# Multimodal fission from self-consistent calculations

Daniel Lay  0000-0001-8947-391X Department of Physics and Astronomy and FRIB
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA  Eric
Flynn  0000-0002-6641-3620 Department of Physics and Astronomy and FRIB
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
Sylvester Agbemava  0000-0003-2432-2402 FRIB Laboratory, Michigan State
University, East Lansing, Michigan 48824, USA  Kyle Godbey
0000-0003-0622-3646 FRIB Laboratory, Michigan State University, East Lansing,
Michigan 48824, USA  Witold Nazarewicz  0000-0002-8084-7425 Department of
Physics and Astronomy and FRIB Laboratory, Michigan State University, East
Lansing, Michigan 48824, USA  Samuel A. Giuliani  0000-0002-9814-0719
Departamento de Física Teórica and CIAFF, Universidad Autónoma de Madrid,
Madrid 28049, Spain  Department of Physics, Faculty of Engineering and
Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United
Kingdom.  Jhilam Sadhukhan  0000-0003-1963-1390 Physics Group, Variable Energy
Cyclotron Centre, Kolkata 700064, India

###### Abstract

Background

When multiple fission modes coexist in a given nucleus, distinct fragment
yield distributions appear. Multimodal fission has been observed in a number
of fissioning nuclei spanning the nuclear chart, and this phenomenon is
expected to affect the nuclear abundances synthesized during the rapid
neutron-capture process ($r$-process).

Purpose

In this study, we generalize the previously proposed hybrid model for fission-
fragment yield distributions to predict competing fission modes and estimate
the resulting yield distributions. Our framework allows for a comprehensive
large-scale calculation of fission fragment yields suited for $r$-process
nuclear network studies.

Methods

Nuclear density functional theory is employed to obtain the potential energy
and collective inertia tensor on a multidimensional collective space defined
by mass multipole moments. Fission pathways and their relative probabilities
are determined using the nudged elastic band method. Based on this
information, mass and charge fission yields are predicted using the recently
developed hybrid model.

Results

Fission properties of fermium isotopes are calculated in the axial quadrupole-
octupole collective space for three energy density functionals (EDFs).
Disagreement between the EDFs appears when multiple fission modes are present.
Within our framework, the UNEDF1${}_{\textrm{HFB}}$ EDF agrees best with
experimental data. Calculations in the axial quadrupole-octupole-hexadecapole
collective space improve the agreement with the experiment for SkM∗. We also
discuss the sensitivity of fission predictions on the choice of EDF for
several superheavy nuclei.

Conclusions

Fission fragment yield predictions for nuclei with multiple fission modes are
sensitive to the underlying EDF. For large-scale calculations in which a
minimal number of collective coordinates is considered,
UNEDF1${}_{\textrm{HFB}}$ provides the best description of experimental data,
though the sensitivity motivates robust quantification of the uncertainties of
the theoretical model.

##  I Introduction

Nuclear fission is expected to play a major role during the rapid neutron-
capture nucleosynthesis process ($r$ process) [1, 2] that occurs in
astrophysical scenarios with sufficiently high neutron densities [3, 4, 5].
Due to the occurrence of fission cycling, fragment yields of neutron-rich
fissioning nuclei are predicted to shape the $r$-process abundances in the
$110\lesssim A\lesssim 170$ region [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17]. Additionally, multimodal fission has been argued to impact the shape of
the rare-earth peak of the $r$-process abundance distributions [18].
Therefore, the ability to efficiently identify possible fission pathways is
important for global calculations of $r$-process nuclei.

Figure 1: (a) Mass and (b) charge yield distributions of 284Cn. The magenta
patterned regions are calculated with the UNEDF1${}_{\textrm{HFB}}$ functional
(our model). Also shown are the results from the BSM [19] (black solid lines),
SPM [14] (blue dashed lines), and GEF [20] (red dash-dotted lines) models.

Since most of the nuclei synthesized during the $r$-process neutron
irradiation phase cannot be measured, missing data for nuclear reaction
network simulations must be obtained theoretically. However, predictions of
current models for fragment yields of neutron-rich fissioning nuclei are not
consistent. As an example, Fig. 1 shows the fragment yields for 284Cn computed
using four models. The framework discussed in this paper predicts strongly
asymmetric fission (cluster decay), while the other models predict symmetric
distributions of varying widths.

