# Fission fragment distributions and their impact on the r-process
nucleosynthesis in neutron star mergers

J.-F. Lemaître<EMAIL_ADDRESS>Institut d’Astronomie et
d’Astrophysique, Université Libre de Bruxelles, Campus de la Plaine CP 226,
BE-1050 Brussels, Belgium  CEA, DAM, DIF, F-91297 Arpajon, France  S. Goriely
Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles,
Campus de la Plaine CP 226, BE-1050 Brussels, Belgium  A. Bauswein  GSI
Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291 Darmstadt,
Germany  Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI
Helmholtzzentrum für Schwerionenforschung, Campus Darmstadt, Germany  H.-T.
Janka  Max-Planck-Institut für Astrophysik, Postfach 1317, D-85741 Garching,
Germany

###### Abstract

Neutron star (NS) merger ejecta offer a viable site for the production of
heavy r-process elements with nuclear mass numbers $A\gtrsim 140$. The crucial
role of fission recycling is responsible for the robustness of this site
against many astrophysical uncertainties. Here, we introduce new improvements
to our scission-point model, called SPY, to derive the fission fragment
distribution for all neutron-rich fissioning nuclei of relevance in r-process
calculations. These improvements include a phenomenological modification of
the scission distance and a smoothing procedure of the distribution. Such
corrections lead to a much better agreement with experimental fission yields.
Those yields are also used to estimate the number of neutrons emitted by the
excited fragments on the basis of different neutron evaporation models. Our
new fission yields are extensively compared to those predicted by the so-
called GEF model.

The impact of fission on the r-process nucleosynthesis in binary neutron
mergers is also re-analyzed. Two scenarios are considered, the first one with
low initial electron fraction is subject to intense fission recycling, in
contrast to the second one which includes weak interactions on nucleons. The
various regions of the nuclear chart responsible for fission recycling during
the neutron irradiation as well as after freeze-out are discussed. The
contribution fission processes may have to the final abundance distribution is
also studied in detail in the light of newly defined quantitative indicators
describing the fission recycling, the fission seeds and the fission
progenitors. In particular, those allow us to estimate the contribution of
fission to the final abundance distribution stemming from specific heavy
nuclei. Calculations obtained with SPY and GEF fission fragment distributions
are compared for both r-process scenarios.

##  I Introduction

The r-process, or the rapid neutron-capture process, of stellar
nucleosynthesis is invoked to explain the production of the stable (and some
long-lived radioactive) neutron-rich nuclides heavier than iron that are
observed in stars of various metallicities, as well as in the solar system
(for a review, see Ref. Arnould  _et al._ (2007); Arnould and Goriely (2020);
Cowan  _et al._ (2020)). In recent years, nuclear astrophysicists have
developed more and more sophisticated r-process models, trying to explain the
solar system composition in a satisfactory way by adding new astrophysical or
nuclear physics ingredients. The r-process remains the most complex
nucleosynthetic process to model from the astrophysics as well as nuclear-
physics points of view. Progress in the modeling of type-II supernovae and
$\gamma$-ray bursts has raised a lot of excitement about the so-called
neutrino-driven wind environment. However, until now a successful r-process
cannot be obtained ab initio without tuning the relevant parameters (neutron
excess, entropy, expansion timescale) in a way that is not supported by the
most sophisticated existing models Wanajo  _et al._ (2011); Janka (2012).
Although these scenarios remain promising, especially in view of their
potential to contribute to the galactic enrichment significantly, they remain
affected by large uncertainties associated mainly with the still incompletely
understood mechanism responsible for the supernova explosion and the
persistent difficulties to obtain suitable r-process conditions in self-
consistent dynamical explosion and neutron-star (NS) cooling models Janka
(2012); Hüdepohl  _et al._ (2010); Roberts  _et al._ (2010); Fischer  _et al._
(2010). In particular, a subclass of core-collapse supernovae, the so-called
collapsars corresponding to the fate of rapidly rotating and highly magnetized
massive stars and generally considered to be at the origin of observed long
gamma-ray bursts, could be a promising r-process site Siegel  _et al._ (2019).
The production of r-nuclides in these events may be associated with jets
predicted to accompany the explosion, or with the accretion disk forming
around a newly born central BH Winteler  _et al._ (2012).

Since early 2000, special attention has been paid to NS mergers as r-process
sites following the confirmation by hydrodynamic simulations that a non-
negligible amount of matter could be ejected from the system. Newtonian
Freiburghaus  _et al._ (1999); Ruffert and Janka (2001); Janka  _et al._
(1999); Korobkin  _et al._ (2012); Roberts  _et al._ (2011), conformally flat
general relativistic Oechslin  _et al._ (2007); Goriely  _et al._ (2011); Just
_et al._ (2015), as well as fully relativistic Kyutoku  _et al._ (2013);
Hotokezaka  _et al._ (2013); Bauswein  _et al._ (2014); Wanajo  _et al._
(2014); Foucart  _et al._ (2014) hydrodynamical simulations of NS-NS and NS-
black hole (BH) mergers with microphysical equations of state have
demonstrated that typically some $10^{-3}\,M_{\odot}$ up to more than 0.1
$M_{\odot}$ can become gravitationally unbound on roughly dynamical timescales
due to shock acceleration and tidal stripping. Also the relic object (a hot,
transiently stable hypermassive NS Baumgarte  _et al._ (2000) followed by a
stable supermassive NS, or a BH-torus system), can lose mass through outflows
driven by a variety of mechanisms Ruffert and Janka (1999); Dessart  _et al._
(2009); Fernandez and Metzger (2013); Siegel  _et al._ (2014); Metzger and
Fernandez (2014); Perego  _et al._ (2014); Just  _et al._ (2015).

Simulations of growing sophistication have confirmed that the ejecta from NS
mergers are viable strong r-process sites up to the third abundance peak and
the actinides Just  _et al._ (2015); Goriely  _et al._ (2011); Wanajo  _et
al._ (2014); Roberts  _et al._ (2011); Bauswein  _et al._ (2013). The
r-nuclide enrichment is predicted to originate from both the dynamical
(prompt) material expelled during the NS-NS or NS-BH merger phase and from the
outflows generated during the post-merger remnant evolution of the relic BH-
torus system. The resulting abundance distributions are found to reproduce
very well the solar system distribution, as well as various elemental
distributions observed in low-metallicity stars Cowan  _et al._ (2020). During
the dynamical phase of the merging scenario, the number of free neutrons per
seed nucleus can reach so high values (typically few hundreds) that heavy
fissioning nuclei are produced. Fissioning results in a robust reproduction of
the solar-system-like abundance pattern of the rare-earth elements, as
observed in metal-poor stars Arnould  _et al._ (2007); Goriely  _et al._
(2011); Goriely (2015); Cowan  _et al._ (2020). This supports the possible
production of these elements by fission recycling in NS merger ejecta. In
addition, the ejected mass of r-process material, combined with the predicted
astrophysical event rate (around 10 My-1 in the Milky Way Dominik  _et al._
(2012)) can account for the majority of r-material in our Galaxy, e.g. Qian
(2000); Metzger  _et al._ (2010); Goriely  _et al._ (2011); Bauswein  _et al._
(2014). A further piece of evidence that NS mergers are r-nuclide producers
indeed comes from the very important 2017 gravitational-wave and
electromagnetic observation of the kilonova GW170817 7,Abbott  _et al._
(2017a, b); Cowperthwaite  _et al._ (2017); Nicholl  _et al._ (2017); Chornock
_et al._ (2017); Waxman  _et al._ (2018); Watson  _et al._ (2019).

In this specific NS merger scenario, the neutron richness was found to be so
high that heavy fissioning nuclei can be produced. For this reason, in this
astrophysical site, fission plays a fundamental role, more particularly by i)
recycling the matter during the neutron irradiation (or if not, by allowing
the possible production of super-heavy long-lived nuclei, if any), ii) shaping
the r-abundance distribution in the $110\leq A\leq 170$ mass region at the end
of the neutron irradiation, iii) defining the residual production of some
specific heavy stable nuclei, more specifically Pb and Bi, but also the long-
lived cosmochronometers Th and U, and iv) heating the environment through the
energy released. More details can be found for instance in Refs. Korneev and
Panov (2011); Korobkin  _et al._ (2012); Goriely  _et al._ (2013a); Goriely
(2015); Eichler  _et al._ (2015); Mendoza-Temis  _et al._ (2015); Giuliani
_et al._ (2018); Mumpower  _et al._ (2018); Zhu  _et al._ (2018); Holmbeck
_et al._ (2018); Wu  _et al._ (2019); Vassh  _et al._ (2019).

Despite the recent success of nucleosynthesis studies for NS mergers, the
details of r-processing in these events is still affected by a variety of
uncertainties, both from the nuclear physics and astrophysics point of view.
For this reason, we present here a new effort to further improve the
description of fission fragment distributions (FFD) obtained on the basis of
the so-called Scission Point Yield (SPY) model Lemaître  _et al._ (2019). Sec.
II is devoted to the description of such improvements. A phenomenological
correction to the scission distance and a new smoothing procedure for deriving
the fission yields are presented in Sec. II.1. Since the FFD of relevance for
the r-process application correspond to the post-neutron-emission one deduced
from pre-neutron FFDs, our procedure for estimating the neutron evaporation is
detailed in Sec. II.2. As r-process is impacted by neutron captures,
sensitivity studies are performed in Sec. II.3 to demonstrate that the neutron
emission depends mainly on the primary fission yields. SPY pre-neutron FFDs
are compared with experimental data and the well-known GEF (“GEneral
description of Fission observables”) calculations Schmidt  _et al._ (2016) in
Sec. II.4.

In addition, it has been shown that weak interactions may strongly affect the
composition of the dynamical ejecta and thus the efficiency of the r-process
Wanajo  _et al._ (2014); Sekiguchi  _et al._ (2015); Goriely  _et al._ (2015);
Foucart  _et al._ (2016); Lehner  _et al._ (2016); Radice  _et al._ (2016);
Bovard  _et al._ (2017); Martin  _et al._ (2018); Goriely (2019); Ardevol-
Pulpillo  _et al._ (2019). In this case, the initial neutron richness of the
ejecta can be significantly reduced and consequently the role of fission,
albeit in basically all calculations a significant fraction of neutron-rich
ejecta is present. In Sec. III, we re-evaluate the role of fission in NS
merger in two distinct scenarios, namely the traditional one neglecting weak
interactions on nucleons and an updated version of such a simulation where
neutrino interactions are included phenomenologically Goriely  _et al._
(2015); Goriely (2019) to reproduce qualitatively the results found in Ref.
Ardevol-Pulpillo  _et al._ (2019). In Sec.III.1, we will present the final
abundance distributions obtained within both scenarios as well as the role of
the fission recycling and link them to fundamental initial properties of
ejected material, namely the initial neutron mass fraction, electron fraction
and entropy. The relevance of the fission recycling and its contribution to
the final r-abundances is thoroughly detailed in Secs. III.2–III.4 for both
scenarios. Finally, conclusions are drawn in Sec. IV.

##  II Fission fragments distributions

###  II.1 SPY model

The SPY model is a static and statistical scission point model Lemaître  _et
al._ (2015, 2019) where a thermodynamic equilibrium at scission is assumed,
hence the evolution between the saddle and the scission points is neglected.
The model is based on two pillars, namely the absolute available energy
balance at the scission configurations and the statistical description of the
available phase space. The available energy balance is performed for all
energetically possible fragmentations of a fissioning system at scission as a
function of the deformation of both fragments. The available energy is defined
as the difference between the potential energy of the fissioning system at
scission and the energy of the excited compound nucleus where both nascent
fragments are supposed to be at rest. The potential energy of the fissioning
system at scission is obtained as the sum of the individual binding energies
of the two fragments and the interaction energy between the fragments composed
of the Coulomb repulsion and the nuclear attraction. The system at scission is
treated as a microcanonical ensemble where all available states are
equiprobable. In this framework, the number of available states of a given
fragmentation is the product of the state densities of the two isolated
fragments. The yield of a fragmentation is the number of available states
associated with this fragmentation regardless of the deformation of the
fragments.

