are associated with the lowest-energy configuration that evolves with the
collective variable [399, 340]. The variety of these definitions along with
the rather arbitrary definitions of collective and intrinsic excitations may
lead to confusions. We discuss below some common pitfalls related to this
topic.

As a first example, theoretical evidence from TDDFT clearly suggests that the
collective motion is strongly dissipative, i.e. the shape of the system
evolves slowly and toward configurations which minimize the collective
potential energy while the system populates single-particle states with high
energies [354, 341, 342]. Overall, TDDFT predicts time scales in the range of
$10^{-20}-10^{-19}$ s to go from the saddle point to scission. On the one
hand, this very slow collective motion justifies the reduction of TDDFT
dynamics to its adiabatic counterparts, specifically, the hypothesis of small
collective momenta that underpin, e.g., the ATDHF theory [368]. On the other
hand, TDDFT dynamics is often qualified of diabatic in the sense of point
(iii) above, because excited single- or quasi-particles states are populated.
One should thus be very careful when employing the adjective of adiabaticity
to refer to slow collective motion.

The definition (ii) of adiabaticity especially focuses on the dissipation of
energy from the collective DoFs to the intrinsic ones. Such dissipation is
known to be a crucial ingredient in the Langevin dynamics as mentioned in
Section 4.2. It appears in the form of the friction and random force terms in
the collective equation of motion. The TDDFT picture does not give direct
insight into such dissipation since it does not rely on an explicit splitting
of the DoFs between collective and intrinsic. It is possible to estimate a
posteriori the kinetic energy associated with any collective variable
represented by a local one-body operator [340]. Such an approach suggested a
strong dissipation of the quadrupole moment kinetic energy during the fission
dynamics. With this in mind, it might seem puzzling that the ATDHF approach
based only on the assumption of slow collective motion (which seems to be
valid) yields a collective equation of motion with no friction or dissipative
terms [57]. In addition, the TDGCM produces primary fragments distributions
similar to the one obtained by Langevin approaches while also introducing no
explicit friction term. This suggest that primary fragments mass/charge
distributions are simply not very sensitive to dissipation. In contrast,
predictions of the excitation energy of the primary fragments, which are not
well described by TDGCM, are a much better probe into the dissipation
mechanism.

In the case of ATDHF, there is another deeper reason for the absence of a
collective dissipation mechanism. Contrary to the current implementations of
TDGCM, which rely on a set of constrained HFB generator states, the original
formulation of the ATDHF(B) approach builds its collective path out of the
time-even component of the states in a TDDFT trajectory. These states can
already contain significant single- or quasi-particle excitations and only the
small energy component related to the residual time-odd part needs to be taken
into account by the collective dynamics. Following this idea, Reinhard et al.
proposed to build the collective path from a variational principle [401] that
decouples by design the collective dynamics from other DoFs. This philosophy
was formalized in the development of the ASCC model, which generalizes ATDHF
theory while solving some of its inconsistencies [204, 403, 196, 139, 66]. To
summarize, the collective phase space constructed formally in these approaches
is in fact completely different from the one used either in current TDGCM
implementations or in Langevin approaches. In other words, it is very possible
that some theories need an explicit treatment of collective dissipation while
some others do not. The presence and strength of dissipative terms in the
collective equation of motion strongly depends on the arbitrary choice of the
collective phase space. One should keep this in mind while discussing
adiabaticity in the sense of point (ii) above or comparing the dissipative
features of various theories.

###  4.6 Ternary Fission

Ternary fission is the rare process where the fission of a heavy element is
accompanied by the emission of an $\alpha$ particle or, much less frequently,
a light nucleus ranging from Lithium ($Z=3$) to Oxygen ($Z=8$) isotopes [404,
405]. More precisely, it is the (quasi)-simultaneous divide of the fissioning
nucleus in three fragments – one of them much lighter than the others. This
phenomenon is usually quantified by the $\alpha$-emission probability
$P_{\alpha}$ or, equivalently, the relative number $N_{\alpha}$ of emitted
$\alpha$ over the entire number of fission events $N_{f}$ [406]. Ternary
fission does not lend itself easily to a description in terms of geometric
collective variables as discussed in Section 2 unless one enforces some amount
of clustering, e.g. via a three-center basis [407]. Neither can such a process
be easily simulated by the simple ansatz of TDDFT, where the nuclear wave
function is a “standard” product state of particles or quasiparticles. In
fact, the proper quantum-mechanical treatment of ternary fission should
probably include a mechanism to rigorously simulate nuclear clustering. For
these reasons, all models of ternary fission tend to be rather
phenomenological. Probably the most advanced one relies on estimating
$P_{\alpha}$ by solving the time-dependent Schrödinger equation for the
$\alpha$ particle moving in a time-dependent mean-field potential, which is
obtained by simply parametrizing with $t$ specific trajectories across the
potential energy surface [408, 409]. Alternative methods involve the random-
rupture neck model, where it is assumed that the neck between the prefragments
ruptures at two different locations nearly simultaneously, the part of the
neck between these two locations forming the light particle [410]. The
rupture-neck model was also combined with statistical theory (see Section 4.1)
to account more realistically for the available excitation energy at scission
[411].

##  5 Emission of Light Particles

After the scission point, nuclear forces between the two fragments quickly
drop to zero while Coulomb repulsion drives the acceleration and kinematics of
the fission fragments. This is the acceleration phase of the fragments. We can
estimate an order of magnitude for the time scale of the acceleration from the
classical dynamics of two point-wise particles. Let us consider an initial
situation where the surfaces of the two fragments are separated by a distance
of a few fermis and neglect the relative velocity between the fragments at
this stage. Integrating the classical equation of motion results in a fast
acceleration of the two fragments that reach 90% of their final speed (2-5% of
the speed of light) within 5 zs. Within this time range, the fragments
typically cover a distance of several tens of fermis.

The large deformation of the prefragments at scission, the effect of the
Coulomb repulsion on the inhomogeneous charge densities in the fragments, as
well as the internal excitation created during the descent from the saddle
point to scission produce two excited fragments far from their ground-state
equilibrium. On average, about 30 MeV of excitation energy is shared between
the two fragments in the fission of an actinide while several theoretical and
experimental arguments indicate the typical spin of the fragments lies in the
range of 5-12 $\hbar$ units (cf. Section 6). The primary fragments lose their
energy mostly by emitting neutrons and $\gamma$ rays. The post-neutron pair of
fragments resulting from this neutron emission phase are called secondary
fragments. Further $\beta$ decay and possible delayed neutron/$\gamma$
emission may follow this process leading to a final pair of nuclei, the
fission products; see Fig. 1.

The characteristics of the emitted fission neutrons and $\gamma$ rays, such as
their multiplicity and energy spectrum, are key nuclear data for energy
applications and give stringent constraints on our understanding of the
primary fragments formation. State-of-the-art theoretical approaches
predicting these observables generally rely on a statistical deexcitation
model to simulate cascades of particle emissions from the excited fragments.
In the specific case of fission, such deexcitation simulations have to be
repeated for every combination of primary fragments pairs and set of relevant
initial properties that may impact the evaporation process. These generally
include the kinetic energy, excitation energy, spin and parity of the primary
fragments. The final observables are then obtained by averaging fission events
both over the large set of primary fragments configurations as well as over
the set of deexcitation cascades for each of these configurations. Each event
is weighted by its associated probability of appearance. This averaging can be
performed in a deterministic way [412] as long as the phase space considered
for fission events is small enough. For more detailed descriptions of
deexcitation cascades including for instance the spin of the emitted particles
the phase space of fission events becomes so large that fragments deexcitation
models often turn to a Monte Carlo estimation of observables. In this case,
the code sample a large but limited number of fission events according to
their probability distribution and the observables are deduced as averages
over these events.

The success of such approaches relies on high-accuracy and high-precision
estimates of the primary fission fragment distributions, especially for the
total kinetic energy, which directly determines the total energy available for
neutron/$\gamma$ emission. As a consequence, it is customary to rely mostly on
experimental data or empirical models for the primary fragments distributions.
In this section we will give an overview of such empirical treatments. Recent
efforts directed toward replacing progressively these inputs by genuine
predictions from microscopic theories will be discussed in Section 6.

