations discussed in Section 2.3 will be relevant when solving
the Langevin equation since the end-point of Langevin trajectories typically
correspond to scissionned configurations.

To obtain meaningful estimates of fission observables, the equations (26) must
be solved many times, possibly with different initial conditions. These
statistical fluctuations of the initial state mimic here the quantum
fluctuations of the system. Because of the stochastic nature of the equation
of motion, each trajectory is unique. If sufficiently many trajectories are
sampled, probability distributions can be reconstructed. The full Langevin
equation has thus been applied to studies of fission rates [255], to the
determination of the charge and mass distribution of fission fragments [319,
320, 321], and to the evaluation of the TKE [322, 320, 323, 324, 325]. Figure
16 shows an example of the latter calculations. Finally, it is possible to
include into the Langevin framework the competition between fission and the
emission of light particles such as neutrons or $\gamma$ rays. This is of
particular interest to study multi-chance fission at high energies [326].

Figure 16: Distribution of TKE as a function of the fragment mass number for
the fission of 236U at excitation energy $E^{*}=6.545$ MeV. The red curve
represents the kinematically allowed maximal value of TKE given by $Q+E^{*}$,
where $Q$ is the $Q$-value of the reaction 235U(n,f). Figures reproduced with
permission from [323] courtesy of Usang; copyright 2017 by The American
Physical Society.

If the motion across the potential is strongly dissipative, it makes sense to
assume that collective velocities are very small. In this case, we can neglect
terms proportional to the square of the velocities as well as the acceleration
terms [327, 328]. The Langevin equation takes the much simpler form of the
Smoluchowski equation

$\sum_{\beta\gamma}\Gamma_{\alpha\beta}\dot{q}_{\beta}=-\frac{\partial
V}{\partial q_{\alpha}}+\sum_{\beta}\Theta_{\alpha\beta}\xi_{\beta}(t).$ (27)

The Smoluchowski equation (27) was used to compute estimates of primary
fission fragment distributions [327, 328, 329, 330, 331, 30]. The formalism
was extended in [330] to describe fission dynamics at higher excitation
energies. In the case of the macroscopic-microscopic model, the main effect of
increasing excitation energy is to destroy shell corrections [95, 332, 99],
although the macroscopic energy should in principle also be modified [96]. One
can incorporate this effect by computing, at each deformation point
$\boldsymbol{q}$ of the standard macroscopic-microscopic PES, a local
temperature $T(\boldsymbol{q})$ based on the value of the relative excitation
energy at this point. This local temperature can then be used to modify the
Strutinsky shell correction and generate PES at finite excitation energy; see
examples in [99, 333]. This approach was used in [322, 323] to analyze the
impact of excitation energy on dissipation. A simplified version of
temperature-dependent shell corrections consists in multiplying the shell
correction by a phenomenological damping factor. Figure 17 shows the impact of
the PES underpinning the random walk calculations at finite excitation
energies. At temperatures $T>1$ MeV, such an approximation works rather well
and has been used in large-scale calculations in [330], but it fails to
account for the more complex dependency of the yields at low excitation
energies. In this regime, an alternative to temperature-dependent shell
correction is to use level densities to guide the evolution of the Metropolis
walk on the surface [334].

Figure 17: Relative fragment charge distribution fo the fission of 234U at
$E^{*}=11$ MeV excitation energy. Each curve was obtained by solving (27) with
different prescriptions for the PES: a pure liquid drop formula ($U_{\rm
macro}$), the standard macroscopic-microscopic potential defined by (1)
($U_{T=0}$) and a modified version of the latter where the shell correction is
multiplied by a damping factor $S(E^{*})$ ($U_{E}$). Figures reproduced with
permission from [330] courtesy of Randrup; copyright 2013 by The American
Physical Society.