All nuclear inputs to the SPY model, i.e. the individual binding energies,
proton distributions and proton and neutron single-particle level schemes of
each fission fragment as a function of its axial deformation are estimated for
about 7000 nuclei from $Z=20$ to $Z=100$ within the framework of the
constrained self-consistent Hartree-Fock-Bogoliubov (HFB) formalism using the
Skyrme BSk27 effective nucleon-nucleon interaction Goriely  _et al._ (2013b).
The nuclear state density is calculated in the framework of the statistical
model of nuclear level densities Lemaître  _et al._ (2019); Decowski  _et al._
(1968); Moretto (1972a, b); Demetriou and Goriely (2001) on the basis of the
discrete single-particle level scheme obtained self-consistently within the
same HFB calculation.

A scission criteria is needed to characterize the scission configurations,
more precisely, to define the relative position of the fragments, hence to
estimate the Coulomb and nuclear energies. This criteria can be based on an
inter-surface distance, as in the initial version of the SPY model Lemaître
_et al._ (2015) or through the proton density at the scission neck which
defines the scission distance as in our new version of the SPY model Lemaître
_et al._ (2019). Presently, with respect to our last study Lemaître  _et al._
(2019), a phenomenological modification of the scission distance is introduced
in order to better reproduce experimental fission yields. The corrected
scission distance ($d_{\mathrm{sc,corr}}$) for a fragmentation
$(Z_{1},N_{1})+(Z_{2},N_{2})$ is defined through the proton density at the
scission neck ($d_{\mathrm{sc}}$), as in Ref. Lemaître  _et al._ (2019),
though with two additional corrections, one per fragment

$d_{\mathrm{sc,corr}}=d_{\mathrm{sc}}+d_{\mathrm{corr}}(N_{1})+d_{\mathrm{corr}}(N_{2})~{}.$
(1)

Each correction term is composed by a positive term and a negative one which
are chosen to be dependent on the difference between the fragment neutron
number ($N$) and some specific neutron numbers ($N_{\mathrm{corr}}$), i.e.

$\displaystyle d_{\mathrm{corr}}(N)=$
$\displaystyle\sum_{N_{\mathrm{corr}}\in\\{47,51,81,85,89\\}}0.2~{}e^{-\frac{(N-N_{\mathrm{corr}})^{2}}{2}}$
(2) $\displaystyle-$
$\displaystyle\sum_{N_{\mathrm{corr}}\in\\{64,66,68,70,80\\}}0.2~{}e^{-\frac{(N-N_{\mathrm{corr}})^{2}}{2}}\
[\mathrm{fm}]~{}.$

Both light and heavy fragments contribute to the distance correction (Fig. 1a)
with a maximal correction of $\pm 0.4$ fm. The available energy is impacted by
a few MeV, for example by around 3 MeV for the neutron-induced fission of 235U
(Fig. 1b). Such an effect is of the same order of magnitude as pairing effects
and can be regarded as a way to correct the approximate pairing effects which
assumes the pairing gap at zero temperature can be described by a five-points
mass difference Madland and Nix (1988) and remains constant with the nucleus
deformation. For nucleosynthesis applications, the distance correction is kept
constant irrespective of the fissioning system, even if a specific adjustment
could be performed.

Figure 1: (Color online) (a) Distance correction associated with the light
fragment, $d(N_{1})$ with $N_{1}\leqslant N_{\mathrm{CN}}/2$ (orange curve),
with the heavy fragment, $d(N_{2})$ with $N_{2}\geqslant N_{\mathrm{CN}}/2$
(green curve), and with the total one $d(N_{1})+d(N_{2})$ (black dashed curve)
for 236U fission. The specific neutron numbers $N_{\mathrm{corr}}$ (Eq. 2) are
indicated by gray arrows. (b) Available energy of fragments from thermal
neutron-induced fission of 235U calculated with (red curve) or without (green
curve) the phenomenological distance correction (Eq. 2). The blue line
corresponds to the the difference between corrected and non-corrected
energies.

This distance correction help to better reproduce the yields of the thermal
neutron induced fission of 235U (Fig. 3a and d, green and blue curves), more
particularly its symmetric component is reduced due to the distance correction
around $N=64,66,68,70$. As seen in Fig. 3c and f, this correction has a
negligible impact on the FFD for the spontaneous fission of 252Cf. However,
yields of the thermal neutron-induced fission of 239Pu around $A=130$ becomes
underestimated (Fig. 3b and e) due to the same correction as in the U case.

As described in Ref. Lemaître  _et al._ (2019), the SPY FFD or yields
distribution present strong staggering patterns that are not found
experimentally. Those can be smoothed, as traditionally done, by a double
Gaussian function

$Y_{\mathrm{smooth}}(Z,N)=\\\
\sum_{i=-4}^{4}\sum_{j=-15}^{15}Y_{\mathrm{raw}}(Z+i,N+j)C_{z}e^{-\frac{i^{2}}{2\sigma_{z}^{2}}}C_{n}e^{-\frac{j^{2}}{2\sigma_{n}^{2}}}$
(3)

where $C_{z}$ and $C_{n}$ are normalization factors and $\sigma_{z}=0.55$ and
$\sigma_{n}=3$ are adopted. In the present paper the width of the Gaussian
function is larger in the neutron direction $\sigma_{n}$ than in the proton
direction in order to be simultaneously consistent with experimental width of
the isobaric and isotopic fission yields distributions (Fig. 3). As there is
not coupling between $Z$ and $N$ in the double Gaussian function (Eq. 3), the
isotopic yields are not affected by the smoothing in the neutron direction.
Similarly to yields, other fragments observables $\mathcal{O}$, such as the
available energy or the kinetic energy, are also smoothed following the same
procedure

$\mathcal{O}_{\mathrm{smooth}}(Z,N)=\\\
\sum_{i=-4}^{4}\sum_{j=-15}^{15}\mathcal{O}_{\mathrm{raw}}(Z+i,N+j)C_{z}e^{-\frac{i^{2}}{2\sigma_{z}^{2}}}C_{n}e^{-\frac{j^{2}}{2\sigma_{n}^{2}}}.$
(4)

Figure 2:  (Color online) Isotopic (top panels) and isobaric (bottom panels)
fission yields of 236U, 240Pu and 252Cf. The black lines with dots represent
experimental (pre-neutron-emission) fission yields for fission of
${}^{236}\mathrm{U}$ $(Q=6.5~{}\mathrm{MeV})$ (a) Schmitt  _et al._ (1984) and
(d) Romano  _et al._ (2010), ${}^{240}\mathrm{Pu}$ $(Q=6.5~{}\mathrm{MeV})$
(b) Mariolopoulos  _et al._ (1981) and (e) Tsuchiya  _et al._ (2000) and
${}^{252}\mathrm{Cf}$ $(Q=0~{}\mathrm{MeV})$ (c) Bocquet and Brissot (1989)
and (f) Zeynalov  _et al._ (2011). The green lines (labeled by “raw”)
correspond to raw fission yields, the blue lines (labeled by
“$d_{\mathrm{corr}}$”) to raw yields including distance corrections and the
orange lines (labeled by “smooth”) to the final smooth yields after distance
corrections.

Figure 3:  (Color online) Evaporated neutron distributions as a function of
the fragment mass number. The black dotted lines represent experimental
evaporated neutron distribution of (a) ${}^{236}\mathrm{U}$
$(Q=6.5~{}\mathrm{MeV})$ Vorobyev, A.S.  _et al._ (2010), (b)
${}^{240}\mathrm{Pu}$ $(Q=6.5~{}\mathrm{MeV})$ Nishio  _et al._ (1995) and (c)
${}^{252}\mathrm{Cf}$ $(Q=0~{}\mathrm{MeV})$ Dushin  _et al._ (2004). The
orange/green lines (labeled by “smooth”/“raw”) are SPY evaporated neutron
distribution using smooth/raw pre-neutron yields.

###  II.2 Neutron evaporation model

The SPY model allows us to investigate the primary fragments, i.e. fragments
formed at scission, hence their de-excitation by neutron evaporation. The
excitation energy of fragments has two components. First, each fragment
carries a fraction of the available energy at scission under an intrinsic
excitation form. Second, since each fragment can be deformed, a deformation
energy may contribute to the excitation energy. No assumption about available
energy sorting at scission between primary fragments is needed within the SPY
framework, contrary to other neutron evaporation models. Indeed, some models
require to introduce an energy sorting parameter $R_{T}(A)$ Litaize  _et al._
(2015); Talou  _et al._ (2011) or a thermal equilibrium to deduce the energy
sorting on the basis of level density parameters with phenomenological
corrections in order to increase the excitation energy of the light fragment
Vogt and Randrup (2013).

In the SPY model, the energy sorting, denoted by $x$ (Eq. 2 in Ref. Lemaître
_et al._ (2019)), results from the competition between the state density of
the light fragment ($\rho_{1}$) and the heavy one ($\rho_{2}$). The intrinsic
excitation energy and deformation energy of each fragment are obtained by
averaging over all possible energy sorting and deformations (Eq. 3 in Ref.
Lemaître  _et al._ (2019)). In the simplest model, the competition between
neutron and gamma evaporation during the de-excitation cascade is not taken
into account, so that the fission fragments are sequentially de-excited by
neutron evaporation as long as it is energetically possible. The kinetic
energy of evaporated neutrons $E_{\mathrm{n}}$ can be assumed to follow a
Maxwellian distribution

$\phi_{M}(E_{\mathrm{n}})=\frac{2}{\sqrt{\pi
T^{3}}}\sqrt{E_{\mathrm{n}}}e^{-E_{\mathrm{n}}/T},$ (5)

where $T$ is the nuclear temperature before neutron evaporation. $T$ can be
extracted from the statistical model of nuclear level densities including BCS
pairing interaction, denoted as $T_{\mathrm{BCS}}$ (see Eqs. 16 and 18 in Ref.
Lemaître  _et al._ (2019) and Refs. Decowski  _et al._ (1968); Moretto (1972a,
b)).

Following this approximation, the distribution of evaporated neutrons are
calculated for the three well-known fissioning systems using both raw and
smooth pre-neutron yields and shown in Fig. 3. In both cases, the number of
emitted neutrons is underestimated for the heaviest fragments $A=150-160$ due
to the low available energy, which, in turn, originates from an overestimate
of the kinetic energy of these related fragmentations, as mentioned in Sec. IV
of Ref. Lemaître  _et al._ (2019). However, for heavy fragments with
$A=130-140$, the emitted neutron number is overestimated which, in this case,
is due to an underestimate of the total kinetic energy of the fission
fragments. The distributions of evaporated neutron deduced from smooth yields
(Fig. 3, orange curves) follow roughly the same pattern as the raw ones except
for the heaviest fragments region where the number of emitted neutron is
larger. This overestimation is due to the smoothing procedure (Eq. 4) which
leads to an increase of the total excitation energy by around 2 MeV, hence
sometimes makes the emission of an additional neutron possible depending on
its kinetic energy. More generally, the number of neutrons emitted by the
light fragment is in better agreement with data, though slightly
underestimated, than for the heavy fragment where $\nu$ is systematically
overestimated. This is linked to the available energy sorting within a
fragmentation which is too favorable for the heavy fragment.

The overestimation the number of neutron emitted by the heavy fragments leads
an overestimation of the total mean number of evaporated neutrons per fission.
As displayed in Table 1, it is overestimated by 0.77-1.1 neutron when smooth
yields are used and by 0.6-1 neutron for raw yields. The overestimation is
higher with smooth yields due to a more important contribution of the heavy
fragments.

Table 1: Total mean number of evaporated neutrons per fission, $\bar{\nu}_{\mathrm{tot}}$, from the evaluated data library ENDF/B-VII.1 Chadwick  _et al._ (2006) and SPY using the raw or smooth yields (see Fig. 3) for the 3 benchmarked nuclei shown in Fig. 3 and obtained either by neutron-induced fission (nif) or spontaneous fission (sf). Nucleus | reaction | ENDF | SPY(raw) | SPY(smooth)
---|---|---|---|---
${}^{235}\mathrm{U}$ | nif | 2.44 | 3.08 | 3.20
${}^{239}\mathrm{Pu}$ | nif | 2.88 | 3.97 | 3.96
${}^{252}\mathrm{Cf}$ | sf | 3.77 | 4.37 | 4.39

To study the sensitivity of the predicted number of evaporated neutrons to the
various model uncertainties, the impact of the neutron kinetic energy
distribution $\phi(E_{\mathrm{n}})$, the nuclear temperature and the neutron-
gamma competition are analyzed in the next sub-sections for the specific case
of 252Cf spontaneous fission.