Code | PbP [412] | GEF [413] | FREYA [414] | CGMF [274] | FIFRELIN [415]
---|---|---|---|---|---
Pre-fission emission | – | n${}_{\text{pe}}$,n${}_{\text{stat}}$,$\gamma$ | n${}_{\text{pe}}$,n | n,$\gamma$ | –
Wide range of fissioning systems | no | yes | no | no | no
Neutron-$\gamma$ competition | no | yes | partial | full | full
Gamma deexcitation | no | yes | yes | yes | yes
Deexcitation model | Weisskopf | Weisskopf | Weisskopf | Hauser-Feshbach | Hauser-Feshbach
Statistical fluctuation | no | no | no | no | yes
Open-source | no | yes | yes | yes | no
Table 1: Summary of some features of fission fragments deexcitation models. In
the pre-fission emission row, npe denotes pre-equilibrium neutrons and
n${}_{\text{stat}}$ corresponds to statistical neutrons. ’Statistical
fluctuation’ denotes taking into account fluctuations of nuclear structure
properties on top of average quantities such as the level density and the
averaged partial widths.

The statistical description of nuclear deexcitation cascades is by itself an
old problem quite independent of fission studies. It started with the seminal
paper of Weisskopf in 1937 concerned with the spectrum of neutrons emitted
from highly excited nuclei [265]. Grover et al. pioneered the use of the
Hauser-Feshbach model of deexcitation for the study of neutron, $\gamma$ and
$\alpha$ evaporation [416, 417, 418] and published the first study quantifying
the impact of the spin of the fission fragments on such decays [419]. Many of
the developments of statistical reaction theory in the 1970ies and 1980ies
briefly mentioned in Section 3.2.2 aiming at going beyond Hauser-Feshbach
theory [272, 273] were not really tested in practical simulation of fission
fragment decay. Interest for this subject surged in the 2000’s, in particular
due to the growing issue of $\gamma$ heating in nuclear reactors [420, 421,
422] as well as new technological applications such as neutron/$\gamma$
interrogation or advanced Monte Carlo simulations of detection devices.

Today, several codes, implementing different models, are dedicated to the
description of fission fragment deexcitation. In this section we will only
review the most commonly-used and complete ones: FREYA [414], GEF [413], CGMF
[274], FIFRELIN [415] and the Point-by Point-(PbP) [412] model. We summarize
in Table 1 the main features of these codes. The more recent code HF3D [423,
424] will not be discussed since it implements a similar physics as CGMF with
a different numerical resolution (deterministic for HF3D versus Monte Carlo
for CGMF). Other codes implement nuclear deexcitation cascades for specific
applications, for example GEMINI++ for the deexcitation of fragments produced
in heavy-ion reactions [425]. Conversely, general-purpose codes such as TALYS
[426] or EMPIRE [427] have a much broader scope of applications and aim at
describing any type of nuclear reactions. For the sake of completeness, let us
also mention the work of Lestone [428] focused on a very precise reproduction
of the prompt fission neutron multiplicities, spectrum and angular
correlations.

###  5.1 Distribution of Primary Fragments Properties

The properties of the primary fission fragments could in principle be obtained
as the result of a statistical or dynamical approach to the large-amplitude
collective motion as outlined in Section 4. However, one should keep in mind
that predictions of prompt particle emission are very sensitive to the
characteristics of the primary fragments. For instance, microscopic theories
can predict the kinetic energy of the fragments at best within a few percent
precision [342]. In the fission of an actinide, this would typically lead to
an uncertainty of a few MeV on the total excitation available for
deexcitation, which would translate into a $0.5-1$ uncertainty on the average
prompt neutron multiplicity. For comparison, measurements of the prompt
neutron multiplicity can typically reach precision below one percent ($\simeq
0.015$ neutrons for 252Cf spontaneous fission in Ref. [429]).

Adopting a pragmatic strategy, most fragments deexcitation models thus start
by estimating the primary fragments distributions either by directly using
some available experimental data or by relying on empirical models previously
calibrated to a collection of experimental data. These distributions of
probability are then used as an input to the statistical deexcitation model.
Note that such an approach discards any quantum correlations between the
various primary fragments configurations. We now highlight the different
ingredients and models used to build the empirical distribution of the primary
fragments configurations.

##### Proton and neutron numbers

The principal characteristics of the primary fragments configurations are the
number of protons and neutrons in each fragment. In the standard approach the
probability distribution that fission produces a fragment with a number of
mass $A_{\rm f}$ and a number of charge $Z_{\rm f}$ given the excitation
energy $E^{*}_{\text{CN}}$ of the compound system is factorized into

$P(A_{\rm f},Z_{\rm f}|E^{*}_{\text{CN}})=Y(A_{\rm
f}|E^{*}_{\text{CN}})P(Z_{\rm f}|A_{\rm f}).$ (32)

The factor $Y(A_{\rm f}|E^{*}_{\text{CN}})$ stands for the primary mass
distribution whereas $P(Z_{\rm f}|A_{\rm f})$ accounts for the charge
probability distribution given the mass number $A_{\rm f}$. The complementary
fragment is always determined through mass and charge conservation (i.e.
ternary fission is neglected)

$A_{f2}=A_{\rm CN}-A_{f1},\qquad Z_{f2}=Z_{\rm CN}-Z_{f1}.$ (33)

Fragments deexcitation models assume a normal distribution of charge with a
mass-dependent mean $\bar{Z}_{f}(A_{\rm f})$ and standard deviation
$\sigma_{Z}(A_{\rm f})$

$P(Z_{\rm f}|A_{\rm f})\propto\text{exp}\left[-\frac{1}{2}\left(\frac{Z_{\rm
f}-\bar{Z}_{f}(A_{\rm f})}{\sigma_{Z}(A_{\rm f})}\right)^{2}\right].$ (34)

The mean of this distribution is either directly determined from the Unchanged
Charge Distribution (UCD) approximation (as in FREYA) or includes an
additional polarization charge effect. In the later case, the polarization of
charge is parameterized as a function of $A_{\rm f}$ (PbP model) or estimated
from the Wahl’s systematics [430] (FIRELIN, CGMF).

As for the mass distributions, a possible strategy relies on using
experimental data, either in tabulated form, or described in terms of a few
fission modes [178]. The codes FIFRELIN and PbP model implement such an
approach that is limited to fissioning systems and energies for which
experimental data is available. To overcome this issue, several deexcitation
models favor a description of the primary mass yields as a sum of a three or
five Gaussians. For a given fissioning system, the position (average mass),
the width as well as the weight of each Gaussian (also known as fission modes)
follow empirical laws fitted to reproduce a set of experimental data spanning
a range of excitation energies. For example, the code CGMF implements a
parameterization of the mass yields with 3 Gaussians whose means and widths
linearly depend on the excitation energy [424, 274]. A similar technique is
adopted in FREYA based on five Gaussians with energy-independent means and a
quadratic dependency of the widths [431]. Such an approach enables systematic
studies of the prompt particles evolution as a function of the input channel
energy. Because of the remaining dependency on experimental data, this
approach only enables calculations on a few well known fissioning systems.

Going one step further, the philosophy of the GEF code is to provide an
empirical yet systematic estimation of the mass yields for a wide range of
energies and fissioning systems. In this model, the fissioning system is
described by an elongation degree of freedom coupled to several transverse
collective DoFs such as the mass of the fragments, and to a thermal bath
describing the remaining degrees of freedom. The distribution of the neutron
and charge number of the fragments is determined from a two-step statistical
process guided by the physics of the collective dynamics,

$P(A_{\rm f},Z_{\rm f})=P(\text{mode})P(A_{\rm f},Z_{\rm f}|\text{mode}).$
(35)

In the first step, a statistical population of the available states in the
neighborhood of the fission outer saddle point yields the probability for the
system to choose a particular fission mode. The probability associated with a
fission mode $P(\text{mode})$ is proportional to

$P(\text{mode})\propto\text{exp}\left(-\frac{E^{0}_{\text{mode}}}{kT}\right),$
(36)

where $E^{0}_{\text{mode}}$ is the minimum of the potential energy of the mode
at an elongation corresponding to the outer barrier, relatively to the energy
of the lowest mode, and $T$ is the temperature of the system. In practice,
five to seven modes are considered. A second step determines the distribution
of the particle numbers in the fragments $P(A_{\rm f},Z_{\rm f}|\text{mode})$
from the properties of the system close to scission. The potential energy
surface in the transverse direction associated with the charge and mass
numbers are described as harmonic potentials functions of $Z$ and $A$ with a
minimum value at $Z_{\text{mode}}$ and $A_{\text{mode}}$. A second statistical
population of states in the neighborhood of scission configuration then leads
to the following distribution in each mode:

$P(A_{\rm f},Z_{\rm f}|\text{mode})=P(A_{\rm f}|\text{mode})P(Z_{\rm f}|A_{\rm
f},\text{mode}),$ (37)

with

$P(A_{\rm f}|\text{mode})\propto\text{exp}\left(-\frac{(A_{\rm
f}-A_{\text{mode}})^{2}}{2\sigma^{2}_{A,\text{mode}}}\right),\quad P(Z_{\rm
f}|A_{\rm f},\text{mode})\propto\text{exp}\left(-\frac{(Z_{\rm
f}-Z_{\text{mode}})^{2}}{2\sigma^{2}_{Z,\text{mode}}}\right).$ (38)

The widths $\sigma_{A,\text{mode}}$ and $\sigma_{Z,\text{mode}}$ depend in
particular on the energy balance estimated for each mode. Note that a special
treatment is applied to the standard 2 mode ($S_{2}$) with a non harmonic
potential. The GEF model finally includes the odd-even staggering of the
fission yields as an additive correction based on an equal-filling assumption
of the intrinsic states. Overall, the determination of the primary fission
yields relies on a set of free parameters that are calibrated to a large body
of experimental data, which includes high-resolution data from inverse
kinematic experiments [432]. This approach is then capable of estimating
systematically and with accuracy the primary fragments yields as a function of
the mass, charge and energy of the fissioning nucleus. This feature is quite
unique, and other fragment deexcitation models sometimes directly take the
yields predictions from GEF as an input to their own deexcitation engine [433,
434, 435].