The hypothesis of strongly-damped motion was originally introduced with semi-
phenomenological arguments based on the wall-and-window model for dissipation
[335, 336]. Early attempts at verifying this hypothesis from more microscopic
arguments related to the nuclear shell structure suggested that dissipation
may be even stronger than anticipated [337]. Theories of dissipation based on
transport models were introduced to provide a better description of the
dissipation tensor in either the Langevin or Smoluchowski equations [194, 338,
339]. In recent years, fully microscopic simulations of fission based on
TDHF+BCS and TDHFB also predict a strongly dissipative dynamics along with a
very small collective kinetic energy [340, 341, 342]. This is illustrated in
Fig. 18, which shows the collective kinetic energy in the descent from saddle
to scission in the real-time simulation of 240Pu,

$E_{\rm flow}(t)=\frac{1}{2}\int
d^{3}\boldsymbol{r}\,m\boldsymbol{v}^{2}(t)\rho(\boldsymbol{r},t).$ (28)

For most of the evolution, this collective energy is of the order of 1-2 MeV,
which is less than 10% of the variations in potential energy and less than 1%
of total energy released at scission.

Figure 18: Collective flow energy $E_{\rm flow}(t)$ as a function of time in
the fission of 240Pu. Each curve correspond to a simulation with different
energy functional and/or pairing characteristics. Error bars indicate
variations with respect to different initial conditions. Figures reproduced
with permission from [342] courtesy of Shi; copyright 2019 by The American
Physical Society.

###  4.3 Time-Dependent Density Functional Theory

The Langevin equations and their different variants provide a computationally
tractable method to study the real-time evolution of a fissioning nucleus in
presence of dissipation and fluctuations but they present a few limitations.
For instance, a classical treatment of dynamics cannot account for tunneling
effects, which is crucial for spontaneous fission. In addition, going beyond a
thermal bath description of the intrinsic DoFs is necessary if one wants to
predict detailed properties of the emerging fragments such as, e.g. the energy
or spin of primary fragments. The goal of microscopic approaches is to
overcome these limitations by describing the quantum dynamics of the many-body
wave function associated with all the nucleonic DoFs. A reduction in
complexity is enforced by a variational principle that limits the possible
many-body wave functions to a subspace of the complete Hilbert space. Assuming
that the many-body wave function remains a coherent state at all times leads
to the time-dependent density functional theory (TDDFT). In this article, we
include under the label TDDFT an arsenal of different methods including TDHF,
THDF+BCS, TDHFB. A detailed discussion of how these methods are related can be
found, e.g. in [136, 343]. Several important aspects of the general formalism
have been presented in review articles and textbooks; see [344, 345, 346, 347,
196, 139, 66]. A more succinct presentation of the theory specifically geared
toward fission theory can be found in [348].

Figure 19:  Real-time evolution of the one-body density of the 240Pu
asymmetric fission. The predictions are obtained from a TDHF+BCS calculation.
The 3D surface highlights the half-saturation density (0.08 fm-3) isosurface
whereas the projected color map corresponds to a localization function of the
nucleons. Figures reproduced with permission from [111] courtesy of Scamps;

Time-dependent density functional theory is especially relevant for fission:
given an initial state, which is typically a deformed, SR-EDF solution, TDDFT
describes the real-time evolution of the nucleus as it deforms all the way
until the point of scission and even beyond; see Fig. 19. For instance, in
Ref. [349] the dynamics is simulated up to fragments separated by a distance
of 30 fm. At this stage the nuclear interaction between the fragments is
negligible and the properties of the primary fragments such as their
excitation energy and spin can be reliably estimated. The simplest version of
TDDFT is the time-dependent Hartree-Fock (TDHF) theory which leads to the
equation of motion

$i\hbar\frac{\partial\rho}{\partial t}=\big{[}h[\rho(t)],\rho(t)\big{]},$ (29)

where $\rho(t)$ is the time-dependent one-body density matrix and $h[\rho(t)]$
the Hartree-Fock mean field – the time-dependent version of the mean field in
$h$ in (8). The time-dependent Hartree-Fock-Bogoliubov (TDHFB), which includes
the pairing degrees of freedom in the dynamics, is obtained from (29) by doing
the substitutions $\rho(t)\rightarrow\mathcal{R}(t)$ and
$h[\rho(t)]\rightarrow\mathcal{H}[\mathcal{R}(t)]$; see Eq. (8). Note that
practical implementations of these theories in nuclear physics always rely on
an energy density functional to encode the nucleon-nucleon interaction.