Multimodal fission is found in regions of the nuclear chart between nuclei
with symmetric and asymmetric fission modes. Examples of nuclei that show
competing fission modes are: 257,258Fm, 260Md, and 258No [21, 22, 23, 24]. The
coexistence of fission pathways is caused by the delicate balance between
competing shell effects and nuclear dynamics, providing a stringent test of
fission models. Much theoretical progress has been made in understanding
multimodal fission. A number of studies have used macroscopic-microscopic
approaches, either with [25, 26, 27] or without [28, 29] inclusion of
collective dynamics. Other studies have used self-consistent density
functional theory (DFT) [30, 31, 32], although fragment yields were not
reported. Investigations using the time-dependent generator coordinate method
have also been carried out [33].

In this work, we use constrained nuclear DFT and the nudged elastic band
method [34] to efficiently identify competing fission pathways for fermium
isotopes and selected superheavy nuclei. We then use the hybrid approach
proposed in Refs. [35, 36] to compute spontaneous fission (SF) fragment
yields. We also study the sensitivity of predictions on the size of the
collective space and the choice of EDF.

The paper is organized as follows: Section II reviews the methodology used to
compute the fragment yields. Sections III and IV discuss the results obtained
for quadrupole-octupole and quadrupole-octupole-hexadecapole collective
spaces, respectively. Conclusions and prospects are summarized in Sec. V.

##  II Model

Potential energy surfaces (PESs) and the collective inertia tensor are
calculated within the nuclear DFT framework [37]. We consider two Skyrme EDFs:
SkM∗ [38] and UNEDF1${}_{\textrm{HFB}}$ [39], in the particle-hole channel.
The SkM∗ EDF is a standard for fission calculations, while
UNEDF1${}_{\textrm{HFB}}$ extends the optimization protocol of UNEDF0 [40] to
include the additional data on fission isomers; it describes pairing
correlations at the Hartree-Fock-Boguliubov (HFB) level. The pairing EDF is
approximated using the mixed-type density-dependent delta interaction [41]. We
also consider the D1S parametrization [42] of the Gogny interaction, which has
been used for studying fission properties of heavy and superheavy nuclei [43].
We restrict our calculations to axially symmetric shapes, as these are
sufficient to describe the outer turning surface defining fission fragment
yields. Specifically, we constrain the mass quadrupole ($Q_{20}$), octupole
($Q_{30}$), and hexadecapole ($Q_{40}$) moments, defined as

$\displaystyle{Q}_{\lambda 0}=\int
d^{3}r\rho{(\mbox{\boldmath$r$})}r^{\lambda}Y_{\lambda 0},$ (1)

where $\rho{(\mbox{\boldmath$r$})}$ is the total nucleonic density at $r$ and
$Y_{\lambda 0}$ are spherical harmonics. To evaluate action integrals, we
introduce dimensionless collective coordinates [44].

To solve the HFB equations, we employ the axial DFT solver HFBTHO (v3.00) [45]
for the UNEDF1${}_{\textrm{HFB}}$ and SkM∗ EDFs, and HFBaxial [46] for the D1S
EDF. These solvers use the harmonic oscillator basis expansion. To ensure
convergence, we set the maximum oscillator number of the harmonic oscillator
basis to 25.

The collective inertia tensor is calculated in the adiabatic time-dependent
HFB framework, employing the perturbative cranking approximation [47, 48, 49].
Given the number of configurations explored, this approach to the inertia
tensor enhances computational efficiency while maintaining reasonable
numerical accuracy. For simplicity, we neglect the dynamical pairing degree of
freedom [50, 44, 51, 52] as we do not carry out a detailed SF lifetime
analysis.

Fission pathways are equivalent to least action paths (LAPs) obtained by
minimizing the collective action integral [53]:

$S(L)=\frac{1}{\hbar}\int_{s_{\rm in}}^{s_{\rm
out}}\sqrt{2\mathcal{M}_{\text{eff}}(s)\left(V_{\text{eff}}(s)-\Delta
E\right)}\,ds\,,$ (2)

where $V_{\text{eff}}(s)$ and $\mathcal{M}_{\text{eff}}(s)$ are the effective
potential energy and collective inertia, respectively, along the path $L(s)$.
The integration limits $s_{\rm in}$ and $s_{\rm out}$ correspond to the
classical inner and outer turning points, respectively, determined by the
condition $V_{\text{eff}}(s)=\Delta E$. The quantity $\Delta E$ represents the
zero-point correction to the ground-state energy. Throughout this work,
following discussion in Ref. [34], we refer to the contour of energy
$V_{\textrm{eff}}=\Delta E$ as the outer turning line, and the collective
coordinates at $s_{\textrm{out}}$ as the exit point. We use the nudged elastic
band method (NEB) [34] to compute LAPs, as this method is capable of
identifying LAPs corresponding to different fission modes. The supplemental
material (SM) [54] provides further details on the PES and LAP calculations.