##### Energy sharing

The excitation energy available in each primary fragments before particle
evaporation is a crucial ingredient, since it is the primary driver of the
prompt neutron emission. To estimate this quantity for any given mass and
charge split, most of the deexcitation codes rely on an energy balance of the
system before the reaction and after the complete acceleration phase of the
fragments. For neutron-induced fission, the energy balance reads [422]

${\rm TXE}+{\rm TKE}+M(Z_{\rm H},N_{\rm H})+M(Z_{\rm L},N_{\rm
L})=M(Z,N)+S_{n}(Z,N)+E_{n},$ (39)

where $M(Z,N)$ is the mass of the nucleus $(Z,N)$, $S_{n}$ the neutron
separation energy in the nucleus $(Z,N)$, $E_{n}$ the energy of the incident
neutron in the center of mass frame, TXE the total excitation energy available
to both fragments, and TKE the total kinetic energy of the fragments. In the
case of spontaneous fission, this formula can be applied with $S_{n}=E_{n}=0$;
for photofission, $S_{n}=0$ and $E_{n}\equiv E_{\gamma}$. The evaluation of
masses and neutron separation energy is performed on a regular basis (see
[436, 437] for the latest one) and data are available in nuclear structure
databases such as ENSDF [29], leaving only the total kinetic energy and the
total excitation energy to determine. Because of the quantum fluctuations in
fission, we can treat TKE as a random variable with a large variance. In
fragments deexcitation codes, its probability distribution is typically
assumed Gaussian [274] with a mean and variance that depend on the mass split

$P(\text{TKE}|A_{\rm f},Z_{\rm
f},E^{*}_{\text{CN}})\propto\operatorname{exp}\left[-\frac{1}{2}\left(\frac{\text{TKE}-\overline{\text{TKE}}(A_{\rm
f})}{\sigma(A_{\rm f})}\right)^{2}\right].$ (40)

These moments of the probability distribution are in general extracted from
experimental measurements of the fragments kinetic energies (e.g. Ref. [438])
and may also include a dependency on the energy of the entrance channel [274].
Combining equations (40) and (39) yield a distribution of probability for the
total excitation energy TXE available for the emission of particles.

Note that such a scheme neglects the possible emission of particles
(especially neutrons) before the complete acceleration of the fragments.
Although this hypothesis is strongly supported by the angular distribution of
the prompt neutrons [385], which is peaked at forward angles, it remains an
open question whether, and to which extent, some of the neutrons are emitted
before the complete acceleration. Numerous papers by Carjan and Rizea attempt
the challenging prediction of scission neutrons and/or pre-acceleration
neutrons based on a time-dependent average nuclear potential approach [439,
440, 441]. Other studies were performed based on static potentials [442] and
even on a full TDHFB calculation [341]. As of today, no clear consensus exists
between the results of these theoretical approaches and experimentally
verifying their predictions remains difficult. In practice, most deexcitation
codes are neglecting this kind of neutron emission.

After the determination of the total excitation energy available, one still
needs to split this energy between the two fragments. This energy sharing
results from the complex competition between various energy reservoirs
(intrinsic energy, deformation energy, etc) that interact with each other
during the fission dynamics. The repartition of the energy strongly depends on
the mass of the fragments and it is expected to be highly correlated with the
prompt neutron multiplicity $\bar{\nu}(A)$. Fragments deexcitation codes
therefore rely on empirical laws for the total excitation energy sharing,
which are guided by, or directly fitted on, this neutron multiplicity. The
code CGMF assumes for instance that the excitation energy is of thermal nature
and build an empirical law for the temperature of the fragments as a function
of its mass number. The code FREYA follows a similar approach with the
introduction of a parameter that governs the ratio of temperatures of the two
fragments. In FIFRELIN, the excitation energy in each fragment is recast into
the sum of a rotational and a thermal contribution, with again a mass-
dependent temperature ratio. The point by point model (PbP) describes the
excitation energy in terms of a thermal excitation and a deformation component
that depends on the mass. Finally, the GEF model shares the major part of TXE
according to a maximum entropy principle at scission and the remaining
collective kinetic energy is split evenly between the fragments. Despite the
apparent diversity of assumptions in all the models, we note that the shape
relaxation and the mass split dependency are the key ingredients that enable
all these models to reproduce a sawtooth shape for the neutron multiplicity
$\bar{\nu}(A)$.

##### Spin-Parity Distribution

As will be shown in Section 5.2, the spin distribution of the fission
fragments plays a very important role in setting the photon multiplicities,
i.e., the average number of photons emitted during the deexcitation. The
statistical theory assumes that the probability for a nucleus to have spin $J$
is approximately given by [443]

$p(J)\propto(2J+1)\exp\left[-\frac{1}{2}\frac{(J+\tfrac{1}{2})^{2}}{\sigma^{2}}\right],$
(41)

where $\sigma$ is called the spin-cutoff parameter. Note that the spin-cutoff
parameter is proportional to the expectation value $\braket{\hat{J}^{2}}$ of
the total angular momentum in the system – the mean of the actual distribution
$p(J)$ [444, 445]. In the Fermi gas model, it is proportional to the moment of
inertia of the system as well as to its temperature (or excitation energy)
[446]. Given this background, we can now summarize the different features of
each deexcitation code:

  * •

CGMF – The spin of each fragment is sampled from the distribution (41). The
spin-cutoff parameter is explicitly parametrized by the product of the nuclear
moment of inertia $\mathcal{I}$ and temperature $T$. The $Z$\- and
$A$-dependence of $\mathcal{I}$ is taken into account but not the variations
as a function of deformation, angular momentum or temperature [274]. There is
no constraint on the relative angular momentum between the fragments.

  * •

FIFRELIN – The treatment of the spin is very similar to CGMF, but with an
additional constraint used to simulate the damping of shell effects with
increasing temperature [447, 448].

  * •

FREYA – This code uses a semi-classical treatment of angular momentum [449].
It assumes that the system can be treated as a dinuclear (binary) system of a
heavy and a light fragment, each characterized by their moments of inertia
$\mathcal{I}_{H}$ and $\mathcal{I}_{L}$ and spins $\boldsymbol{J}_{H}$ and
$\boldsymbol{J}_{L}$, as well as a relative angular momentum $\boldsymbol{L}$.
Angular momentum is thus explicitly conserved in this treatment. The total
rotational energy of the dinuclear system is then expressed in terms of
specific rotational modes, which are sampled separately based on a statistical
formula [450]. The dependency on nuclear structure inputs is encoded in the
nuclear moments of inertia which, in most applications of FREYA, was
approximated by its $A$-dependent rigid-body value. Recently, this
approximation was improved following guidance from microscopic calculations
[449].

  * •

PbP – The spin of the fragments is not taken into account.

  * •

GEF – As for CGMF and FIFRELIN, the spin of the fragments is sampled from
(41), but this is only done for the post-statistical emission spin. The
statistical neutron and $\gamma$ emission is spin-independent.

Without any strong physical arguments as for the parity of the primary
fragments, all these models assume equiprobable parities.