###  II.3 Sensitivity studies of the evaporation model

####  II.3.1 Impact of the kinetic energy distribution of the neutron

There are various models for the kinetic energy distribution of the neutron
$\phi(E_{\mathrm{n}})$ Kawano  _et al._ (2013). The chosen one is related to
the Maxwell-Boltzmann distribution describing the distribution of speeds of
particles in an idealized gas which yields to the kinetic energy distribution
of neutron $\phi_{M}$ (Eq. 5). Another possible description relies on the
evaporation model of Weisskopf and Ewing Weisskopf (1937); Weisskopf and Ewing
(1940) yielding to

$\phi_{W}(E_{\mathrm{n}})=\frac{E_{\mathrm{n}}}{T_{\mathrm{n}}^{2}}e^{-E_{\mathrm{n}}/T_{\mathrm{n}}}$
(6)

for small $E_{\mathrm{n}}$ Kawano  _et al._ (2013) and where $T_{\mathrm{n}}$
is the nuclear temperature after the neutron emission. The impact of the
neutron kinetic energy distribution on the number of evaporated neutron of
252Cf spontaneous fission is negligible, as shown in Fig. 4, (orange and light
green curves, labeled respectively by “$\phi_{M},T_{\mathrm{BCS}}$” and
“$\phi_{W},T_{\mathrm{BCS}}$”) where the difference is seen to rarely exceed
0.1 neutron (Fig. 4b, light green curve, labeled by
“$\phi_{W},T_{\mathrm{BCS}}$”).

Figure 4: (Color online) (a) Evaporated neutron distribution of spontaneous
fission of ${}^{252}\mathrm{Cf}$ using the SPY raw yields. (b) Difference with
respect to the evaporated neutron distribution computed with the Maxwell
distribution (Eq. 5) and temperature from the BCS state density.

####  II.3.2 Impact of the nucleus temperature

As an alternative to the BCS theory of nuclear level densities, the
phenomenological Fermi Gas approximation Bethe (1937), excluding shell and
pairing effects, can also be considered to estimate the nuclear temperature,

$T_{\mathrm{FG}}=\sqrt{U/a}$ (7)

where $U$ is the excitation energy and $a\sim A/8$ is the empirical high-
energy limit of the level density parameter Tōke and Światecki (1981). As seen
in Fig. 4 (cyan and black curves, labeled respectively by
“$\phi_{M},T_{\mathrm{FG}}$” and “$\phi_{W},T_{\mathrm{FG}}$”), both models
for the nuclear temperature ($T_{\mathrm{BCS}}$ or $T_{\mathrm{FG}}$) give
rather similar predictions of the evaporated neutron distribution. For the
sake of coherence, $T_{\mathrm{BCS}}$ is adopted in the subsequent study.

####  II.3.3 Impact of neutron-gamma competition

In the previous calculation of the number of emitted number of neutrons, it
was assumed that neutron emission dominates over the electromagnetic de-
excitation. To test the impact of such an approximation, TALYS nuclear
reaction code Koning and Rochman (2012) is now used to describe the
competition between the strong and electromagnetic channels in the de-
excitation of the the primary fragments (Fig. 4, green curve, labeled by
“TALYS”). Note that all de-excitation channels are taken into account in the
Hauser-Feshbach and pre-equilibrium framework. The difference between the
evaporated neutron distribution from TALYS and the one obtained with the
simplified Maxwellian approach described above is shown in Fig. 4 (green
curve, labeled by “TALYS”) and does not exceed 0.3 neutron.

These results indicate that the primary fragments de-excitation can be
essentially approximated by a sequence of neutron emissions, followed finally
by gamma emission when no more neutron can be evaporated. This simplified
description based on energetic considerations seems to be sufficient within an
accuracy of about 0.1–0.2 neutron. This conclusion is only valid for low
excited primary fragments up to 20–25 MeV. The crucial point to improve the
description of the emitted neutron distribution concerns the SPY model and
particularly the kinetic energy of the fission fragments which directly affect
the available energy and mean deformation of the fragments.

In the following, if not mentioned otherwise, the smooth yields (Eq. 3) are
adopted and neutron evaporation are deduced using the Maxwell distribution
prescription (Eq. 5) with the BCS temperature.

(a) Peak multiplicity for the raw pre-neutron isobaric yields with corrected
distance (Eq. 1). (b) Mean prompt neutron multiplicity per fission according
to the prescription given in Sec. II.2, deduced from the smooth pre-neutron
isobaric yields with corrected distance (Eq. 1). (c) Peak multiplicity for the
smooth pre-neutron isobaric yields with corrected distance (Eq. 1). (d) Mean
available energy release per fission deduced from the smooth pre-neutron
isobaric yields with corrected distance (Eq. 1). (e) Peak multiplicity for the
smooth pre-neutron isobaric yields without corrected distance. (f) Isolines of
mean available energy release per fission (orange lines) from Fig. 5(d) and
mean prompt neutron multiplicity per fission (green lines) from Fig. 5(b).

Figure 5: (Color online) Systematics in the ($N,Z$) plane of major fission
observables for some 3000 nuclei with $70\leqslant Z\leqslant 124$ for an
initial excitation energy of $Q=8\ \mathrm{MeV}$.

###  II.4 Systematics

####  II.4.1 Over all fissionable nuclei

Systematic SPY calculations were performed for isotopic chains between $Z=70$
and 124 from the proton to the neutron driplines with an initial excitation
energy $Q=8\ \mathrm{MeV}$ (Fig. 5). With $Q=8~{}\mathrm{MeV}$, only few
nuclei with $Z<80$ can fission. The smoothing procedure (Eq. 3) may impact
locally the peak multiplicity (Fig. 5(c)) when compared those obtained with
raw yields (Fig. 5(a)). In particular, peaks can be merged if there are close
each other or if they are small enough. However, the location of transition
between the various fission modes are seen not to be globally affected by the
smoothing procedure and does not distort significantly the peaks location of
FFDs and their widths as shown in Fig. 3.

Four zones can be identified where the peak multiplicity is affected by the
distance correction (Eq. 2), as seen when comparing Figs. 5(c) and 5(e)
obtained with or without this correction. In general terms, a
negative/positive distance correction makes fragments closer/farther which
induced a decrease/increase of the available energy, hence of the fission
probability. The first one is the shift of the symmetric/asymmetric fission
transition toward neutron-deficient nuclei from $N_{\mathrm{CN}}\approx 140$
to $N_{\mathrm{CN}}\approx 133$. The symmetric mode is disfavored by the
negative distance correction for $N_{\mathrm{frag}}=64,66,68,70$ leading to
the appearance of the asymmetric mode while for
$N_{\mathrm{frag}}=47,51,81,85,89$ the positive distance correction favors the
asymmetric mode. In the second zone, located at $155\lesssim
N_{\mathrm{CN}}\lesssim 165$, a second asymmetric mode emerges, less
asymmetric than the one without distance correction. It is induced by a
negative distance correction for the light fragment of the most asymmetric
fission mode, i.e. $N_{\mathrm{frag}}=64,66,68,70$ and by a positive distance
correction for the heavy fragment of the new slightly asymmetric fission mode
($N_{\mathrm{frag}}=85,89$). The peak multiplicity increases from 2 to 4 when
the distance corrections are important. For more neutron-rich nuclei, in the
third zone $165\lesssim N_{\mathrm{CN}}\lesssim 180$, a symmetric mode emerges
for high distance corrections. It is induced by a negative distance correction
due to the light fragment of the asymmetric fission mode
($N_{\mathrm{frag}}=64,66,68,70$) which disfavors the asymmetric mode. The
symmetric mode is further accentuated by the positive distance correction for
$N_{\mathrm{frag}}=81,85,89$. Finally, for nuclei in the fourth zone
$180\lesssim N_{\mathrm{CN}}\lesssim 200$, the positive distance correction
for $N_{\mathrm{frag}}=85,89$ induces a slightly asymmetric fission mode on
top of the symmetric one. Although some fission yields distributions are
impacted by the phenomenological correction to the scission distance,
globally, SPY predictions for neutron-rich nuclei remains rather robust with
respect to such a correction, as seen in Figs. 5(c) – 5(e).

The mean number of neutrons emitted per fission $\bar{\nu}_{\mathrm{tot}}$
ranges from 2 up to 37 neutrons depending on the fissioning system (Fig.
5(b)). It depends mostly on the neutron richness of the fissioning nucleus.
Its evolution with proton number is not trivial despite the strong variation
of the mean available energy of fragmentations (Fig. 5(d)). For a given
fissioning nucleus, an increase of the excitation energy ($Q$) enhances the
intrinsic excitation energy of fission fragments, hence more neutrons are
emitted. However, this relation between available energy and emitted neutrons
is not sufficient to explain the relation between emitted neutrons and mean
available energy of fragmentations. Along a line of constant available energy
(Fig. 5(f)) the number of evaporated neutron is multiplied by at least five
between proton-rich and neutron-rich fissioning nuclei. Similarly, along a
constant line of emitted neutron number (Fig. 5(f)), the available energy
increases by a factor up to hundred between light and heavy nuclei. These
features are due to the neutron binding energy of fission fragments, i.e. the
more neutron-rich the fissioning nucleus, the more-neutron rich the fission
fragments and the lower their neutron binding energy, hence the more neutrons
can be evaporated. Even if the fission of a heavy nucleus releases more
available energy compared to a light one, the number of evaporated neutrons
can be similar because the neutron binding energies of fragments from heavy-
nucleus fission is higher (less neutron-rich) than the fragments from the
lighter one.

####  II.4.2 SPY vs GEF

(a) Evaluated isotopic yields (formed by Coulomb excitation) from Ref. Schmidt
_et al._ (2000) converted into isobaric yields using the UCD (Unchanged Charge
Density) hypothesis. For ${}^{230}\mathrm{Th}$ post-neutron yields from
ENDF/B-VII database Chadwick  _et al._ (2006).  (b) Yields of
${}^{232-233}\mathrm{U}$ are from Schmidt  _et al._ (2000) (converted into
isobaric), ${}^{234}\mathrm{U}$ from Baba  _et al._ (1997),
${}^{235-237}\mathrm{U}$ (formed by 0.5 MeV neutron) from Chadwick  _et al._
(2006), ${}^{236}\mathrm{U}$ from Romano  _et al._ (2010),
${}^{238}\mathrm{U}$ and ${}^{239}\mathrm{Np}$ and ${}^{244}\mathrm{Cm}$ from
Ramos  _et al._ (2018), ${}^{239}\mathrm{U}$ Zeynalov  _et al._ (2011),
${}^{238}\mathrm{Np}$ fromHambsch  _et al._ (2000), ${}^{240}\mathrm{Pu}$ from
Thierens  _et al._ (1981), ${}^{242-243}\mathrm{Pu}$ and ${}^{255}\mathrm{Es}$
(formed by thermal neutron) from Chadwick  _et al._ (2006),
${}^{243}\mathrm{Am}$ from Rubchenya  _et al._ (2004), ${}^{244}\mathrm{Am}$
from Goverdovskij  _et al._ (1995), ${}^{246}\mathrm{Cm}$ and
${}^{253}\mathrm{Es}$ and ${}^{254-256}\mathrm{Fm}$ (spontaneous fission) from
Chadwick  _et al._ (2006), ${}^{248}\mathrm{Cm}$ from Zhao  _et al._ (2003),
${}^{250}\mathrm{Cf}$ from Unik  _et al._ (1974), ${}^{252}\mathrm{Cf}$ from
Zeynalov  _et al._ (2011), ${}^{256}\mathrm{Cf}$ and ${}^{258}\mathrm{Fm}$
from Hoffman  _et al._ (1980a), ${}^{244-246-248}\mathrm{Fm}$ from Hoffman
_et al._ (1980b), ${}^{259}\mathrm{Lr}$ from Hamilton  _et al._ (1992),
${}^{262}\mathrm{Rf}$ from Lane  _et al._ (1996).

Figure 6: (Color online) Isobaric fission yields from SPY model (green curves)
compared to those calculated by the GEF model (blue curves) Schmidt  _et al._
(2016) and experimental pre-neutron (black curves) or post-neutron (gray
curves) yields. For each FFD, the x axis (mass number) ranges from $A=70$ to
$A=160$ by step of 10 and the y axis (FFD) from 0 to 15% by step of 2%. The
background color refers to the peak multiplicity.