Figure 20: Top: Schematic evolution of single-particle orbitals as a function
of deformation. The thick line represent the last occupied state and the up
and down arrows the Cooper pairs of nucleons on that level. Bottom:
corresponding evolution of the total nuclear energy. Each crossing of the s.p.
level result in the transition to a new intrinsic configuration and change of
the potential energy. Figures reproduced with permission from [342] courtesy
of Shi; copyright 2019 by The American Physical Society.

Negele and collaborators reported the first application of the TDHF formalism
to fission back in 1978 [80]. However, these initial simulations included a
large number of rather strong simplifications; the earliest realistic
application of TDHF to fission can probably be attributed to Simenel and Umar
[350]. One important result of their work was the quantitative prediction of
vibrational modes in the fission fragments and the determination of a
mechanism to estimate the TKE of the reaction and the excitation energy of the
fragments. The implementation and release of efficient TDDFT codes without
symmetry assumptions [351, 352, 353] together with the inclusion of
superfluidity at the BCS [354] and full HFB level [341] were key to popularize
this approach. Among the most important early results was the confirmation
that pairing degrees of freedom are indispensable to reach the point of
scission: pure TDHF calculations initialized around the saddle point of the
fission barrier in actinides failed to reach that point unless a significant
boost in energy was imparted to the system [355, 356]. Including pairing
degrees of freedom, even at the TDHF+BCS approximation with BCS occupation
frozen after a given time, completely removed the need to introduce such
boosts [354]. This mechanism naturally results from the ability of pairing
correlations to pass through single-particle level crossings, as schematically
depicted in Figure 20. For this reason, pairing has been dubbed the “fission
lubricant”. The TDDFT approach successfully predicts many properties of the
most probable primary fission fragments such as their kinetic energy, with a
precision of a few percents [341], their deformation close to scission [111],
and more recently the spin of the fragments [349].

The TDHF and TDHFB formalisms are often called semi-classical, in the sense
that they do not allow collective quantum tunneling: if a TDHF(B) evolution
was initialized in a configuration $\rho(t=0)$ ($\mathcal{R}(t=0)$) inside a
potential well determined from constrained HF(B) calculations, the system
would remain inside that well [57]. Various extensions of the TDDFT methods to
imaginary time have been proposed to handle quantum tunneling but have rarely
been applied in practical calculations [221, 222, 357, 223, 225, 226]; see
also Section 3.1.

In spite of its advantages, current implementations of TDDFT are also not well
adapted to describing large fluctuations of observables, i.e. the quantity
$\braket{\Delta\hat{O}^{2}(t)}=\braket{\hat{O}^{2}(t)}-\braket{\hat{O}(t)}^{2}$
[345, 358, 347]. This limitation stems from the restriction of the many-body
wave function to a Slater or Bogoliubov state that do not encode enough
correlations. As a result TDDFT predictions capture well the features of the
average fragmentation but systematically underestimate the associated quantum
fluctuations. For this reason, they are not the most efficient tools to
generate, e.g., fission fragment distributions. There have been several
attempts to go beyond “simple” TDHF(B). In the stochastic mean field theory,
random fluctuations of the density matrix are used to generate an ensemble of
trajectories (in the space of density matrices) that are similar to TDHF [359,
360, 361]. This trick allows extracting reasonable estimates of kinetic energy
distributions but the quantum coherence of the system is lost. Other on-going
investigations along this line simply introduce various phenomenological terms
simulating dissipation and fluctuations in the TDHFB equation [362]. The Time-
dependent RPA as well as the Balian-Veneroni variational principle are other
approaches that could be used, in principle, to get additional fluctuation
while retaining quantum coherence [347, 363], but it has only been applied in
heavy-ion collisions so far [190, 364]. Another possibility consists in mixing
TDDFT trajectories in the quantum-mechanical sense: early attempts were based
on TDHF trajectories [365] while more recent ones rely on full TDHFB solutions
(albeit in a simplified system) [366].