If $S_{i}$ is the action along the $i$’th competing fission path, the relative
normalized probability of this mode is

$P_{i}\approx\frac{\exp(-2S_{i})}{\sum_{j=1,n}\exp(-2S_{j})}.$ (3)

The parameter $\Delta E$ can be estimated by means of the generator coordinate
method [55, 56, 57]. However, $\Delta E$ is often treated as a free parameter
that can be adjusted to lifetime measurements [58, 59, 60]. Here, $\Delta E$
is only used to determine the prescission configuration $\mathcal{C}$ with the
expectation that the fission fragment yields estimated at $\mathcal{C}$ are
independent of $\Delta E$. We demonstrate $\Delta E$ independence in SM [54],
and we take $\mathcal{C}$ to be the exit point configuration computed with
$\Delta E=0$.

Fission isomers (FIs) are local minima in the PES at deformations between the
g.s. and the outer turning line. In some transactinide nuclei, FIs can have
lower potential energy than the g.s. configuration. If $\Delta E$ is chosen
relative to the g.s., the FI may be classically accessible. However, Eq. (2)
assumes a tunneling configuration. To assess the influence of the FI on the
fragment yields, we set $\Delta E=0$ relative to the configuration with lower
energy. While in some cases, LAPs starting from the g.s. may bypass the FI
[44], this does not occur for the cases considered in this work. So, via Eq.
(3), $P_{i}$ is independent of the action calculated in between g.s. and FI.

Figure 2: Potential energy surfaces of 254,256,258,260Fm (in MeV) calculated
using UNEDF1${}_{\textrm{HFB}}$ (left), SkM∗ (center), and D1S (right) EDFs.
The symmetric (dashed lines) and asymmetric (solid and dotted lines) least
action paths are drawn from the ground state (filled circle) to the fission
isomer (asterisk), to the exit point (open square). The white contour denotes
the outer turning line $V_{\textrm{eff}}=\Delta E=0$. Gray contours are marked
at 1 MeV intervals for $0<V_{\textrm{eff}}<5$ MeV.

Once the prescission configuration $\mathcal{C}$ has been determined, the
fission fragment yield is computed using the hybrid method developed in [35,
36]. This approach identifies prefragments using the nucleonic localization
functions (NLFs) at $\mathcal{C}$. The remaining nucleons, the so-called neck
nucleons, are distributed between the two prefragments according to a
microcanonical probability distribution. For details on the prefragment
selection, see SM [54] and [61]. The combined fragment yields are computed by
taking the weighted sum of the yields and their corresponding $P_{i}$.

##  III Multimodal fission in 2D

In this section, we investigate fission in the two-dimensional collective
space $(Q_{20},Q_{30})$. We first study the benchmark chain of Fm isotopes,
which is known to transition from asymmetric to symmetric fission with
increasing neutron number [24, 62]. Next, we investigate selected superheavy
nuclei which are expected to show stronger model dependence because of their
reduced fission barriers.

###  III.1 Fermium isotopes

The even-even Fm isotopes with $154\leq N\leq 160$ are known to undergo a
transition from asymmetric fission characteristic of lighter actinides to
symmetric fission as $N$ increases towards $N=164$. This transition is related
to strong shell effects in prefragments as they approach the doubly magic
nucleus 132Sn. The resulting bimodal fission in intermediate isotopes was
investigated in numerous papers [63, 28, 64, 65, 30, 66, 67, 68, 69, 32, 70,
71, 26, 33, 27, 35, 36, 72, 73].

Figure 2 shows the PESs and LAPs calculated for 254,256,258,260Fm with
different EDFs. In general, the topologies of PESs agree well between various
EDFs. The fission barriers obtained with SkM∗ and D1S are higher than those of
UNEDF1${}_{\textrm{HFB}}$. Except for 260Fm with UNEDF1${}_{\textrm{HFB}}$
(Fig. 2(d)), all the PESs predict a FI at $100<Q_{20}<160$ b and $Q_{30}=0$.