SPY FFDs with corrected distance are compared in Fig. 6 to experimental data
or with the evaluation from the ENDF/B-VII database Chadwick  _et al._ (2006),
if no experimental data is available. Evaluated data (Fig. 6, gray curves)
correspond to post-neutron FFDs while experimental data can be pre-neutron or
post-neutron emission (Fig. 6, black curves). SPY FFDs (Fig. 6, green curves)
are also compared to GEF (version 2019/1.2) thermal neutron-induced fission
yields Schmidt  _et al._ (2016); gef . The excitation energies of experimental
or evaluated data may not be exactly the same as those considered in SPY or
GEF calculations but a variation of the excitation energy by a few MeV only
impacts moderately the shape of the FFD.

In the Po to U region (Fig. 6(a)), the transition between symmetric fission
for neutron-deficient nuclei to asymmetric fission for neutron-rich nuclei is
rather different between both SPY and GEF predictions. SPY FFDs are in good
agreement with experimental data Schmidt  _et al._ (2000) for Th and Pa
isotopes where the transition is well reproduced, in contrast to what is
obtained by the GEF model. The transition from symmetric to asymmetric fission
predicted by GEF occurs for more neutron-deficient isotopes compared to
experimental data.

For Th isotopes, GEF transition occurs between ${}^{216}\mathrm{Th}_{126}$ and
${}^{226}\mathrm{Th}_{136}$ for which no symmetric peak is found anymore. From
an experimental point of view, the transition seems to start at
${}^{222}\mathrm{Th}_{132}$ where the symmetric peak broadens while beyond
${}^{229}\mathrm{Th}_{139}$ the FFD is completely asymmetric. The symmetric to
asymmetric transition of Th predicted by SPY is slightly more complex (Fig.
5(e) and Fig. 6(a)); there are two peaks in addition to the symmetric
component for ${}^{222-226}\mathrm{Th}_{132-136}$ which gives a peak
multiplicity of two and even four for ${}^{226}\mathrm{Th}_{136}$. For more
neutron-rich Th, this substructure disappears into one symmetric peak which
become smaller for more neutron-rich isotopes and completely vanishes for
${}^{235}\mathrm{Th}_{145}$ (Fig. 5(e)). This kind of transition from
symmetric to asymmetric through a substructure at the top of the vanishing
symmetric peak is also visible on the SPY Pa FFDs.

The FFD broadening in the Rn to Ac isotopic chains is well reproduced by SPY
even if the FFDs of neutron-deficient isotopes tend to be narrower than the
experimental ones. A strong odd-even staggering in GEF FFD is observed for
even isotopes of Po and Rn which makes FFD plots completely blue filled (Fig.
6(a)).

For actinides from U to Rf (Fig. 6(b)), FFDs from SPY and GEF model are rather
similar and in good agreement with experimental data, particularly for U and
Es isotopes. Compared to experimental data for Am and Cm isotopes, the
symmetric part of the yields distributions is underestimated by both models.
The peaks are narrower with SPY compared to GEF model. However, for neutron-
rich nuclei (like ${}^{258}\mathrm{Fm}$), SPY predicts a purely asymmetric
fission, contrary to GEF and evaluated data. This asymmetric splitting
predicted by SPY model is due to the doubly magic nuclei ${}^{132}\mathrm{Sn}$
which is disfavored. Indeed, even if ${}^{132}\mathrm{Sn}$ has more available
energy than other fragmentations, the available states for these
fragmentations is significantly lower and consequently their yields is lower
than other fragmentations leading to an asymmetric fission.
${}^{132}\mathrm{Sn}$ affects fissioning systems in the region around
${}^{258}\mathrm{Fm}$ (Fig. 5(e)) where a doubly asymmetric fission is found
by SPY model whereas GEF model predicts an symmetric mode. For the same
reasons, the symmetric fission of 259Lr and 262Rf are not reproduced with the
SPY model contrary to the GEF model.

Figure 7:  (Color online) Representation in the ($N$,$Z$) plan the decay mode
with the highest decay rate (a) between $\beta$ decay followed by 0, 1, 2 or 3
neutron emission ($\beta kn$), $\beta$-delayed fission ($\beta$df),
spontaneous fission (sf) or $\alpha$ decay; (b) the corresponding lowest
lifetime $\tau_{\min}=1/\lambda_{\max}$ among the various decay mode given in
(a). (c) Temperature at which fission and radiative neutron capture are
equally favorable, above this temperature the neutron induced fission
dominates the neutron capture. (d) Nuclei for which neutron-induced fission is
dominant over the $(n,\gamma)$ channel for $T<0.2~{}\mathrm{GK}$ or for which
the main decay mode is spontaneous fission or $\beta$-delayed fission are
depicted by the gray area (region of efficient fission. Blue hatched areas are
the fission seeds areas : the “fission bottleneck” ($A<292$ and $N>184$)
hatched by blue circles, the “fission roof” ($A\geqslant 320$) hatched by blue
vertical+horizontal lines, the “decay roof 0” ($A<278$) hatched by blue
vertical lines, the “decay roof 1” ($A\geqslant 278$ and $N\leqslant 184$)
hatched by blue horizontal lines and the “decay roof 2” ($292\leqslant A<320$
and $N>184$) hatched by blue diagonal lines. The double black lines show the
$Z=82$, $N=126$ and $N=184$ shell closures. See text for more details.

##  III Impact of fission on the r-process nucleosynthesis in NS mergers

The NS-NS merger simulation was performed with a general relativistic Smoothed
Particle Hydrodynamics scheme Oechslin  _et al._ (2007); Bauswein  _et al._
(2013, 2010) representing the fluid by a set of particles with constant rest
mass, the hydrodynamical properties of which were evolved according to
Lagrangian hydrodynamics. We only consider dynamical (prompt) ejecta expelled
during the binary NS-NS merger in the present paper since this ejecta
component provides generally the most favorable conditions for fission
recycling due to the low initial electron fraction. A symmetric
1.365$M_{\odot}$ – 1.365$M_{\odot}$ binary system compatible with the total
mass detected in the GW170817 event Abbott  _et al._ (2017a) has been modelled
on the basis of the so-called SFHO equation of state Steiner  _et al._ (2013).
R-process calculations are performed on 500 ejected “particles” (i.e. mass
elements) representative of the dynamical ejecta. The r-process
nucleosynthesis is calculated with a reaction network including all 5000
species from protons up to $Z=110$ lying between the valley of
$\beta$-stability and the neutron drip line. All charged-particle fusion
reactions on light and medium-mass elements that play a role when the nuclear
statistical equilibrium freezes out are included in addition to radiative
neutron captures and photodisintegrations. The reaction rates on light species
are taken from the NETGEN library, which includes all the latest compilations
of experimentally determined reaction rates Xu  _et al._ (2013).
Experimentally unknown reactions are estimated with the TALYS code Goriely
_et al._ (2008); Koning and Rochman (2012) on the basis of the Skyrme HFB
nuclear mass model, HFB-21 Goriely  _et al._ (2010). On top of these
reactions, $\beta$-decays as well as $\beta$-delayed neutron emission
probabilities are also included, the corresponding rates being taken from the
relativistic mean-field model of Ref. Marketin  _et al._ (2016). Fission
processes, including neutron-induced, spontaneous and $\beta$-delayed fission,
are carefully introduced in the network together with the corresponding FFD.
Fission barriers obtained with the BSk14 Skyrme interaction are adopted here
to estimate fission probabilities Goriely  _et al._ (2007, 2009). The new
smooth FFDs from SPY model, as described in the previous section, as well as
GEF predictions for comparison, are used in the present study. Note that all
fission products with a yield larger than typically $10^{-5}\%$ are linked to
the parent fissioning nucleus in the network calculation, i.e. about 500
fission fragments are included for each fissioning nucleus. The neutron-rich
fission fragments located outside the network, i.e. across the neutron drip
line, are assumed to instantaneously emit neutrons down to the neutron drip
line. The total mass fraction as well as number of nucleons is conserved at
all time and the emitted neutrons are recaptured consistently by all the
existing species. Finally, $\alpha$-decays are taken into account for all
heavy species, the rates being extracted from Ref. Koura  _et al._ (2002).

Despite the recent success of nucleosynthesis studies for NS mergers, the
details of r-processing in these events is still affected by a variety of
uncertainties. In particular, the exact impact of neutrino is not yet
understood in all details, the main reason of which is the not yet manageable
computational complexity associated with full neutrino transport in a
generically three-dimensional (3D), highly asymmetric environment with nearly
relativistic fluid velocities and rapid changes in time. It was long assumed
that neutrino interactions could not, at least not drastically, affect the
initial neutron richness of the ejecta. However, recent NS-NS merger
simulations Wanajo  _et al._ (2014); Sekiguchi  _et al._ (2015); Radice  _et
al._ (2016); Foucart  _et al._ (2016); Lehner  _et al._ (2016); Bovard  _et
al._ (2017); Martin  _et al._ (2018); Ardevol-Pulpillo  _et al._ (2019)
including the effect of weak interactions on free nucleons demonstrate that
neutrino reactions can significantly affect the neutron-to-proton ratio in the
merger ejecta, with direct consequences in particular on the amount of
synthesized low-mass ($A<140$) r-nuclides produced in the dynamical ejecta
along with the heavier species. To broadly study the impact of weak
interactions of free nucleons on the r-process nucleosynthesis, a parametric
approach in terms of prescribed neutrino luminosities and mean energies,
guided by hydrodynamical simulations, was followed in Ref. Goriely  _et al._
(2015). A similar approach is also considered here to test the role of fission
during the r-process nucleosynthesis in mainly two scenarios. The first one
(hereafter scenario I) neglects all weak interactions on free nucleons and
assumes the initial electron fraction of the ejected material is consequently
not modified at the beginning of the expansion. In this case, the mean initial
electron fraction at the time of the network calculation (i.e. when the
temperature has dropped below 10 GK) amounts to $\langle Y_{e}\rangle=0.03$.
This case is likely to be relevant for NS-BH mergers or highly asymmetric NS-
NS mergers, where the lower mass component develops an extended tidal tail,
from which considerable amounts of cold, unshocked matter can be centrifugally
ejected before neutrino exposure of these ejecta may play an important role
Bauswein  _et al._ (2014); Just  _et al._ (2015); Foucart  _et al._ (2014). In
the second case (scenario II), the equilibrium between electrons, positrons
and neutrinos is assumed to hold down to the fiducial density of
$\rho_{\mathrm{eq}}=10^{12}$ g cm-3, and from that density on, electron,
positron and electron neutrino and antineutrino captures on nucleons

$\displaystyle\nu_{e}+n\rightleftharpoons p+e^{-}$ (8)
$\displaystyle\bar{\nu}_{e}+p\rightleftharpoons n+e^{+}$

are systematically included assuming the (anti)neutrino luminosities and mean
energies remain constant in time, as detailed in Ref. Goriely  _et al._
(2015), and compatible with the results obtained within the Improved Leakage-
Equilibration-Absorption Scheme (ILEAS) method (Ardevol-Pulpillo  _et al._ ,
2019) (see in particular their Fig. 14). To do so, we adopt the values of
$L_{\nu_{e}}=0.3\times 10^{53}$ erg/s; $L_{{\bar{\nu}}_{e}}=10^{53}$ erg/s;
$\langle E_{\nu_{e}}\rangle=8$ MeV; $\langle E_{{\bar{\nu}}_{e}}\rangle=12$
MeV. The resulting mean electron fraction at the time of the network
calculation reaches $\langle Y_{e}\rangle=0.23$ in relatively good agreement
with the 0.26 value obtained in Ref. Ardevol-Pulpillo  _et al._ (2019).

Without weak interaction, the initial electron fraction, $Y_{e}^{0}$, of the
various mass elements is low and characteristic of the NS inner crust, i.e.
less than 0.06; consequently the number of free neutrons per seed nuclei can
reach a few hundreds. With such a neutron richness, heavy fissioning nuclei
can be efficiently produced during the r-process. Weak interactions on
nucleons (Eq. 8) increase the initial electron fraction in the ejecta to
values ranging in this scenario II from 0.06 to 0.38 and make the production
of heavy fissioning nuclei and fission recycling during the r-process less
favorable, though not impossible.

On the basis of the BSk14 fission barriers adopted here, the production of
super-heavy neutron-rich nuclei is possible up to $Z=110$ if the neutron
irradiation is large enough to overcome the $N=184$ shell closure. Beyond
$Z=110$, the production of elements is unlikely because these elements fission
spontaneously (Fig. 7a) with a very short lifetime (Fig. 7b). This is the main
reason why the reaction network has been limited to elements $Z\leqslant 110$.
Nuclei reaching this limit ($Z=110$ and $A\geqslant 320$) define a “fission
roof” area ($A\geqslant 320$) (Fig. 7d).