###  5.2 Prompt Particle Emission

Starting from a highly exited state, the fission fragments first deexcite
through successive emission of particles. As pictured in the left panel of
Fig. 22, the dominant type of particle emitted depends on the excitation
energy and spin of the fragment. For fission fragments, the $\alpha$ emission
channel is usually closed and the proton emission can be neglected. At
energies above 10 MeV, the neutron emission channel is open and state-of-the-
art theories agree on the fact that it is largely favored. Just above the
neutron separation energy, the nucleus enters an area of competition between
the neutron and $\gamma$ emission. As we go lower in energy, the $\gamma$
emission channel becomes dominant up to energies of the order of 100 keV. At
first, this $\gamma$ emission is mostly of $E2$ character and is statistical
in nature and heavily fragmented [240, 451], i.e., one cannot recognize
individual rotational bands. When the nucleus has reached the lowest possible
energy at a given spin $I$, what is known as the Yrast line [98], discrete
$\gamma$ transitions take place all the way down to the ground state. Several
rotational bands possibly connected by fast $E1$ transitions may be populated.
At energies of the order of $\sim$ 100 keV, the internal conversion phenomenon
becomes important: the electromagnetic quanta emitted by the nucleus has a non
negligible probability to interact with the electrons of the atom, leading to
the emission of an internal electron accompanied by X-rays.

Figure 22: Left: Schematic of particle emission as a function of the
excitation energy. Right: Yrast plot showing decay mechanisms in the $E(I)$
plane. Neutron emission (in red) dominates at high excitation energy. Below
the neutron separation energy $S_{n}$ (grey shaded area), statistical $\gamma$
emission (dark green) takes over until the Yrast line is reached, at which
point discrete $\gamma$ transitions (green) occur until final deexcitation.
Fast inter-band transitions, typically $E_{1}$ connect different rotational
bands.

This prompt emission of particles takes place over a wide range of time
scales. Arguments related to the statistical evaporation of the neutrons lead
to typical emission times ranging from 10-18 to 10-13 s [452]. This upper
bound is actually the official definition of prompt neutrons for the United
State Department of Energy [453]. The prompt $\gamma$ emission mostly takes
place within a few nanoseconds. During this cascade, isomeric states of the
fission fragments can be populated leading to $\gamma$ transition at post-
scission times of up to several microseconds. The relative population of these
isomeric states, which is quantified by the branching ratios, has a strong
impact on various fission properties [444, 454, 455, 456].

State-of-the-art theoretical approaches aiming at predicting the deexcitation
process of the compound nucleus and/or the fission fragments are based on a
statistical deexcitation approach. In such a model, the dynamics of the
deexcitation process is not explicitly described. Starting from an initial
excited state $|i\rangle$ of the nucleus, the statistical model predicts the
probability for the emission of a series of particles leading to a final state
$|f\rangle$ for the residual nucleus.

In the statistical model picture, a deexcitation cascade is seen as a multi-
step process in which the system jumps from one intermediate state to another
while emitting each time one particle. A cascade of deexcitation is therefore
defined by (i) a series of transitions
$|i_{k-1}\rangle\rightarrow|i_{k}\rangle$ between nuclear states or
resonances, (ii) the particles $p_{k}$ emitted at each transition along with
their physical properties (kinetic energy, spin/multipolarity, etc). Owing to
the quantum nature of the deexcitation dynamics, any possible deexcitation
cascade or channel, generically denoted by the letter $c$ below, may happen
with a given probability. In standard approaches this probability is
factorized as a product of individual transition probabilities

$P(c)=P(i=i_{0}\rightarrow i_{1},p_{1})\times\cdots\times P(i_{k-1}\rightarrow
i_{k},p_{k})\times\cdots\times P(i_{n-1}\rightarrow i_{n}=f,p_{n}).$ (42)

The probability $P(i_{k-1}\rightarrow i_{k},p_{k})$ represents the transition
probability from the nuclear state $|i_{k-1}\rangle$ to the nuclear state
$|i_{k}\rangle$ with the emission of a particle $p_{k}$. Such a factorization
translates the Bohr hypothesis, which is assumed to be valid at each step of
the cascade. In particular, it neglects the possible quantum interferences
between the cascades. After determining the probabilities associated with all
possible deexcitation cascades, one computes a target observable as a
statistical average over all these events. For example, the average $\gamma$
multiplicity of the process is determined as

$M_{\gamma}=\sum_{c\in\mathcal{C}}P(c)M_{\gamma}(c),$ (43)

where $P(c)$ is the probability of occurrence of the cascade $c$ and
$M_{\gamma}(c)$ the number of $\gamma$ rays emitted during this cascade. A key
aspect of the method is the proper determination of the set of all possible
cascades, which is related both to the structure of the nuclei involved and to
the probabilities $P(i_{k-1}\rightarrow i_{k},p_{k})$ driven by the strength
of the particle emissions.

At low excitation energies (below a few MeV), the structure of most of the
fission fragments is partially known from $\gamma$ spectroscopy. The ENSDF
database records the first discrete energy levels, their spin and parity,
electronic conversion coefficient and possibly their transition rates to other
discrete levels. Deexcitation models such as FIFRELIN, CGMF or FREYA include
this experimental structure information in their deexcitation engine through
tabulated databases such as the RIPL library [446]. At higher excitation
energies the structure becomes unknown and ultimately neighboring neutron
resonances are overlapping. To tackle this energy range, deexcitation models
rely on a statistical description of nuclear structure [457, 458, 459]
identical to the ideas developed for the compound nucleus in the context of
neutron-induced reactions (see Section 3). The level scheme is expressed in
terms of a level density that encodes the average number of states/resonances
in the neighborhood of a given excitation energy. An additional description of
the fluctuation of the the level spacing built from the Gaussian orthogonal
ensemble (GOE) may complete the level density information. A statistical
description also applies to the probabilities of transfer from one level to
the other. The transmission coefficient $T_{a}(\epsilon)$ encodes the average
probability to emit a particle characterized by the output channel $a$ and the
energy $\epsilon$. On top of this mean value, one can take into account the
so-called Porter-Thomas fluctuations [460] or Ericson fluctuations [461] at
higher energies. These concepts form the backbone of fragment deexcitation
models. Their detailed implementation varies significantly from one model to
another with two major trends: models based on a Weisskopf-Ewing deexcitation
spectrum and models based on the Hauser-Feshbach framework.

##### Weisskopf-Ewing evaporation

Starting from an excited state of the nucleus, the Weisskopf-Ewing approach
expresses the energy spectrum of a particle evaporation. It originates from
the seminal paper of Weisskopf focused on neutron emission far above the
neutron separation energy [265]. The Weiskopf evaporation spectrum
$\chi(\epsilon)$ in the center of mass frame reads

$\chi(\epsilon)\propto\sigma_{\text{CN}}(\epsilon)\,\epsilon\,\text{exp}\left(-\frac{\epsilon}{T}\right),$
(44)

where $\epsilon$ is the kinetic energy of the emitted particle (e.g.
neutrons), $\sigma_{\text{CN}}(\epsilon)$ is the inverse cross section for the
formation of the excited compound nucleus by the process of the particle
absorption and $T$ is the temperature of the residual nucleus (after neutron
emission). Such a spectrum results from the summation over all possible
transitions for the particle energy $\epsilon$ [187]. This implies that the
detailed structure of the nuclei involved in the deexcitation cascade does not
need to be computed within this approach. Also, the spin/multipolarity of the
particle and the spin-parity of the residual state are not explicit taken into
account.

The deexcitation engine of the codes FREYA, PbP and GEF relies on the
Weiskopf-Ewing evaporation spectrum. For the neutron emission, the probability
$P(i_{k-1}\rightarrow i_{k},p_{k})$ of a transition is proportional to Eq.
(44). The PbP model relies on an optical model calculation for the inverse
cross section $\sigma_{\text{CN}}(\epsilon)$ whereas FREYA assumes it is
constant and GEF takes its energy dependency as $1/\sqrt{\epsilon}$. The
Weisskopf-Ewing approach does not provide the relative probability of the
neutron emission versus $\gamma$ emission. These deexcitation models therefore
assume that successive neutron emissions take place first, followed by the
emission of $\gamma$ rays (see Fig.22). The excitation energy at which the
nucleus switches to $\gamma$ emission is set to $S_{n}$ in the case of the PbP
model. To better account for the neutron/$\gamma$ competition, FREYA
implements an empirical shift of this energy threshold toward higher energies.
Another option explored by GEF consists in using an empirical formula for the
relative neutron versus $\gamma$ probability that only depends on the
excitation energy of the fragment. Once the neutron evaporation is completed,
the $\gamma$ emission can be treated in a similar manner. For instance, FREYA
uses the following Weiskopf-Ewing spectrum for $\gamma$ emission

$\chi_{\gamma}(\epsilon)\propto\epsilon^{2}\,\text{exp}\left(-\frac{\epsilon}{T}\right).$
(45)

Contrary to a complete Hauser-Feschbach approach, such Weisskopf-Ewing
evaporation is expected to fail when the level density of the residual nucleus
starts to be discrete. The work of Marcath [462] compares the predictions
obtained from FREYA and the Hauser-Feshbach code CGMF to experimal measurement
on 252Cf spontaneous fission. It highlights quantitative differences between
the models predictions but an overall agreement of both models to experimental
data.