As discussed in the previous section, the strongly dissipative nature of
fission is another result that had been invoked as a reasonable hypothesis by
many authors and was observed in the context of TDHFB simulations [340, 341,
342]. However, whether this strong dissipation would remain when exploring a
larger variational space than in current implementations of TDDFT, e.g., a
space that also contains collective correlations, is yet to be determined.
Finally, note that dissipation is also largely caused by pairing correlations:
the same mechanism that allows the system to go across single-particle
crossings explains both that the system can deform sufficiently up to scission
and that the collective energy is dissipated into changes of intrinsic
configurations [367].

###  4.4 Time-Dependent Generator Coordinate Method

Because of the many successes of collective approaches, from the seminal paper
of Bohr and Wheeler to the predictions of fission isomers, spontaneous fission
half-lives or fission fragment distributions, there is a rather large
consensus that fission can be well described by invoking a few collective
variables such as shape deformations. This idea motivates the search for
approaches reducing the complete quantum many-body dynamics to an evolution in
a small-dimensional collective space. The ATDHF theory mentioned in Section
3.1 in the context of spontaneous fission [368, 369, 370, 371], and its ATDHFB
extension with pairing [372], are the poster child of this philosophy. They
reduce the complexity of TDDFT, which involves all the one-body degrees of
freedom, to a classical collective dynamics on a manifold parameterized by a
few collective variables [57]. Another approach leading to a quantum
collective dynamics is the time-dependent generator coordinate method (TDGCM)
and especially its Gaussian overlap approximation (GOA). Note that re-
quantizing the ATDHFB theory leads in fact to an equation of motion similar to
the TDGCM+GOA. The theoretical framework of the TDGCM is presented in great
details in [307]. In this section, we will thus discuss only the case of the
TDGCM under the Gaussian overlap approximation which is, as of today, the only
quantum microscopic approach that has been tested in predictions of actual
primary fragments distributions.

The first step of the TDGCM approach consists in building an ensemble $S$ of
generator states (sometimes referred to as a collective path) parameterized by
a few collective variables
$S=\\{\ket{\Phi(\boldsymbol{q})}|\boldsymbol{q}\in\mathbb{R}^{N}\\}$. In the
context of fission the generator states are most often time-even Bogoliubov
states obtained by solving the constrained HFB equations as discussed in
Section 2.2. The collective variables are typically related to the first
multipole moments of the one-body density in order to encode the physics of
deformation. With this family of generator states, the TDGCM proceeds with a
variational principle applied to the Hill-Wheeler ansatz for the wave function

$|\psi(t)\rangle=\int_{\boldsymbol{q}}f(\boldsymbol{q},t)|\phi(\boldsymbol{q})\rangle
d\boldsymbol{q}.$ (30)

The time-dependent mixing function $f(\boldsymbol{q},t)$ is the direct
variational unknown but it is often more convenient to deal with a transformed
function $g(\boldsymbol{q},t)$ that could be interpreted as a wave packet in
the collective space. All operators associated with physical observables can
also be mapped onto operators acting on the collective space (a subspace of
functions of the $\boldsymbol{q}$ variable) [171]. Under the Gaussian overlap
approximation in its simplest form, the equation of evolution for the
collective wave function reduces to

$i\hbar\frac{\partial}{\partial
t}g(\boldsymbol{q},t)=\left[-\frac{\hbar^{2}}{2}\sum_{\alpha\beta}\frac{\partial}{\partial
q_{\alpha}}B_{\alpha\beta}(\boldsymbol{q})\frac{\partial}{\partial
q_{\beta}}+V(\boldsymbol{q})\right]g(\boldsymbol{q},t),$ (31)

where $V(\boldsymbol{q})$ is the potential energy and
$\mathsf{B}(\boldsymbol{q})\equiv B_{\alpha\beta}$ is the collective inertia
tensor. The potential energy and collective inertia have the same meaning as
in Section 4.2 and Section 3.1. An important point is that these quantities
derive from the variational principle and are, formally, the zero- and second-
order derivatives of the reduced Hamiltonian kernels between the generator
states.

The adaptation of the TDGCM formalism to fission dynamics was performed in the
1980’s by the group at CEA,DAM,DIF [373, 146]. These initial developments led
to the first prediction of fission fragment mass distributions in a completely
microscopic theory [374]. Advances in computing capabilities and new
algorithmic developments [375, 376] renewed interest in this approach in the
2010ies. Since then, the theory has been applied largely to compute the
charge, mass and isotopic primary mass distributions of fission fragments
[377, 378, 379, 380, 381, 382, 383, 384]. Figure 21 shows an example of such
calculations for the primary mass distribution $Y(A)$ in 236U and 240Pu
extracted from the solution to the TDGCM+GOA evolution equation.