To produce such super-heavy neutron-rich nuclei, the matter has to flow
through the fissioning region formed by (i) $\beta$-delayed fission of
${}^{286}_{94}\mathrm{Pu}_{192}$, (ii) spontaneous fission of
${}^{286}_{95}\mathrm{Am}_{191}$, (iii) $\beta$-delayed fission of
${}^{288-289}_{97}\mathrm{Bk}_{191-192}$, and (iv) spontaneous fission of
elements with $Z\geqslant 98$ (Fig. 7a). ${}^{286}_{95}\mathrm{Am}_{191}$ and
$Z\geqslant 98$ elements have lifetime shorter than $10^{-4}$ s, which makes
neutron capture by these nuclei unlikely. The $\beta$-delayed fission lifetime
of ${}^{286}_{94}\mathrm{Pu}_{192}$ and
${}^{288-289}_{97}\mathrm{Bk}_{191-192}$ ranges from $10^{-3}$ to $10^{-2}$ s.
The neutron-induced fission of ${}^{287-291}_{98}\mathrm{Cf}_{189-193}$ and
${}^{285}_{95}\mathrm{Am}_{190}$ isotopes also happens to be efficient and
contributes to fission recycling (Fig. 7c). The flow through Pu isotopes is
negligible due to the fast $\beta$-delayed neutron emission of
${}^{278}_{94}\mathrm{Pu}_{184}$ with a lifetime $\tau_{\beta kn}=6.6\times
10^{-4}$ s and ${}^{279}_{94}\mathrm{Pu}_{185}$ with $\tau_{\beta
kn}=4.7\times 10^{-3}$ s (Fig. 7b). In addition, the neutron separation energy
of ${}^{279}\mathrm{Pu}$ equals to 0.28 MeV which makes the neutron capture by
${}^{278}\mathrm{Pu}$ unfavorable. Consequently, matter mainly flows through
the Cm isotopic chain to produce super-heavy neutron-rich elements. This
region forms a “fission bottleneck” ($A<292$ and $N>184$) where
${}^{287}_{96}\mathrm{Cm}_{191}$ represents the “crossing point” (Fig. 7d).
However, the production of super-heavy neutron-rich nuclei is no more possible
for temperatures higher than 2 GK (i.e. $k_{B}T=0.17$ MeV) because the
neutron-induced fissions become more favorable than the radiative neutron
captures of ${}^{285-288}_{96}\mathrm{Cm}_{189-192}$ and
${}^{285}_{97}\mathrm{Bk}_{188}$ (Fig. 7c) which close the “fission
bottleneck”.

At the end of the neutron irradiation, heavy neutron-rich elements
$\beta$-decay up to the region of efficient fission. Elements accumulated
during the neutron irradiation along the $N=184$ neutron shell closure
$\beta$-decay up to reach a region of efficient fission. This can be split in
two zones : the “decay roof 0” ($A<278$) and “decay roof 1” ($A\geqslant 278$
and $N\leqslant 184$) which correspond to the region reached by nuclei $Z<94$
and $Z\geqslant 94$, respectively. Elements with $292\lesssim A\lesssim 320$
accumulated after the “fission bottleneck” $\beta$-decay up to reach another
region of efficient fission named “decay roof 2” ($292\leqslant A<320$).

In the following subsection (Sec. III.1), we present the final abundances of
the ejected material obtained in both scenarios as well as their sensitivity
to the initial conditions of the trajectories. The fission contribution to the
r-process and its impact on the final abundances will be studied in both
scenarios in Sec. III.2. The contributions of the various fissioning regions
to the fission recycling is studied in Sec. III.3, while, in Sec. III.4, the
impact of the FFDs on the final abundances are studied with a specific
comparison made when adopting SPY or GEF FFDs.

###  III.1 Final abundances

Figure 8:  (Color online) Final composition (in terms of molar fraction) of
matter ejected in scenarios I and II and obtained either with SPY or GEF FFDs.
For comparison, the solar r-abundance distribution Goriely (1999) (blue dotted
circles) is shown and arbitrarily normalized.  Figure 9:  (Color online) (a-e)
Final abundances of the trajectories computed within scenario II classified
into 5 groups : “1st peak”, “between 1st and 2ndpeak”, “2nd peak”, “rare earth
& no fiss.” and “fiss.”. The green curves with dots are the final abundances
averaged over all trajectories computed within scenario II (a-e). The gray
curves with dots correspond to the mean final abundances within each group.
(f) Final abundances of the trajectories computed within scenario I (black
curves) and the final abundances averaged over all trajectories (gray curve
with dots). (g) Distribution of the trajectories in the ($S^{0}$,$Y_{e}^{0}$)
plane and iso-$X_{n}^{0}$ curves (green curves). (h) Mean value of the initial
neutron mass fraction $X_{n}^{0}$ (green curve) and mean mass
$\left<A\right>_{A\geqslant 50}$ (gray curve) of each group and the associated
ejecta mass fraction of groups for the scenario II (color bars).

The final abundance distributions of the ejected material are shown in Fig. 8
for both scenarios I and II. Calculations are performed adopting either the
SPY or GEF predictions of the FFDs. While in scenario I, only nuclei with
$A\,\,\raise 1.4pt\hbox{$>$}\kern-7.59995pt\lower
2.79999pt\hbox{$\sim$}\,\,120$ are essentially produced, weak interactions on
nucleons also allow the production of lighter species with $80\,\,\raise
1.4pt\hbox{$<$}\kern-7.59995pt\lower 2.79999pt\hbox{$\sim$}\,\,A\,\,\raise
1.4pt\hbox{$<$}\kern-7.59995pt\lower 2.79999pt\hbox{$\sim$}\,\,120$ in
scenario II, but also decrease the production of the Pb peak nuclei as well as
the long-lived actinides. In this case, the choice of the FFD model (SPY or
GEF) is seen to have a negligible impact on the final abundance distribution.
The same conclusions hold if we consider our original version of the SPY FFD
Lemaître  _et al._ (2019). Without weak interactions, with the low initial
electron fractions, the number of neutron per seed is rather high (a few
hundreds), so that the production of the heaviest nuclei is efficient. In this
case, nuclear fission plays a fundamental role during both the neutron
irradiation and at freeze-out Goriely (2015) and the FFD model directly
affects the abundance distribution, particularly around the second peak
$A=130$ where SPY FFDs give higher abundances than GEF (see Sec. III.4). This
nucleosynthesis has been described in detail in Refs. Bauswein  _et al._
(2013); Goriely  _et al._ (2011, 2013a); Just  _et al._ (2015); Goriely  _et
al._ (2015); Goriely (2019). Note that the corrections introduced in the
present study (Sec. II.1) in comparison with our 2019 version Lemaître  _et
al._ (2019) have a small impact on the final abundance distribution. They tend
to slightly increase the second r-abundance peak at $A\simeq 130$ and decrease
its low-$A$ tail. Such differences are significantly smaller than the one
observed between the SPY and GEF predictions.

It is well accepted Hoffman  _et al._ (1997) that the efficiency of the
r-process nucleosynthesis can be related to three main physical properties of
the ejected material, namely the electron fraction $Y_{e}$, the entropy per
baryon $S$ and the expansion timescale $\tau$ Qian and Woosley (1996). In
scenario I, trajectories are characterized by a low initial electron fraction
$Y_{e}^{0}$ ranging between 0.02 and 0.06 at the beginning of the
nucleosynthesis, i.e. when the temperature has dropped below 10 GK and the
density below the drip density ($\rho_{\mathrm{drip}}\simeq 4\times 10^{11}$
g/cm3) (Fig. 9g). Initial entropies $S^{0}$ reach values up to
$170~{}k_{B}/\mathrm{nucleon}$, though most of them remain below
$50~{}k_{B}/\mathrm{nucleon}$. As already mentioned, in scenario II, the
initial electron fraction covers a much larger range between 0.06 and 0.38 due
to the weak interactions (Eq. 8).

Both the initial electron fraction and entropy affect the initial mass
fraction of free neutrons $X_{n}^{0}$ (Fig. 9g, green curves) which is
consequently a parameter of first relevance in the analysis of the efficiency
of the r-process nucleosynthesis. In scenario I, $X_{n}^{0}$ ranges between
0.85 and 0.98, as seen in Fig. 9g, while in scenario II it varies between 0.02
and 0.82. The lower $Y_{e}^{0}$, the larger $X_{n}^{0}$ by definition, but
also the higher the initial entropy, the larger $X_{n}^{0}$ due to the release
of free neutrons through photodissociation. Therefore, $X_{n}^{0}$ engulfs
information on both the initial electron fraction and entropy. The sudden
variation of $X_{n}^{0}$ observed at
$S^{0}=8\mathrm{-}10~{}k_{B}/\mathrm{nucleon}$, independently of $Y^{0}_{e}$,
is linked to the initial abundances of trans-alpha elements ($A>4$) :
$\sum_{A>4}Y^{0}(A)$. The matter is not completely dissociated for
trajectories with a low initial entropy $S^{0}<10~{}k_{B}/\mathrm{nucleon}$,
which makes the initial abundance of trans-alpha elements (with a mean mass
$\left<A\right>_{A>4}^{0}$ ranging from 50 to 85) non-negligible and reduces
the initial neutron mass fraction. For $S^{0}>10~{}k_{B}/\mathrm{nucleon}$,
the initial abundances of trans-alpha elements is negligible, i.e. the ejected
nuclear matter is completely dissociated into protons, neutrons and alphas.
The low entropy effect is more important for trajectories in scenario II where
for $18\%$ of the ejected mass, $\sum_{A>4}Y^{0}(A)>0.001$, whereas in
scenario I, only $5\%$ of the ejecta has $\sum_{A>4}Y^{0}(A)>0.001$. Trans-
alpha elements can boost the r-process depending on the neutron irradiation.
Initial and final abundance distributions are more or less similar for
trajectories with a low initial neutron mass fraction ($X_{n}^{0}<0.3$) and a
low initial entropy ($S^{0}<10~{}k_{B}/\mathrm{nucleon}$); in both cases, the
r-process is weak due to the low neutron irradiation.

The relevance of the initial neutron mass fraction $X_{n}^{0}$ in the analysis
of the efficiency of the r-process can be clearly seen in Figs. 9. The various
trajectories can be classified according to their final abundance
distributions (Figs. 9a-f). The trajectories computed within scenario I are
classified in the group labeled “no $\nu$” while those computed within
scenario II are divided into five groups : the group labeled “fission”
contains trajectories with the highest fission recycling
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>0.01$, more details about the
definition of this fission recycling indicator will be given in Sec. III.2).
Among the trajectories with $\sum_{125\leqslant A\leqslant
135}Y^{\mathrm{f}}(A)>0.01$, if $\sum_{190\leqslant A\leqslant
200}Y^{\mathrm{f}}(A)<0.001$ then they are classified in the group “2nd peak”
else in the group “rare earth & no fission”. The trajectories with
$\sum_{75\leqslant A\leqslant 85}Y^{\mathrm{f}}(A)>0.1$ are classified in the
group “1st peak”. Remaining trajectories correspond to the group “between 1st
and 2ndpeak”. The two first groups “1st peak” and “between 1st and 2ndpeak”
represent each 7% and 6%, respectively, of the ejected mass within scenario II
(see Fig. 9h). The other groups “2nd peak”, “rare earth & no fission” and
“fission” represent 41%, 30% and 16%, respectively. The mean initial neutron
mass fraction increases with the group (Fig. 9h, green curve), heavy elements
production is possible only with a high initial neutron mass fraction.
Similarly, the mean mass of elements heavier than 50
($\left<A\right>_{A\geqslant 50}$) also increases with the group (Fig. 9h,
gray curve) but reaches a maximum around 170 when fission occurs in groups
“fission” and “no $\nu$” corresponding to the saturation regime induced by the
fission recycling (see Sec. III.2). In the ($S^{0},Y_{e}^{0}$) plane (Fig.
9g), the trajectories of the various groups are distributed according the
iso-$X_{n}^{0}$ lines which clearly demonstrate the relevance of the initial
neutron mass fraction $X_{n}^{0}$ in the analysis of the r-process efficiency.