##### Hauser-Feshbach Formalism

As briefly descried in Section 3.2.2, the Hauser-Feshbach approach associates
with each open channel (neutrons, $\gamma$, fission, etc.) a partial width
$\Gamma$ that is inversely proportional to the characteristic time of the
corresponding transition. The probability of a transition is directly
proportional to its associated partial width

$P(i_{k-1}\rightarrow i_{k},p_{k})=\frac{\Gamma(i_{k-1}\rightarrow
i_{k},p_{k})}{\sum_{i^{\prime}p^{\prime}}\Gamma(i_{k-1}\rightarrow
i^{\prime},p^{\prime})}.$ (46)

Contrary to the Weisskopf-Ewing approach, the partial widths depend here on
all characteristics of the emission channel, including the spin of the emitted
particle as well as the spin-parity of the residual nucleus. In addition, this
approach naturally accounts for the competition between different kinds of
emissions. Computing the probability distribution of one emission requires the
knowledge of the structure (discrete levels or statistical level scheme) of
all possible residual nuclei. The implementation of one decay step amounts to
(i) listing all possible emission channels starting from the initial state,
(ii) determining the structure of all potential residual nuclei, (iii)
computing the partial widths of all possible transitions, (iv) computing the
sum of these partial widths to normalize the distribution of probability. In
the context of fission, the codes CGMF an FIFRELIN implement such an Hauser-
Feschbach scheme.

At low excitation energy, the partial widths may be known from $\gamma$
spectroscopy and directly incorporated into Eq. (46). For higher excitation
energies, one relies on a statistical description of these quantities. For
neutron emission, an optical model calculation determines the average partial
width from

$\bar{\Gamma}_{n}(i\rightarrow
f,l,j)=\frac{T_{l,j}(\epsilon)}{2\pi\rho(E_{i},J_{i},\pi_{i})}.$ (47)

The transmission coefficient $T_{l,j}(\epsilon)$ corresponds to a neutron
emission with an orbital moment $l$, a total spin $j$ and a kinetic energy
$\epsilon$. In the denominator, $\rho(E_{i},J_{i},\pi_{i})$ stands for the
spin-parity-dependent level density of the mother nucleus. The average
$\gamma$ partial width associated with the emission of a $\gamma$ ray of
multipolarity $XL$ ($X=E,M$) and energy $\epsilon$ is given by

$\bar{\Gamma}_{\gamma}(i\rightarrow
f,XL)=\frac{\epsilon_{\gamma}^{2L+1}f_{XL}(\epsilon)}{\rho(E_{i},J_{i},\pi_{i})}.$
(48)

The $\gamma$-strength function $f_{XL}$ contains the information related to
the inverse cross section reaction. Although it can be inferred from the
results of QRPA calculations [463, 464, 465], it is most often parameterized
as a Lorentzian shape corrected for various physical effects (Pigmy resonance,
low energy shift). Such a parameterization is rooted in the Brink-Axel
hypothesis that establishes a simple link between the $\gamma$-strength
function depending only on the energy and multipolarity, and the
photoabsorption cross section from any excited state [466]. More elaborate
models such as the Enhanced Generalized LOrentzian (EGLO) overcome this
assumption by including a temperature dependency in the gamma strength [446].

These partial widths correspond to probabilities of transition averaged over
the detailed nuclear structure near the initial and final states $i_{k-1}$ and
$i_{k}$. Significant variations from these averages can rise due to the
Porter-Thomas fluctuations [460] (fluctuation in the strength of the decay) or
Ericson fluctuations at higher energies [461]. Different methods are possible
to take into account these fluctuations [467]. Although their impact on the
deexcitation observables is weak in general, it proved to be of significant
importance in the area where the number of open neutron channels comes close
to unity.

###  5.3 Pre-fission Particle Emission

If the entrance channel of the reaction supplies sufficient energy to the
system, the emission of particles before fission happens is likely. For
example, in the neutron-induced fission of actinides with fast neutrons, the
emission of neutrons before fission leads to the well-known phenomenon of
second- and third-chance fission: not only does the original compound nucleus
with $Z$ and $N$ neutrons undergoes fission, but so does the isotope with
$N-1$ neutrons (second-chance), $N-2$ neutrons (third-chance), etc. Since such
processes carry away part of the energy and spin of the initial system, it is
essential to take them into account to obtain realistic estimates of the
prompt emission characteristics. The fission models discussed here may take
into account this phenomenon with different degrees of refinement.

As already mentioned in Section 3, fission competes with other light particles
channels and the ratio of the total widths of each channel gives their
respective probability. For instance the fission probability reads

$P_{f}=\frac{\Gamma_{f}}{\Gamma_{f}+\Gamma_{n}+\Gamma_{\gamma}}.$ (49)

Codes like FIFRELIN or PbP do not implement this competition mechanism and
assume a probability one for fission. This restricts their usage at energies
below the threshold of emission before fission. Other codes like GEF and FREYA
account for pre-fission emission of neutrons and/or $\gamma$ rays. To do so
they parameterize the total widths of pre-fission emissions channels with
analytical formulas. GEF evaluates for instance the neutron and $\gamma$
channels from the compound nucleus mass, temperature of the intrinsic DoFs at
different steps of the fission process as well as the fission barrier [413].
In FREYA, only neutron emission is considered before the fission process (no
$\gamma$). Both statistical and pre-equilibrium neutrons emission are
possible. The ratio $\Gamma_{n}/\Gamma_{f}$ is determined based on the level
density of the residual nucleus. For the case of fission, the density
considered is at the top of the fission barrier. (Eq. 2 of [414]); see [468]
for more detailed information. Finally, the code HF3D, a deterministic
equivalent of CGMF, computes the fission widths coming from the Hill-Wheeler
formula (20) like in TALYS [426] and EMPIRE [427]. In these calculations, the
fission width competes with possible neutron- and $\gamma$-emission widths
depending on the energies and possibly multipolarities of the emission: this
is a fully fledged Hauser-Feschbach calculation. Additional details are given
in [424].

###  5.4 Applications of Fragments Deexcitations Models

The deexcitation codes developed specifically to describe the decay of fission
fragments are obviously important tools to evaluate the prompt fission
spectrum or provide constraints on the theoretical models aiming to describe
the physics of scission. In recent years, their range of application has also
grown to encompass more specific applications in nuclear technologies or
transport codes. In this section, we summarize some of these developments.

####  5.4.1 Benchmarks of Prompt Observables

The fission fragments (FF) deexcitation models have been extensively compared
to the available experimental data on prompt neutron and $\gamma$ rays
characteristics during the last decade. Most of the relevant literature is
focused on the spontaneous fission of 252Cf, because of the quality of the
experimental data, and on the thermal neutron-induced fission of 235U and
239Pu because of their importance in nuclear technology. Global benchmarks of
the models with experimental data can be found in Refs. [415] for FIFRELIN,
[468] for FREYA, [412] for PbP, [413] for GEF and [469, 470] for CGMF. A few
benchmark comparisons between some of these models have also been published
[471].

Figure 23: Left: Average prompt neutron multiplicity of 235U(nth,f) as a
function of the primary fission fragment mass. Results predicted by the BeoH
code compared to experimental data. Figure reproduced with permission from
[424] courtesy of A. Lovell; copyright 2013 by The Amercian Physical Society.
Right: Prompt neutron spectrum from the thermal induced fission on 235U. The
spectrum is represented as its ratio to a Maxwellian with temperature
$T=1.341$ MeV. The predictions obtained with the FF deexcitation models
FIFRELIN, CGMF, FREYA and PbP are compared using the same primary FF mass and
charge yields. Figure reproduced with permission from [472] courtesy of R.
Capote; copyright by Elsevier.

One of the key observables in FF deexcitation is the average neutron
multiplicity as a function of the mass/charge of the primary fragment. The
number of emitted neutrons is indeed closely related to the available
excitation energy in the primary FFs and presents a so-called sawtooth shape
as shown in the left panel of Fig. 23. This sawtooth behavior is mostly
understood as coming from the shape relaxation of the fragments from their
configuration close to scission to their ground-state deformation. Typically,
heavy fragments close to 132Sn are assumed to be produced only with a low
deformation at scission, which translates into a low excitation energy after
shape relaxation. This reasoning explains the dip in the observed neutron
multiplicity $\bar{\nu}(A)$ near $A=132$. Reproducing the sawtooth shape in
the neutron multiplicity has led to the current implementations of excitation
energy sharing between the fragments enumerated in Section 6.4. All
deexcitation models discussed in this section reproduce qualitatively this
sawtooth behavior. A widespread discrepancy compared to experiment was the
prediction of a minimum in the neutron multiplicity at $A=132$ instead of the
the measured A=$130$ in the fission of actinides. This deviation was recently
attributed to the assumption of shell closure at A=$132$ which is actually
shifted by a few mass units if one also considers octupole deformations in the
fragments [111].