Figure 21: Left: Fission fragment mass distribution of 236U for thermal
neutrons obtained by solving the TDGCM+GOA equation with the SkM*
parameterization of the Skyrme EDF, compared with experimental data taken from
[385, 386, 387, 388, 389, 390, 391, 392, 393]. The raw TDGCM+GOA results are
convoluted following three different methods that yields the blue circles, the
red squares and the orange triangles. Right: Same quantity for 240Pu. Figures
reproduced with permission from [384] courtesy of Verriere; copyright 2021 by
The American Physical Society.

Obviously, results of TDGCM+GOA calculations will depend on the choice and
number of collective variables. We already mentioned the recent introduction
of pairing collective variables [214, 217, 216] for spontaneous fission; very
few TDGCM calculations have been performed in such extended spaces [394]. Let
us highlight an interesting attempt to use the collective variables $D=|z_{\rm
CM}^{(1)}-z_{\rm CM}^{(2)}|$ and $\xi=|A_{1}-A_{2}|/A$, with $z_{\rm
CM}^{(i)}$ the coordinate of center of mass of fragment $i$ and $A_{i}$ is
total number of particles [377, 37]. Such coordinates, which were inspired by
classical shape parameterizations, are much better adapted to describing
fission fragments than the standard multipole moments and cover most of the
set of all possible fragmentations. The role and importance of the collective
inertia tensor also remains an open problem, and it is not clear if it has as
large an impact for induced fission as it has for spontaneous fission.
Similarly, the role of time-odd collective momenta, which have been neglected
so far, is worth investigating since they provide the proper asymptotic values
of the collective inertia (at least in the case of translations) [197, 171]. A
preliminary study of these terms in the static case of particle number
projection suggests a large effect [395]. An attempt to even remove the need
for inertia by solving the complete TDGCM without approximation was also
reported in Ref. [396].

Finally, a major limitation of the current implementations of the TDGCM is to
rely only on constrained HFB states with the lowest energy for a given
constraint. This is the cause for discontinuities in the manifold of generator
states that break down the GOA [397, 398]. In addition, TDDFT calculations
have shown that the most probable path toward fragmentation involves HFB
states with significant intrinsic excitation energy. The very fact that
fission fragments emit light particles (cf. Section 5) proves that they are
produced in excited states. With the standard family of generator states, the
ansatz (30) can not model such effects, which prevents the TDGCM from
describing properly the dynamics in the neighborhood of scission. In a
pragmatic approach, the description of fission dynamics with the TDGCM should
therefore always be stopped with di-nuclear configurations before scission,
where the two prefragments still significantly interact by nuclear forces. A
possible way to overcome this issue consists in expanding the collective space
to include intrinsic excitations, e.g., sets of two-quasiparticle states built
on top of constrained HFB vacuua [399]. Such an extension would allow
incorporating both some amount of collective dissipation in the theory and the
possibility to describe excited nascent fragments. Let us also mention an
alternative approach that consists in deriving a quantum theory of transport
that combines a GCM description of collective dynamics with quantum statistics
to account for thermal excitations and dissipation [400].

###  4.5 Comments on Adiabaticity and Dissipation

The TDGCM, ATDHFB and adiabatic self-consistent collective (ASCC) methods are
often referred to as adiabatic theories of large-amplitude collective motion.
However, while this adjective has a clear definition in classical
thermodynamics, it is often employed with a different (or several different)
meaning(s) in fission theory and the theory of large-amplitude collective
motion in general. In the literature, the concept of adiabaticity has been
used as a proxy for three very different hypotheses, which are sometimes
combined: (i) the collective motion, i.e., the change of shape leading to
fission, is very slow [278, 367], (ii) the coupling between such collective
degrees of freedom and single-particle, or intrinsic degrees of freedom is so
small that it can be neglected [401, 197, 402] and (iii) the intrinsic DoFs