###  III.2 Fission recycling

Figure 10: (Color online) Initial neutron mass fraction $X_{n}^{0}$ (black
curve with dots) and the mass distribution of the ejecta (blue curve)
depending on the fission recycling $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$. The
dashed line separates abundances computed within scenario I
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>1$) to the ones computed within
scenario II ($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<1$).

(a) For scenario I where the mean value of the total fissioning cumulative
mass fraction $\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}=1.45$.

(b) For scenario II where the mean value of the total fissioning cumulative
mass fraction $\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}=0.03$.

Figure 11:  (Color online) ($\alpha$) Final abundances obtained with SPY FFDs,
averaged for each bin of $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ colored by
their $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ value and compared to the average
final abundances (green solid curve with dots). ($\beta$) Same final
abundances in the ($A,X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$) plan colored by
their abundances. The mean mass $\langle A\rangle_{A\geqslant 50}$ for each
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ bins are represented by red crosses.

Fission recycling for a given trajectory can be quantified by the so-called
total cumulative fissioning mass fraction

$\displaystyle X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$
$\displaystyle=\sum_{Z,A}\sum_{\mathrm{r=fiss}}\int\frac{dX_{C}}{dt}(Z,A,t,r)dt$
$\displaystyle=\sum_{Z,A}\int X(Z,A,t)\left[N_{n}\langle\sigma
v\rangle_{\mathrm{n,f}}^{Z,A}+\lambda_{\mathrm{sf}}^{Z,A}+\lambda_{\beta\mathrm{df}}^{Z,A}\right]\;dt$
$\displaystyle=\sum_{Z,A}X^{\mathrm{cum}}_{\mathrm{fiss}}(Z,A)$

where $\frac{dX_{C}}{dt}(Z,A,t,r)$ corresponds to the reaction flux, expressed
in terms of the mass fraction $X$ for a given nucleus $(Z,A)$, at a given time
$t$ and for a given fission reaction $r$.
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ consequently corresponds, at a given
time, to the cumulative mass fraction of fissioning nuclei that are destroyed
by fission reactions and gives a quantitative indicator of the amount of mass
recycled by fission processes. Fission can take place by spontaneous (at a
rate $\lambda_{\mathrm{sf}}$) or $\beta$-delayed (at a rate
$\lambda_{\beta\mathrm{df}}$) fission or by neutron-induced fission (at a rate
$N_{n}\langle\sigma v\rangle_{\mathrm{n,f}}$, where $\langle\sigma
v\rangle_{\mathrm{n,f}}$ is the corresponding astrophysical rate and $N_{n}$
the neutron density). Consequently, there are three factors contributing to
the fission recycling : (i) the quantity of fissionable nuclei, (ii) the
fission rates and (iii) the amount of time over which fission reactions may
occur.

The total fissioning cumulative mass fraction
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ increases with initial neutron mass
fraction $X_{n}^{0}$ (Fig. 10, black curve with dots). The fission recycling
can therefore be explained in term of initial neutron mass fraction
$X_{n}^{0}$, i.e. the more neutron-rich the initial ejecta, the larger the
production of heavy elements, hence the more efficient the fission.

####  III.2.1 Scenario I

Within scenario I, $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>1$ for all ejected
trajectories due to the high initial neutron mass fraction ($X_{n}^{0}>0.85$).
In Fig. 11(a), the final abundances of the 500 trajectories have been sorted
and binned as a function of $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ which is
found to range between 1.19 and 1.82 with a mean value over all trajectories
of $\langle X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}=1.45$.
The final abundances distribution for a given trajectory can be directly
linked to its total fissioning cumulative mass fraction
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$. An overall shift of final abundances
towards heavier elements can be expected with increasing fission recycling but
the mean $A\geqslant 50$ mass, $\langle A\rangle_{A\geqslant 50}$, does not
increase with $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ (Fig. 11(a)$\beta$, red
crosses), and stays rather constant between 166 and 172. This value remains
constant because the final abundances are dominated by the second ($A=130$)
and third ($A=195$) peaks.

The final abundance of light elements ($A\leqslant 4$), mainly
$\alpha$-particles, is significant with $0.1<\sum_{A\leqslant
4}Y^{\mathrm{f}}(A)<0.84$ depending on the trajectory. It increases with the
initial entropy $S^{0}$, inducing an overall downward shift of the final
abundances of heavier elements ($A>4$). As $X_{n}^{0}$ is linked to
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ (Fig. 10), the initial entropy $S^{0}$
of trajectories within a given bin of $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$
covers almost the whole range of possible $S^{0}$ (since $X_{n}^{0}$ isolines
evolve in the $(S^{0},Y_{e}^{0})$ plane, as seen in Fig. 9g). This $S^{0}$
admixture of trajectories within a bin of
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ smooths the overall shift of the final
abundances of heavy elements ($A>4$). However, fission recycling has a
significant impact : the final $A<195$ abundances decrease with
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$, whereas for elements $A>195$ they
increase (Fig. 11(a)$\alpha$). With increasing values of
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$, the abundances in the mass region
$140<A<170$ drop by an order of magnitude, but around $A=202$ they increase
from $4\times 10^{-4}$ up to $2\times 10^{-3}$, while for $A<80$ elements,
they drop below $10^{-5}$ (Fig. 11(a)).

Neglecting the effect related to the variation of light elements abundance, a
higher initial neutron mass fraction $X_{n}^{0}$ induces more fission
recycling which implies a decrease of the final $A<80$ abundances. A high
neutron irradiation leads to a significant production of super-heavy elements
($A>300$) whose fission fragments combined to new neutron captures shift the
third peak to higher masses.

Part of the nuclei that accumulated during neutron irradiation along the
neutron drip line (including the neutron shell closure $N=184$) with
$208<A\leqslant 278$ are the progenitors of the trans-lead elements. Among
these accumulated nuclei, those with $260\lesssim A\leqslant 278$ ($79\lesssim
Z\leqslant 94$) predominantly $\beta$-decay up to reach the “decay roof 0”
where they fission. Those with $208<A<232$ $\beta$\- and $\alpha$-decay up to
the third abundance peak, while the remaining intermediate nuclei with
$232\leqslant A\lesssim 260$ $\beta$\- and $\alpha$-decay up to the third
abundance peak or $\beta$\- decay and reach the “decay roof 0” where they
fission or decay into the long-lived ${}^{232}\mathrm{Th}$ and
${}^{235-238}\mathrm{U}$ nuclei. Final $A>204$ abundances decrease with
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ because their progenitors are the
nuclei accumulated along the neutron shell closure $N=184$ during the neutron
irradiation. This accumulation is reduced for high neutron irradiations which
enable to overcome the neutron shell closure $N=184$, hence to increase the
fission recycling.

####  III.2.2 Scenario II

With the low initial neutron mass fraction $X_{n}^{0}$ found in scenario II,
about $55\%$ of the ejected mass is not subject to fission recycling
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<10^{-7}$) (Fig. 10, blue curve) and the
maximum value of $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}=0.6$. The fission
recycling is reduced by a factor 50 with a mean value of the total fissioning
cumulative mass fraction $\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}$ that now amounts
to 0.03 to be compared with 1.45 for scenario I. The present scenario is well
suited to study low fission recycling effects on the final abundances. For
this reason, trajectories are sorted in $\log_{10}$ bins of their total
fissioning cumulative mass fraction, as shown in Fig. 11(b). The bins, except
for the first one, represent each about $5\%$ to $10\%$ of the ejected mass
(Fig. 10).

The final abundance of the light $A\leqslant 4$ elements in bins are more or
less similar ranging from 40% to 55% with no global trend which induces a
smaller shift of the final abundances due to trans-alpha elements in
comparison with scenario I.

$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ is linked to the initial neutron mass
fraction $X_{n}^{0}$, as observed in Fig. 10 (black curve with dots) where
$X_{n}^{0}$ is seen to increase linearly with
$\log_{10}\left(X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\right)$ from 0.5 to 0.75.
In other words, $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ increases exponentially
with $X_{n}^{0}$. An increase with $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$,
hence with $X_{n}^{0}$, of the mean mass $\langle A\rangle_{A\geqslant 50}$
from 105 up to 168 is observed in Fig. 11(b)$\beta$ (red crosses), i.e. the
higher $X_{n}^{0}$, the heavier the final nuclei produced.

For trajectories with a low initial neutron mass fraction, $X_{n}^{0}\approx
0.5$ and $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<10^{-7}$, $A>130$ nuclei,
including lanthanides, are not significantly produced in comparison with the
other trajectories (Fig. 11(b)) and production beyond $A>180$ is negligible.
Fission recycling does not take place in this case because there are not
enough neutrons to overcome significantly the $N=82$ neutron shell closure.
For higher initial neutron mass fraction $X_{n}^{0}\approx 0.6$
($10^{-7}<X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<10^{-5}$), nuclei in the rare
earth and third peak regions start to be produced, but those with $A>200$ are
still highly underproduced and fission recycling remains negligible.

Abundances of third peak elements increase in the range
$10^{-5}<X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<10^{-1}$ ($0.65<X_{n}^{0}<0.71$)
as does the mean mass $\langle A\rangle_{A\geqslant 50}$ which increases
linearly with $\log_{10}\left(X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\right)$ up
to 168. In this $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ range, abundances of
rare earth nuclei ($145<A<180$) increase to reach a maximum at
$10^{-4}<X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<10^{-2}$ before decreasing for
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-2}$. The $N=126$ and $N=184$
neutron shell closures start to be overcome for
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-4}$ leading to the synthesis of
$A>204$ nuclei in a non-negligible amount. Fission recycling starts to play a
role for trajectories with $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-4}$ (or
equivalently with $X_{n}^{0}>0.65$), which represent $29\%$ of the ejected
mass in scenario II.

In the last bin of total fissioning cumulative mass fraction, i.e.
$0.1<X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<0.61$ and an average initial neutron
mass fraction $X_{n}^{0}=0.76$, the third r-abundance peak becomes higher than
the second one and $A>204$ nuclei dominate the composition. The mean mass
$\langle A\rangle_{A\geqslant 50}$ does not increase significantly, a
saturation regime is reached with $\langle A\rangle_{A\geqslant 50}\approx
170$ as observed for scenario I (see Sec. III.2.1); the r-process dynamics for
such trajectories is similar to the one taking place in scenario I.

To complete the discussion of Sec. III.2.1, fission recycling plays a dominant
role when $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>0.1$ corresponding to
$X_{n}^{0}>0.75$ which concerns $9.5\%$ of the ejected mass within scenario
II. A saturation regime is reached $\langle A\rangle_{A\geqslant 50}\approx
170$, when a high initial neutron mass fraction $X_{n}^{0}$ induces more
fission processes feeding back the $80<A<200$ mass region.

###  III.3 Fission seeds

Figure 12:  (Color online) Repartition of the fission recycling
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ between the 5 fission seed areas
defined in Fig. 7d. The dashed line separates scenario I
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>1$) from scenario II
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<1$).

(a) For scenario I. (b) For scenario II.

Figure 13:  (Color online) Fission seed contribution ($x_{\mathrm{FS}}$)
depending on the mass of fission seed (the total height). Each bins contains
the isotopic distribution of $x_{\mathrm{FS}}$ for a given A.

We now turn to the mean contribution of fissioning nuclei, or fission seeds,
to the fission recycling and to the final abundance distribution. The
contribution of a given fissioning nucleus $(Z,A)$ to the fission recycling,
$x_{\mathrm{FS}}(Z,A)$, can be defined, for a given set of trajectories, as
the fraction of the total fissioning cumulative mass fraction $\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}$ due to this
specific fissioning nucleus ($Z,A$), i.e.

$x_{\mathrm{FS}}(Z,A)=\frac{\langle
X^{\mathrm{cum}}_{\mathrm{fiss.}}(Z,A)\rangle_{\mathrm{traj}}}{\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}}~{}.$ (10)

In the specific NS merger cases studied here, $\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}$ reaches rather
similar values of 1.45 and 1.41 in scenario I when using SPY or GEF FFDs,
respectively and 0.03 for both FFDs within scenario II. This similitude is due
to the fact that SPY and GEF FFDs are not fundamentally different (see Fig.
15) and the properties of the fission seeds are not related to their FFDs (see
Fig. 7). Consequently, only results with SPY FFDs are presented here.