Along with the neutron multiplicity, the prompt neutron spectrum is of
particular interest for nuclear technology applications. On this specific
topic, Ref. [472] reviews in details predictions obtained from the available
deexcitation models with regard to measurements and needs for evaluated data.
Despite the efforts of the community, deexcitation models do not yet predict
the prompt fission neutron spectrum (PFNS) with the accuracy required for such
applications. This is illustrated in the right panel of Fig. 23, which shows
the large discrepancy between the PFNS predicted by four state-of-the-art
deexcitation models. Recent studies suggested that modifications both of the
width of the FF distribution and of the spin assignment of a few low-lying
states in the FF structure could improve significantly the PFNS predictions
[423].

Figure 24: Left: Prompt $\gamma$ multiplicity of 252Cf(sf) predicted by FREYA
and compared with two experimental results. Figure reproduced with permission
from [468] courtesy of R. Vogt; copyright 2013 by the American Physical
Society. Right: Comparison of the prompt $\gamma$ spectrum of 252Cf predicted
by with FIFRELIN to experimental data. Figure reproduced with permission from
[415] courtesy of O. Litaize; copyright by Springer.

Figure 24 is representative of typical predictions for the prompt $\gamma$
multiplicity and spectrum. In general, the total prompt $\gamma$ ray
multiplicity is less known than the neutron multiplicity and differences in
theoretical predictions are more pronounced. The total $\gamma$ multiplicity
as a function of the primary fragment mass is particularly sensitive to the
initial distribution of spins of the fragments [469, 473]. This makes
deexcitation models a good tool to constrain our knowledge of the primary
fragments spins from $\gamma$ multiplicity measurements as discussed in
Section 6.3. The right hand side of Fig. 24 compares the low-energy part of
the prompt $\gamma$ spectrum of 252Cf spectrum obtained from FIFRELIN with
three sets of experimental data. The overall magnitude of the spectrum
reproduces the experimental spectrum within $\pm$1 $\gamma$ per fission. It is
also very noticeable that the deexcitation codes based on the Hauser-Feshbach
method reproduce very well the low-energy structures in the spectrum. This is
directly related to the fact that the low-energy structure of all possible FFs
is taken into account as an input of these codes, usually by reading it from
existing data libraries [446].

####  5.4.2 Probes into the Scission Mechanism

Beyond their ability to predict the prompt particles multiplicities and
spectra, most of the deexcitation models provide information on other
observables such as the multipolarity of the $\gamma$ transitions as well as
their correlations. Comparing these results with a large body of experimental
data puts stringent constraints on the empirical modeling of the properties of
the primary fragments and enables to infer information on the physics of
scission itself. A recent example is the study of the 237Np(n,f) reaction as a
function of incident neutron energy [473]. This paper shows that different
models with incompatible energy sharing mechanism and primary fragment spin
distributions reproduce equally well the experimental neutron multiplicity
$\bar{\nu}(A)$ (cf. Fig. 25) but differ significantly on their predictions on
the neutron-$\gamma$ correlation $M_{\gamma}(\bar{\nu})$.

Figure 25: Neutron multiplicity as a function of the primary fragment mass for
two excitation energies of the fission system 238Np. Increasing the excitation
energy of the system mostly increases the neutron multiplicity of the heavy
fragment. This behavior is correctly reproduced when assuming an energy and
mass dependent temperature ratio between the two fragments. Figure reproduced
with permission from [473] courtesy of O. Litaize; copyright 2013 by the
American Physical Society.

Many studies have also attempted to unfold the primary fragments spin
distribution by matching particle emission predictions to known measurements.
Work along this line have focused on the $\gamma$ multiplicity [470, 469, 474,
450] as well as well as on the isomeric ratios in the fragments [454, 475,
474]. We discuss in more details theoretical efforts to predict from
microscopic theory the primary fragments spins in Section 6.3.

####  5.4.3 Evaluated Data for Nuclear Technology

One possible application for deexcitation codes consists in contributing to
the evaluation of nuclear data. The fact that a model predicts the neutron and
$\gamma$-rays spectra and multiplicities in a consistent way is a unique
opportunity to augment the nuclear databases with correlated evaluations as
well as their associated uncertainties. The joint effort of Ref. [472] is a
first step in this direction, focusing on the evaluation of the PFNS based on
various models including the PbP model and FREYA. Covariance matrices of the
uncertainty of the PFNS at various outgoing neutron energies were computed
within FREYA. In practice,

Whereas alternative models exist to evaluate the prompt neutron spectrum,
deexcitation models are basically the only existing tools capable of
predicting prompt $\gamma$ observables. In 2000’s, the needs for better
nuclear data associated with the prompt $\gamma$ production were officially
formulated in the Nuclear Data High Priority Request List of the Nuclear
Energy Agency of the OECD [421]. This surge of interest was mainly due to the
topic of $\gamma$-ray heating in the structural materials of GEN-IV reactors,
especially in large reflectors [476]. For this reason, prompt $\gamma$
multiplicity distributions predicted with the CGMF code have been incorporated
in the ENDF-BIII.0 [477]. In this work the deexcitation code also supported
the evaluation process of the prompt $\gamma$ spectrum. Similarly, in the
JEFF-3.3 libraries, the code FIFRELIN was used to complete the experimental
prompt fission $\gamma$ spectrum data below and above experimental energy
thresholds [27]. These efforts illustrate a recent trend of including
deexcitation codes into the evaluation process. Further extensions of the
scope of such codes suggested by the work of Ref. [475] would be the
evaluation of the independent yields and of the branching ratios toward
metastable states in the deexcitation cascade.

####  5.4.4 Fission Event Generators in Neutron Transport Codes

In general-purpose neutron transport codes such as MCNP or TRIPOLI, fission
events and their associated emission of prompt particles are taken into
account in an average way. The number of prompt neutrons generated is
typically sampled between the two integers closest to the average neutron
multiplicity instead of a complete distribution ranging from 0 up to 10
neutrons. The correlations in energy or emission angles between the emitted
particles are neglected. This simple modeling of the prompt particles emission
turns out to be sufficient for criticality problems in large systems. On the
other hand, neutron transport codes are nowadays targeting new applications
requiring more precise inputs related to fission. This includes simulation of
detectors for experimental nuclear physics and neutron interrogation
technologies [478, 479]. Such applications often involve smaller geometries
for which the anisotropy of transported particles becomes important. The
development of these applications together with the capability of FF
deexcitations codes to predict angular correlations (see for instance [480])
is an incentive to use these models as fission events generators in neutron
transport codes.

As of today, the codes FREYA and CGMF can be used as fission event generators
in the MCNP6.2 [481] code and were also interfaced with the MCNPX-PoliMi
transport code [471]. On the other hand, the codes FREYA and FIFRELIN can also
be used within the TRIPOLI-4 transport code [482, 483]. Only a few
applications have been published so far. These include the simulation of a
prompt neutron-neutron angular correlation detection apparatus [483], which
would not have been possible without a fission event generator, and the study
of the neutron/photon multiplicities correlations measurement [462].

####  5.4.5 Experimental Data Analysis

Fission fragments deexcitation models often provide important insights into
the interpretation of experiments thanks to their ability to estimate the
sensitivity of various observables to an isolated physical effect. For example
the effect of the Doppler shift of the prompt $\gamma$-ray emission due to the
velocity of the FFs on the low energy $\gamma$ spectrum can be explicitly
calculated [484]. Taking into account this effect reproduces nicely the
experimental prompt $\gamma$-ray spectrum of 252Cf measured by Verbinsky et
al. [485].

Fission events generators are capable of estimating the impact of experimental
parameters such as the coincidence time window or the detection energy
threshold on the measured $\gamma$ multiplicity. These two parameters turn out
to be critical for the precise determination of the $\gamma$ multiplicity.
Typically, the late-time $\gamma$ emission from a few nanoseconds to a few
microseconds after scission represents a significant part of 3% to 7% of the
total number $\gamma$ rays [456, 486].

Finally, some specific features of the overall prompt particles spectra may
sometimes be assigned to a specific FF by a deexcitation code analysis.
Interpreting an experiment measuring the high-energy $\gamma$ spectrum between
0.8 and 20 MeV of 235U(n,f) with the code CoH, the authors of Ref. [487]
showed that the high-energy $\gamma$ ($>$ 10 MeV) mostly come from a few FFs.
They were also able to assign a bump in the spectrum near 4 MeV to emission
from the heavy fragments near the shell closure $Z=50$, $N=82$.

Beyond these applications in the context of fission, fragments deexcitation
codes contains general deexcitation models and have thus been used in other
fields such as the physics of neutrino detection [488, 489].

###  5.5 Delayed Emission

After the prompt emission, $\beta$ decay in the fission fragments can still
populate daughter nuclei with a significant amount of energy. Reaching the
ground state of the fission products requires the emission of delayed neutrons
and/or $\gamma$ rays. A typical fission of actinide will emit approximately
one delayed neutron every 100 fissions [430]. Most of the deexcitation codes
reviewed in this section do not simulate this delayed emission yet. There have
been a few studies performed with the HF3D code to simulate the delayed
neutron emission [475, 490]. They especially focus on the prediction of the
cumulative fission yields as well as of isomeric ratios.

##  6 Toward a Consistent Description of Fission

The deexcitation of fission fragments is largely dictated by the properties of
the primary fragments after their complete acceleration. We have shown in
Section 5 that state-of-the-art fragment deexcitation models rely on empirical
estimates of these properties, which are calibrated on experimental
measurements. Yet, both technological and astrophysical applications require
robust predictions of the light-particle emission resulting from the fission
of many short-lived isotopes over a broad range of energies. In many cases,
experimental data for such systems either is very incomplete or simply does
not exist. This provides a strong incentive to use theoretical models of the
formation of primary fragments to provide inputs to fragments deexcitation
codes. The past decade has witnessed tremendous progress in this direction
especially in the context of the EDF framework. In this section, we report on
such recent achievements, which bring us closer to a consistent description of
both the formation and deexcitation of fission fragments.

The EDF framework, whether explicitly time-dependent or not, provides
potentially powerful tools to characterize the primary fission fragments
starting from (effective) nuclear forces among nucleons. The most important
problem is to identify nearly-isolated fragments with good quantum numbers and
well-defined excitation energy. Until very recently, most work focused on
estimating the properties of the fragments close to scission were based on
average properties only (i.e. expectation value of one-body operators). For
instance the expectation value of any one-body, spatially local operator
$\hat{O}$ such as, e.g., particle number, angular momentum, multipole moments,
etc., in the left fragment can be computed simply from

$\braket{\hat{O}}_{\rm
L}=\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dy\int_{-\infty}^{z_{\rm
N}}dz\,\hat{O}(\boldsymbol{r})\rho(\boldsymbol{r}),$ (50)

where $z_{\rm N}$ refers to the position of the neck between the two
prefragments along the axis of elongation. Several pioneering studies of
neutron-induced fission relied precisely on such average estimates in the
neighborhood of scission [491, 374, 177, 492, 176, 161, 493]. The principal
deficiency of this procedure is that some results are not realistic. For
example, the particle numbers in the fragments typically take non-integer
values. In addition, static calculations do not deal with isolated fragments
but with entangled prefragments [176]. Although special procedures can be
designed to mitigate and somewhat quantify this effect, they do not completely
solve the problem [176, 161]. In this respect, TDDFT provides much cleaner
estimates since one can wait until the fragments have fully separated and only
interact with one another through the Coulomb repulsion [350]. In the
following, we are reviewing the recent improvements on the determination of
these primary fragments properties crucial for the simulation of light
particles emission.

###  6.1 Number of Particles

The number of particles in the fragments is the most basic property one would
expect a fission model to provide. Yet, the naive application of (50) almost
always gives fragments with non-integer numbers of particles – a problem made
worse in static theories where results depend on the definition of scission;
see Section 2.3. The solution is to extend well-known particle number
projection (PNP) techniques to the case of a fission fragment following an
idea originally proposed by Simenel for heavy-ion collisions [494]. In this
case, the “only” difference with standard projection is the definition of the
particle number operator, which now reads, e.g., for the left fragment,

$\hat{N}_{L}=\sum_{\sigma}\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dy\int_{-\infty}^{z_{\rm
N}}dz\,c^{\dagger}(\boldsymbol{r},\sigma)c(\boldsymbol{r},\sigma),$ (51)

where $(c^{\dagger}(\boldsymbol{r},\sigma),c(\boldsymbol{r},\sigma))$ are the
usual particle creation and annihilation operators. The first application of
this technique to fission was reported in [354]. It was later extended and
formalized in [495] and applied to estimate particle numbers for a large set
of static scission configurations of 240Pu and 236U [496, 384].

Figure 26: Left: Charge fragmentation probabilities for the light fragment for
a scission configuration in 240Pu. Each curve corresponds to a different size
of the neck between the prefragments. Figure reproduced with permission from
[495] courtesy of Verriere; copyright 2019 by The American Physical Society.
Right: Calculated post-neutron fission-fragment average neutron-to-proton
ratio $\braket{N}/Z$ versus fragment charge compared with experimental data
and the unchanged charge distribution (dashed line). Curves with blue and
black symbols rely on different estimates of the average number of emitted
neutrons. Figure reproduced with permission from [497] courtesy of Schmitt;
copyright 2021 by The American Physical Society.

The left panel of Fig. 26 illustrates that every single scission
configuration, which is characterized by an average number of particles as
estimated from (50), contains in fact of superposition of states with
different, integer-valued particle numbers. For the smaller values of the neck
radius, the method predicts the appearance of an odd-even staggering in the
charge distribution. This technique can be applied to both proton and neutron
numbers in every scission configuration. When combined with the time evolution
of the collective packet in the TDGCM to populate these scission
configurations (see Section 4.4), this method enabled the first prediction of
two-dimensional $Y(A,Z)$ isotopic primary fragments distributions within the
EDF framework [384].

Another method to obtain integer-valued particle numbers in the fragments has
been explored in the context of the microscopic-macroscopic models. The idea
is to rely on collective variables that specifically represent the average
number of particles in each fragment on a discrete mesh [498]. In addition to
giving integer numbers, this technique was used to probe different
neutron/proton ratios in the fragments and therefore computing isotopic yields
[497, 498]. The right panel of Fig. 26 shows an application of this technique
to compute the ratio of the mean number of neutrons per element in the
fragments, as a function of the fragment charge number. Results clearly
indicate that the popular Unchanged Charge Distribution (UCD) [499], which
assumes that this ratio does not depend on the fragment charge number and is
the same as in the fissioning nucleus, is not valid. This result, well known
from experiments, was also reproduced by microscopic calculations with PNP
[384].

###  6.2 Deformations of the Fragments

The deformation of fission fragments is another important property, which has
a direct impact on the number of emitted neutrons [500] and on the spin
distribution of the fragments; see also next section. One of the major
advantages of EDF over macroscopic-microscopic approaches is that these
deformations are predicted at scission, simply owing to the variational nature
of the HFB equation and the fact that deformations are outputs of the
calculations [177, 492]. By contrast, models based on parametrizing the
nuclear shape treat deformations as inputs: to obtain deformed fission
fragments, one has to explicitly set these deformations.

Figure 27: Identification of the heavy pre-fragment in asymmetric fission of
actinides. Localization functions of the fissioning nucleus at scission (upper
half of the contour plot in each frame) and localization function of the
octupole deformed 144Ba (lower-half of the contour plots). Figures reproduced
with permission from [111] courtesy of Scamps; copyright 2021 by Nature.

Figure 27 is a spectacular confirmation of this property. Systematic TDDFT
simulations of the fission of actinide nuclei showed that the heavy fragment
has a sometimes substantial octupole deformation [111]. This explains why the
charge of the most likely heavy fragment in actinide fission is not centered
around $Z=50$, as would naively be expected from considerations of spherical
shell effects in the fragments, but rather around $Z=52-54$, depending on the
fissioning system. This small shift is caused by the well-known octupole shell
closure at $Z=56$ [501].

###  6.3 Spin Distributions

In addition to particle number, projection techniques have also been recently
proposed to estimate the spin of fission fragments using angular momentum
projection (AMP), following again an idea borrowed from theoretical models of
heavy ion collisions [425]. The extension of AMP to fission fragments is very
similar to particle number and involves redefining the angular momentum
operator so that it acts only in the subspace containing a single fragment.
the method was tested both in static HFB calculations for nearly all mass
fragmentations of 240Pu [502] and in TDDFT for the most likely fission [349].
Note that in this context the projection is always performed after the
variational determination of the (TD)HFB many-body state. Results of static
calculations suggest that the average spin of the light fragment is higher
than the one of the heavy fragment and that shell effects at scission are very
important to set the initial amount of angular momentum in the fragment. This
is shown in Fig. 28. The left panel shows the average spin of the fragments
before and after statistical emission of photons, while the right panel shows
that the deformations of the fragments at scission can be very different from
the equilibrium deformations of the corresponding nucleus. Surprisingly,
static calculations (constrained HFB) and full TDHFB simulations give very
similar results – even if the fragments are much more excited in TDHFB than
they are in static HFB calculations. The additional insight of the time-
dependent calculations is to show that the intrinsic spins of the two FFs are
essentially perpendicular to each other.