####  III.3.1 Location in the $(N,Z)$ plane of the contributing fission seeds

Due to the specific $\beta$-decay and neutron capture properties of nuclei
above Th, in particular the “fission bottleneck”, the “fission roof” and the
$N=184$ neutron shell closure, not all the trans-Th nuclei play a role in
fission recycling. The contribution of a given fission seed area (see Fig. 7d
defining the 5 areas of relevance for the nuclear physics adopted in the
present study) to the fission recycling $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$
is the sum of $x_{\mathrm{FS}}(Z,A)$ for all nuclei located in such a specific
fission seed area. Such contributions evolve with the fission recycling, as
shown in Fig. 12. The contribution of the “decay roof 0”, corresponding to the
fission of nuclei originally accumulated along the neutron shell closure
$N=184$ and $Z<94$ and reaching the fission efficient region by $\beta$-decay
after the freeze-out, decreases with increasing
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ since the fission recycling increases
with the initial neutron mass fraction $X_{n}^{0}$. With increasing
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ and $X_{n}^{0}$, matter accumulated
along $N=184$ isotone spreads up to Fm thanks to the successive $\beta$-decays
and neutron captures along this shell closure. More nuclei above $Z=94$ are
produced and increase the contribution of the “decay roof 1” and consequently
decrease the one of the “decay roof 0”. For
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-6}$, the neutron shell closure
$N=184$ starts to be overcome and nuclei in the “fission bottleneck” area
contribute to the fission recycling. In the range
$10^{-6}<X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<10^{-1}$, the contribution of
the “fission bottleneck” is roughly constant representing 20% of the total
cumulative fissioning mass fraction. The matter starts to flow through the
“fission bottleneck”, via Cm isotopes, and reach the “fission roof” for
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-4}$. For high fission recycling
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>1$ as in scenario I), the total
cumulative fissioning mass fraction is dominated by the “fission bottleneck”
and “fission roof” areas, representing around 60% and 30%, respectively. The
contribution of the “decay roof 2” is low because matter does not pile up in
this region due to the absence of shell closure.

####  III.3.2 Scenario I

Sixteen fission seeds are found to contribute for more than $1\%$ to the
fission recycling (Fig. 13(a)) within scenario I. They are distributed mainly
in the “fission bottleneck” and the “fission roof” areas, representing
respectively 57% and 31% of the total cumulative fissioning mass fraction
(Fig. 13(a)).

A major part of the flow crossing the $N=184$ shell closure is concentrated
along the Am chain, fed by the favorable $\beta$-decay of
${}^{278}_{94}\mathrm{Pu}_{184}$, where ${}^{285}_{95}\mathrm{Am}_{190}$ with
$x_{\mathrm{FS}}=45\%$ is the major contributor to the fission recycling. This
favorable fission is due to the low fission barriers of a few nuclei dropping
below 4 MeV according to the BSk14 Skyrme HFB calculations Goriely  _et al._
(2009). They are also main contributors in the “fission roof” area, the main
one in this area and the second major one of all the contributors is
${}^{344}_{109}\mathrm{Mt}_{233}$ with $x_{\mathrm{FS}}=12\%$. In this case,
the primary fission barrier drop below 2 MeV. As the main fission seeds are in
the $275<A<290$ and $330<A<346$ ranges, the FFDs of these fission seeds affect
the final production of nuclei in the $80<A<210$ range.

####  III.3.3 Scenario II

As the initial neutron mass fraction is low in this case, the fission
recycling remains low too. The contribution of the “fission bottleneck” and
the “fission roof” amount to 37% and 6%, respectively, to be compared with 57%
and 31% in scenario I. The main fission seeds area is the “decay roof 1” (Fig.
13(b)) because the low neutron irradiation makes it difficult to overcome the
$N=184$ neutron shell closure. It represents 45% of
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ and the main contributors are Md and No
isotopes. The contribution of the “decay roof 0” increases from $2\%$ in
scenario I to the non-negligible value of $9\%$ in scenario II.

The main fission seed is still ${}^{285}_{95}\mathrm{Am}_{190}$ with
$x_{\mathrm{FS}}=25\%$ with additional ones located in the
$276<A_{\mathrm{CN}}<288$ mass region with $x_{\mathrm{FS}}=2-10\%$. The
fission recycling occurs mostly after freeze-out in scenario II. The post-
freeze-out fission seeds, located in the “decay roof 0 and 1”, represent about
54% of $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$.

###  III.4 Fission progenitors

Figure 14:  (Color online) Contribution of the fission fragments to the final
production of nuclei with mass number $A$
$x_{\mathrm{FFP}}^{\mathrm{f}}(A,X_{\mathrm{fiss,tot}}^{\mathrm{cum}})$
obtained with SPY FFDs. Gray curve delimits
$x_{\mathrm{FFP}}^{\mathrm{f}}(A,X_{\mathrm{fiss,tot}}^{\mathrm{cum}})=0.1$
and white one
$x_{\mathrm{FFP}}^{\mathrm{f}}(A,X_{\mathrm{fiss,tot}}^{\mathrm{cum}})=0.5$.

We now turn in the analysis of the so-called fission progenitors, i.e. the
nuclei for which fission contributes in a non-negligible way to their final
abundance. Through a detailed tracking of the fission processes taking place
during the r-process, the impact of fission on the final abundances can be
quantified by the analysis of the fission fragments progenitors and their
related fission seeds. For each trajectory, the progenitors of a given fission
fragment can be identified and the fission contribution to the production of
this final nucleus can be estimated. More precisely, the contribution stemming
from the fission fragments $x_{\mathrm{FFP}}(Z,A,t_{i})$ (FFP stands for
fission fragments progenitors) to a nucleus $(Z,A)$ with the molality
$Y_{m}(Z,A,t_{i})$ (in mol/g) at time $t_{i}$ is defined as

$x_{\mathrm{FFP}}(Z,A,t_{i})=\frac{x_{\mathrm{FFP}}(Z,A,t_{i-1})Y_{m}(Z,A,t_{i-1})+\frac{\Delta
Y_{m,\mathrm{P}}}{\Delta t}(Z,A,t_{i-1})\Delta
t\sum_{r}x_{\mathrm{FFP}}(Z_{r},A_{r},t_{i-1})x_{\mathrm{RP}}(Z,A,r,t_{i-1})}{Y_{m}(Z,A,t_{i-1})+\frac{\Delta
Y_{m,\mathrm{P}}}{\Delta t}(Z,A,t_{i-1})\Delta t}$ (11)

where $\Delta t=t_{i}-t_{i-1}$ is an arbitrary time step taken here to be of
the order of $\approx t_{i}/10$ and $\frac{\Delta Y_{m,\mathrm{P}}}{\Delta
t}(Z,A,t_{i-1})$ is the total production rate (in mol/g/s) of nucleus $(Z,A)$
between $t_{i-1}$ and $t_{i}$, expressed in terms of the molality $Y_{m}$ (if
normalized to 1, it corresponds to the molar fraction $Y$). The reaction
fraction $x_{\mathrm{RP}}(Z,A,r,t_{i-1})$ is the fraction of the total
production rate due to the reaction $(r)$. The reactions considered here are
$(n,\gamma)$, $(\gamma,n)$, $(\beta,\gamma)$, $(\beta,n)$, $(\beta,2n)$,
$(\beta,3n)$, $(\gamma,\alpha)$, (sf) (i.e. spontaneous fission),
$(\beta,\mathrm{f})$ and $(n,\mathrm{f})$. In other words, the contribution of
fission fragments to the production of a nucleus ${}_{Z}^{A}\mathrm{X}$ at
time $t_{i}$ is the weighted average between the previous contribution
$x_{\mathrm{FFP}}(Z,A,t_{i-1})$ and the contributions
$x_{\mathrm{FFP}}(Z_{r},A_{r},t_{i-1})$ due to the production of this element
by a given reaction
$r~{}:{}_{Z_{r}}^{A_{r}}\mathrm{Y}\rightarrow{}_{Z}^{A}\mathrm{X}+...$,
between $t_{i-1}$ and $t_{i}$, at a rate of $\frac{\Delta
Y_{m,\mathrm{P}}}{\Delta t}(Z,A,t_{i-1})x_{\mathrm{RP}}(Z,A,r,t_{i-1})$. If a
nucleus is exclusively produced by fission, $x_{\mathrm{FFP}}=1$.

This quantity is well suited to quantify the impact of fission fragments on
final abundances because it is not affected by the decay of post-neutron
fission fragments thanks to the time tracking. To obtain the overall final
$x_{\mathrm{FFP}}$, hereafter $x_{\mathrm{FFP}}^{\mathrm{f}}$, (Figs. 16, red
dotted curves), the tracking (Eq. 11) is performed up to $t=1$ yr when the
residual fission does not contribute anymore to the production of light
nuclei.

The contribution of fission fragments to the final production depends on the
fission recycling indicator $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$ (Fig. 14,
gray curve) and starts to be significant, i.e.

$\max_{A}\left[x_{\mathrm{FFP}}^{\mathrm{f}}(A,X_{\mathrm{fiss,tot}}^{\mathrm{cum}})\right](X_{\mathrm{fiss,tot}}^{\mathrm{cum}})>0.1,$

for $X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-4}$ ($X_{n}^{0}>0.65$) which
corresponds to the value given in Sec. III.2.2. For
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}>10^{-2}$, fission contributes at least
to 50% of the production of the final nuclei with $108<A<120$ and $139<A<155$.
For a high fission recycling ($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\approx
1$), this contribution can reach nearly 100% for the elements with $A\approx
110$.

In addition to the strong sensitivity to the fission recycling,
$x_{\mathrm{FFP}}^{\mathrm{f}}(A)$ is obviously affected by the FFDs model
adopted, even if the fission recycling does not depend significantly on the
FFDs model. This sensitivity to the FFDs model is studied for both scenario in
the two subsections below. In order to clarify the link between
$x_{\mathrm{FS}}$ and $x_{\mathrm{FFP}}^{\mathrm{f}}$, we also introduce here
two new quantities, namely the relative contribution of each fission seed to
the production of a final nucleus $A$ by fission

$x^{\mathrm{f}}_{\mathrm{FS/FFP}}(A,A_{\mathrm{FS}})=\frac{x_{\mathrm{FFP}}^{\mathrm{f}}(A,A_{\mathrm{FS}})}{x_{\mathrm{FFP}}^{\mathrm{f}}(A)}$
(12)

where $x_{\mathrm{FFP}}^{\mathrm{f}}(A,A_{\mathrm{FS}})$ is obtained by
tracking only fission fragments produced by the specific fission seed
$A_{\mathrm{FS}}$, $x_{\mathrm{RP}}(Z,A,t_{i-1})$ being replaced by
$x_{\mathrm{RP}}(Z,A,A_{\mathrm{FS}},t_{i-1})$ in Eq. 11. and the contribution
of a compound nucleus $(Z_{\mathrm{CN}},A_{\mathrm{CN}})$ to the fission
recycling, which can be expressed from Eq. 10 as

$x_{\mathrm{CN}}(Z_{\mathrm{CN}},A_{\mathrm{CN}})=\sum_{(Z,A)_{\mathrm{CN}},r=\left\\{\mathrm{sf,\beta
df,nif}\right\\}}x_{\mathrm{FS}}(Z_{r},A_{r},r)$ (13)

where $x_{\mathrm{FS}}(Z_{r},A_{r},r)$ is the contribution to the fission
recycling $\langle
X_{\mathrm{fiss,tot}}^{\mathrm{cum}}\rangle_{\mathrm{traj}}$ of the fission
seed $(Z_{r},A_{r})$ fissioning by a given fission mode $r$ (spontaneous,
$\beta$-delayed or neutron induced). In Eq. 13, the summation is performed in
such a way that the mass of the compound nucleus formed from the fission seed
is equal to $A_{\mathrm{CN}}$ and the proton number to $Z_{\mathrm{CN}}$.

(a) Around the neutron shell closure $N=184$, these nuclei are responsible for
64/86% of the fission recycling in scenario I/II.  (b) Around the “fission
roof”, these nuclei are responsible for 30/5% of the fission recycling in
scenario I/II.

Figure 15: (Color online) Post-neutron FFDs from SPY model computed at a
constant excitation energy $Q=8$ MeV (green curves) and from GEF model for
thermal neutron-induced fission (blue curves). For each box, the x axis ranges
from $A=80$ to 210 by step of 10 and the y axis from 0 to 15% by step of 2%.
Colored background correspond to the contribution of the compound nucleus
$x_{\mathrm{CN}}$ in scenario I/II. Only contributions $x_{\mathrm{CN}}>0.001$
are displayed. FFDs in thick orange boxes correspond to the
$N_{\mathrm{CN}}=184$ neutron shell closure.