Figure 28: Left: Average angular momentum of the fission fragments of 240Pu
before (top) and after (bottom) statistical emission. Right: spin
distributions of select isotopes near the doubly-close shell 132Sn (top).
Figures reproduced with permission from [502] courtesy of Marevic; copyright
2021 by The American Physical Society.

These theoretical results partly confirm recent measurements of the spin
distribution after the emission of statistical neutrons and $\gamma$ in three
actinide nuclei [503]. Specifically, the overall dependency of the spin on the
fragment mass is very similar, and both theory and measurements suggest that
the spin of the light fragment in the most probable fragmentation is larger
than the one of the heavy fragment. Such conclusions were also obtained
independently from studies of the fission spectrum with the code FIFRELIN
[473], where a light fragment spin higher than the heavy one naturally emerges
from the fit of the spin-cutoff parameter to the light and heavy neutron
multiplicities; see Section 5.2 for more details. Although several
parameterizations of this code may lead to similar results, it shows at least
that $\langle J_{L}\rangle>\langle J_{H}\rangle$ is not incompatible with a
qualitative reproduction of both prompt-neutron and prompt-$\gamma$
observables. Contrary to the argument put forward in [503], the most advanced
theoretical studies of the deexcitation of these fragments did not show
evidence that statistical photon emission leads to a constant shift of the
average angular momentum [504, 474]. All three different theoretical
approaches explain this result by the fact that the spin of the fragments is
largely dependent on their deformation at scission [504, 502, 349].

###  6.4 Excitation Energy

The determination of the excitation energy of fission fragments is more
difficult. Two approaches are possible: (i) estimate the total kinetic energy
TKE (or rely on existing measurements), which sets the value of TXE, and model
how this total excitation energy is shared among the fragments, i.e., with a
statistical model; (ii) try to directly estimate the individual excitation
energy of each fragment $E_{\rm H}^{*}$ and $E_{\rm L}^{*}$. In this case, the
total excitation energy is simply ${\rm TXE}=E_{\rm H}^{*}+E_{\rm L}^{*}$ and
can be used to reconstruct the value of the TKE.

Adiabatic models based on static PES give by definition cold fragments where
the excitation energy comes entirely from deformation effects and the fact
that a fragment $(Z,N)$ may have a different deformation from its ground-state
value. An estimate of the value of the excitation energy can still be set “by
hand” based on the total energy balance of the reaction at scission similar to
(39). In this case, the first step consists in determining the total kinetic
energy of the static configurations in the neighborhood of scission. Since
most of the kinetic energy carried away by the fragments derives from the
Coulomb repulsion between them, the most simple estimate of TKE is given by

${\rm TKE}\approx\frac{e^{2}}{4\pi\epsilon_{0}}\frac{Z_{\rm H}Z_{\rm L}}{R},$
(52)

with $R$ the distance between the two fragments, which can estimated from the
position of the two centers of mass [505]. Refinements of this formula involve
adding the pre-scission kinetic energy of the fragments, which can be
estimated as the collective energy term
$\frac{1}{2}\sum_{\alpha\beta}\dot{q}_{\alpha}\dot{q}_{\beta}$ in the context
of the Langevin equation (26); see, e.g., [506, 322, 323]. The remaining total
excitation energy of the system then needs to be shared between the fragments.
This mechanism of energy sharing, also called energy sorting, is often
described based on a statistical argument [507]: in the Fermi gas model, the
excitation energy of the system $E^{*}$ is related to the temperature through
$E^{*}=aT^{2}$, where $a$ is the level-density parameter [56]. If we assume
statistical equilibrium between the two fragments at scission, then the
temperatures satisfy $T_{\rm H}=T_{\rm L}$, and one can easily estimate
$E_{\rm H}^{*}$ and $E_{\rm L}^{*}$. Simulations of the fission spectrum
prompted a discussion of the validity of the equality between the two
temperatures and the need to phenomenologically parametrize the ratio
$R_{T}=T_{\rm H}/T_{\rm L}$ [508, 447, 509, 510, 511]. Recently, yet another
energy sharing mechanism was proposed, based on more realistic calculations of
level densities in the fragments [512, 513].

The second approach to estimating directly the excitation energy of each
fragment is the direct result of progress in microscopic fission theory,
especially real-time TDDFT simulations of fission events. In such cases, the
total energy of the system is conserved, and simulations can be run until the
two fragments are well separated (up to $\simeq 30$ fm between the fragments
in Ref. [349]). At that point, the excitation energy of the fragment is simply
the difference between the computed value and the ground-state binding energy.
In addition, the TKE can be computed directly as the relative kinetic energy
of the two moving fragments, rather than through the Coulomb energy proxy.
This approach was first outlined in [350] and applied in [355] in the context
of TDHF theory. With the inclusion of pairing in full TDHFB calculations,
estimates of TKE and fragment excitation energy in the low-energy fission of
nuclei such as 240Pu became possible [342] up to percent precision. Results
for the most likely fission suggest the two fragments do not have the same
temperature.

##  7 Conclusion

In spite of its applications in nuclear engineering and its essential role in
answering fundamental science questions about the stability of superheavy
elements or the formation of heavy elements in the Cosmos, the phenomenon of
fission remains shrouded in mysteries. In terms of sheer complexity, it has
few rivals as it compounds the challenges of any quantum many-body theory with
those of open quantum systems, nuclear forces and out-of-equilibrium
processes. From a nuclear theory perspective, fission is a deep and
unforgiving probe into many properties of atomic nuclei: small errors in
computing nuclear deformation properties can lead to orders of magnitude
discrepancies on actual observables.

In this article, we attempted to give as thorough a review on fission theories
as possible. We highlighted that there are, broadly speaking, three main
phases in the process: the probability for a nucleus to fission, the large-
amplitude collective motion leading to the production of primary fragments and
the deexcitation of these fragments after they are formed. The first phase
requires the tools of reaction theory to account for the competition between
fission per se and every other decay channel. Even though most of the
theoretical formalism was developed between the 1950ies and the early 1980ies,
lack of proper nuclear structure inputs has been the major bottleneck and
progress, while real, has been relatively slow. In contrast, it is fair to say
that the description of large-amplitude nuclear collective dynamics has seen
the most spectacular progress in the last two decades. This was largely caused
by the possibility to perform large-scale, precise energy density functional
calculations with realistic energy functionals on supercomputers. The last
phase of the fission process, the prompt and delayed emission of particles
from the fragments, is described by the same methods of nuclear reaction
theory as the entrance channel. In recent years, several codes have been
published to perform complete simulations of fission events and thereby
connect with transport codes used in technological applications.

Looking ahead, we see many opportunities to bridge these three phases of
fission into a single, consistent theoretical framework. For example, the
recent studies on the number of particles, excitation energy or spin
distribution of the fission fragments that were discussed in Section 6 have
shown that one can start replacing phenomenological inputs of deexcitation
models with microscopic predictions based on actual simulations of fission
dynamics. Similarly, both the statistical and R-matrix formalisms for fission
cross sections summarized in Secs. 3.2.2 and 3.2.3 rely, most generally, on
nuclear structure theory inputs that could be computed with more advanced
models than used until now. This global theoretical framework that we envision
is bound to be based on the EDF approach, at least in the foreseeable future:
large-amplitude collective motion, $\gamma$ emission or absorption and $\beta$
decay are naturally described within this framework. It is reasonable to think
that formal developments and computational advances should soon enable the
calculation of the complete sets of nuclear wave functions with good quantum
numbers needed to describe the entrance channel. Putting together all these
ingredients will without a doubt require considerable effort, but is also the
promise of a bright future for fission theory.

## Acknowledgments

The authors would like to thank O. Litaize, N. Pillet, J. Randrup, P. Romain,
P. Tamagno and I. Thompson for reading parts of the manuscript and making
relevant suggestions. This work was supported in part by the NUCLEI SciDAC-4
collaboration DE-SC001822 and was performed under the auspices of the U.S.
Department of Energy by Lawrence Livermore National Laboratory under Contract
DE-AC52-07NA27344. Computing support came from the Lawrence Livermore National
Laboratory (LLNL) Institutional Computing Grand Challenge program.

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