####  III.4.1 Scenario I

(a) For scenario I. (b) For scenario II.

Figure 16:  (Color online) ($\alpha$ and $\gamma$) Contribution of the fission
fragments to the final production ($x^{\mathrm{f}}_{\mathrm{FFP}}$) (red
dotted curves) and fission seeds contributions to the production of fission
fragment progenitors ($x^{\mathrm{f}}_{\mathrm{FS/FFP}}$) (color scale)
computed with SPY($\alpha$) or GEF ($\gamma$) FFDs . ($\beta$ and $\delta$)
Mean fission yields $\left<Y_{\mathrm{FF}}\right>_{\mathrm{FS}}$ for SPY
($\beta$) or GEF model ($\delta$) averaged over fission seeds from the
“fission bottleneck” and the “fission roof” (green curves, denoted as pre-
freeze-out) or from “decay roofs” (orange curves, denoted as post-freeze-out).

In scenario I, fission processes contribute significantly to the production of
nuclei with $80<A<200$ for both FFDs models (Fig. 16(a), red dotted curves)
due to the high fission recycling. The shape of
$x^{\mathrm{f}}_{\mathrm{FFP}}(A)$ is affected by the FFDs model (Figs.
16(a)$\alpha$ and $\gamma$, red dotted curves). However the link between
$x^{\mathrm{f}}_{\mathrm{FFP}}$ and FFDs is no straightforward.

Two mean fission yields distributions
$\left<Y_{\mathrm{FF}}\right>_{\mathrm{FS}}$ can be defined, the first one is
associated with the “fission bottleneck” and the “fission roof”, denoted as
pre-freeze-out mean fission yields, corresponds to the mean fission yields
distribution during the neutron irradiation, before freeze-out (Figs. 16,
green curves). The second one, denoted as post-freeze-out mean fission yields,
is associated with the “decay roofs” (Figs. 16, orange curves), after freeze-
out.

For both FFDs model, the pre-freeze-out mean fission yields is dominated by
the FFDs of 286Am ($x_{\mathrm{CN}}=0.45$) and 345Mt ($x_{\mathrm{CN}}=0.12$)
formed by the neutron capture of 285Am and 344Mt. With SPY FFDs (see Figs. 15,
green curves), these two main contributors fission asymmetrically. The three
peaks pattern seen in the pre-freeze-out
$\left<Y_{\mathrm{FF}}\right>_{\mathrm{FS}}$ (Fig. 16(a)$\beta$) is due to (i)
the light fragments of 286Am feeding the peak around $A=120-130$, (ii) the
heavy fragments of 286Am plus the light fragments of 345Mt feeding the peak
around $A=145-160$, and (iii) the heavy fragments of 346Mt responsible for the
peak $A=180-190$. In contrast, the GEF model (see Figs. 15, blue curves)
predicts a more asymmetric fission of 286Am but a more symmetric one for
346Mt. As a result, a double hump pattern is obtained for the pre-freeze-out
$\left<Y_{\mathrm{FF}}\right>_{\mathrm{FS}}$ (Fig. 16(a)$\delta$) where the
peak around $A=100-120$ is due to to the light fragments of 286Am and the one
around $A=150-180$ to the heavy fragments of 286Am plus the symmetric
fragments of 346Mt.

The main contributors to the post-freeze-out mean fission yields, it i.e. to
the fission recycling in the “decay roofs” areas, are isotopes from Fm to Lr
with $A_{\mathrm{CN}}=278$ to 284 where $x_{\mathrm{CN}}$ reaches values up to
0.016 for 283No (see Fig. 15(a)). Those nuclei are characterized by an
admixture of slightly asymmetric fission and a triple-peak fission where the
maximum yields are found around $A=130$ for SPY FFDs. GEF FFD is more
asymmetric leading to post-freeze-out mean fission yields roughly constant in
the $A=110-160$ mass range.

During the neutron irradiation, fission fragments captures neutrons. The
$A<130$ fission fragments accumulate along the neutron shell closure $N=82$
and the relative contribution from the “fission bottleneck”
$x^{\mathrm{f}}_{\mathrm{FS/FFP}}$ increases at $A=120-130$. The fission
fragments produced above $N=82$ ($A\gtrsim 130$) spread out up to $N=126$
($A\lesssim 195$). For $A>160$, the fission contribution to the final
production $x^{\mathrm{f}}_{\mathrm{FFP}}$ and the relative contribution of
the “fission bottleneck” $x^{\mathrm{f}}_{\mathrm{FS/FFP}}$ is higher with GEF
than with SPY since the pre-freeze-out distribution is more asymmetric due to
the more asymmetric FFD of 246Am. At freeze-out, there is no more fission
fragments produced by the “fission bottleneck” and the “fission roof”. The
final abundance peak $A=130$ predicted by SPY model (Fig. 8) is due to the
high post-freeze-out mean fission yields around $A=130$ (Fig. 16(a), orange
curves).

$x^{\mathrm{f}}_{\mathrm{FFP}}$ is locally reduced around $A=130$ due to the
accumulation of non-recycled material along the neutron shell closure $N=82$.
This reduction is limited with SPY model since the pre- and post-freeze-out
mean fission yields are higher than in the GEF case around $A=130$. A low
$x^{\mathrm{f}}_{\mathrm{FFP}}$ for $80<A<100$ could be expected since the
mean fission yields are negligible for both FFD models. However, in this case,
there is no matter accumulation in this mass region, so that the low
$80<A<100$ production ($Y^{\mathrm{f}}<10^{-4}$) occurring after freeze-out is
essentially due to the asymmetric fission of seeds located in the “decay roof
0”. This leads to a non-negligible value of $x^{\mathrm{f}}_{\mathrm{FFP}}$
for $80<A<100$. The drop of $x^{\mathrm{f}}_{\mathrm{FFP}}$ for $A>180$ is due
to the accumulation of matter along the neutron shell closure $N=126$.

####  III.4.2 Scenario II

In scenario II, the reduced fission recycling leads to a non-negligible
$x^{\mathrm{f}}_{\mathrm{FFP}}$ that does not exceed 30% only around
$130<A<185$, irrespective of the FFDs model adopted (Fig. 16(b), red dotted
curves).

For both FFDs models, the post-freeze-out mean fission yields is roughly
similar than the one within scenario I since the main contributors from the
“decay roof” regions are the same in both cases (see Fig. 15(a)) where
$x_{\mathrm{CN}}$ reaches values up to 0.08 for 282No. The fission seed
contribution from the “fission roof” is small, so that the $A=180-190$ pre-
freeze-out mean fission yields are negligible with the SPY model. The GEF pre-
freeze-out mean fission yields is roughly similar to the one found in scenario
I, though the $A=100-120$ peak is now found higher than the $A=160-180$ one.

The low neutron irradiation makes the contribution of the “fission bottleneck”
and the “fission roof” relatively small within this scenario, leading to a
total $x_{\mathrm{FS}}<0.25$ for 90% the ejected mass (i.e.
$X_{\mathrm{fiss,tot}}^{\mathrm{cum}}<0.1$) (Fig. 12). The role played by the
pre-freeze-out mean fission yields is consequently reduced. It remains more
difficult to overcome the neutron shell closure $N=82$ and more non-recycled
material accumulates along this shell closure. For this reason, the final
abundance around $A=130$ is higher than in scenario I with a negligible
fission contribution to the $A\leqslant 130$ abundances. As the SPY post-
freeze-out mean fission yields is peaked around $A=120-160$,
$x^{\mathrm{f}}_{\mathrm{FFP}}$ is maximal for $A=135-160$. Compared to SPY,
GEF pre- and post-freeze-out mean fission yields are higher for $A\gtrsim 160$
which leads to a higher $x^{\mathrm{f}}_{\mathrm{FFP}}$ in this mass region.

##  IV Conclusion

In the present study, new improvements have been brought to our scission-point
model, called SPY, to derive the fission fragment distribution for all
neutron-rich fissioning nuclei of relevance in r-process calculations. These
improvements include a phenomenological modification of the scission distance
and an updated smoothing procedure of the distribution to take into account
effects that are neglected in the original model. Such corrections lead to a
better agreement with experimental fission yields. In particular, SPY FFDs are
now in overall good agreement with the experimental symmetric/asymmetric
transition in the Th and Pa isotopic chains. Those yields at scission are also
used to estimate the number of neutrons emitted by the excited fragments on
the basis of different neutron evaporation models. Our new fission yields
significantly differ from to those predicted by the widely used GEF model.

The impact of fission on the r-process nucleosynthesis in neutron mergers is
also re-analyzed. Two scenarios are considered, the first one with low initial
electron fraction is subject to intense fission recycling, in contrast to the
second one which includes weak interactions on free nucleons resulting in less
pronounced fission recycling. The efficiency of the r-process nucleosynthesis
and the corresponding abundance distribution in a given mass element has been
discussed essentially in terms of the initial neutron mass fraction
$X_{n}^{0}$ which engulfs information on both the initial electron fraction
and entropy. The contribution fission processes may have to the final
abundance distribution has been studied in detail in the light of newly
defined quantitative indicators, namely indicators for the fission recycling
($X_{\mathrm{fiss,tot}}^{\mathrm{cum}}$), fission seeds ($x_{\mathrm{FS}}$)
and fission fragment progenitors ($x^{\mathrm{f}}_{\mathrm{FFP}}$). The
various regions of the nuclear chart responsible for fission recycling during
the neutron irradiation as well as after freeze-out have been identified for
the nuclear physics adopted in the present calculations, what we defined as
the “decay roofs”, the “fission bottleneck” and the “fission roof”. Within the
“fission bottleneck” region, Am isotopes around $A=285$ play a key role by
limiting the nuclear flow to higher elements and strongly contributing to the
fission recycling, mainly due to the low fission barriers predicted by the
BSk14 interaction in this region. Above $Z=110$, the so-called “fission roof”
prohibits the production of super-heavy elements. “Decay roofs” essentially
recycle heavy material after the neutron irradiation freezes out when the
material accumulated at the $N=184$ shell closure decays back to the valley of
$\beta$-stability.

When considering an efficient r-process (scenario I), fission fragments are
found to contribute almost entirely to the final abundances of nuclei with
$100\,\,\raise 1.4pt\hbox{$<$}\kern-7.59995pt\lower
2.79999pt\hbox{$\sim$}\,\,A\,\,\raise 1.4pt\hbox{$<$}\kern-7.59995pt\lower
2.79999pt\hbox{$\sim$}\,\,180$, while in scenario II where weak interactions
on nucleons initially limit the number of free neutrons, such a contribution
does not reach more than 30% and is concentrated in the $A=140-180$ region
depending on the FFD model adopted. Calculations based on the SPY and GEF FFDs
have been compared for both r-process scenarios.

The present study allows us to have a deep insight on the role of fission
processes in hydrodynamical r-process simulations on the basis of sound well-
defined quantitative indicators. Fission remains an important nuclear process
taking place in binary NS mergers, though major nuclear as well as
astrophysical uncertainties still need to be further addressed before
concluding on the definite role of fission processes. These concern in
particular the determination of the fission probabilities for all the nuclei
potentially involved during the r-process as well as a detailed description of
neutrino absorption and hydrodynamics in simulations. Much progress remains to
be done along these two directions.

## Acknowledgments

J.-F.L. and S.G. acknowledge financial support from FNRS (Belgium). This work
was supported by the Fonds de la Recherche Scientifique (F.R.S.-FNRS) and the
Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO) under the EOS Project nr
O022818F. The present research benefited from computational resources made
available on the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles,
infrastructure funded by the Walloon Region under the grant agreement nr
1117545. H.-T.J. acknowledges financial support from Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through
Sonderforschungsbereich (Collaborative Research Center) SFB-1258 “Neutrinos
and Dark Matter in Astro- and Particle Physics (NDM)” and under Germany’s
Excellence Strategy through Cluster of Excellence ORIGINS
(EXC-2094)—390783311, and from the European Research Council through Grant
ERC-AdG No. 341157-COCO2CASA. A.B. acknowledges support by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme under grant agreement No. 759253, and by Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID
279384907 - SFB 1245 and - Project-ID 138713538 - SFB 881 (“The Milky Way
System”, subproject A10).

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