Figure 3: Fission fragment mass (left) and charge (right) yields for
254,256,258,260Fm calculated with UNEDF1${}_{\textrm{HFB}}$ (magenta vertical
patterns), SkM∗ (black horizontal patterns), and D1S (green $\times$
patterns). Experimental yields (circles) [74, 75, 76, 77] are shown where
available.

Since triaxiality is ignored in this study, in most cases, one path along
$Q_{30}\approx 0$ connects the g.s. and the FI. The reflection-symmetric path
continues past the FI, and then a bifurcation resulting in a coexistence of
symmetric and asymmetric pathways takes place. The asymmetric paths are smooth
at the bifurcation point. Some path segments are straight due to the flatness
of the PES. For 258Fm calculated with SkM∗, Fig. 2(g), we observe an
additional asymmetric LAP that ends close to the symmetry axis, due to the
geometry of the outer turning surface. However, within our hybrid approach,
this exit point corresponds to a wide symmetric yield distribution.

The relative probability $P_{s}$ of the symmetric mode is given in Table 1.
All EDFs transition from asymmetric-dominant to symmetric-dominant fission
path with increasing $N$. Competition between the modes is expected in
256,258Fm for UNEDF1${}_{\textrm{HFB}}$, 258Fm for SkM∗, and 256Fm for D1S.
The competition between symmetric and asymmetric fission modes seen in Table 1
is generally consistent with SkM∗ results of Refs. [32, 58]. The small
differences are most likely due to (i) our inclusion of the multidimensional
collective inertia tensor in computing the collective action and (ii) our
precise minimization of $S(L)$.

Table 1: The relative probability $P_{s}$ of the symmetric mode for the 2D calculation of Fm isotopes.  |  254Fm |  256Fm |  258Fm |  260Fm
---|---|---|---|---
UNEDF1${}_{\textrm{HFB}}$ |  $\sim$0 | $0.12$ | $0.79$ | $1$
SkM∗ |  $\sim$0 |  $\sim$0 | $0.17$ | $1$
D1S |  $\sim$0 | $0.88$ | $1$ | $1$

Figure 3 shows the calculated yields, together with experimental data [74, 75,
76, 77]. The experimental data for 254,256Fm show an asymmetric distribution,
while for 258Fm, the yields are primarily symmetric, with a small asymmetric
shoulder. The error bands in the calculated distributions originate from the
two-particle uncertainty [35]. Our UNEDF1${}_{\textrm{HFB}}$ results are in
close agreement with the data for 254Fm. Competition between modes for
256,258Fm is present for each EDF, although the D1S model overestimates the
symmetric contribution. All EDFs predict overlapping symmetric yields for
260Fm. The transition from asymmetric to symmetric fission is clearly present,
albeit at different neutron numbers for the EDFs. Despite the overestimation
of the symmetric mode for 256Fm, UNEDF1${}_{\textrm{HFB}}$ provides the best
description overall.

Figure 4: Panels (a)-(c): similar as in Fig. 2 but for 258Rf, 262Sg, and 262Hs
using UNEDF1${}_{\textrm{HFB}}$. Panels (d)-(f) and (g)-(i): mass and charge
yields, respectively, for UNEDF1${}_{\textrm{HFB}}$ (magenta vertical
patterns) and SkM∗ (black horizontal patterns). Figure 5: The least action
paths in 3D for 258Fm using SkM∗ (a) and UNEDF1${}_{\textrm{HFB}}$ (b). The
outer turning surface is shown. A 2D PES is shown for constant $Q_{40}=16$ b2
($Q_{40}=32$ b2) for SkM∗ (UNEDF1${}_{\textrm{HFB}}$). Neutron localizations
for the identified precission configurations are shown in the insets.

###  III.2 Superheavy nuclei

The superheavy nuclei $(Z>104)$ are stabilized only by quantum shell effects
[78] and hence they are short-lived. The dominant decay modes observed in this
region are $\alpha$ decay and SF [79, 80, 81, 82, 62, 83]. Some isotopes of Fl
and Ts are observed to undergo SF after several $\alpha$ emissions [84, 85].
Neutron-rich superheavy nuclei may decay directly via SF [86, 11, 20, 87, 88,
19, 14] marking the end of the $r$-process path. Multimodal fission has been
studied in the superheavy region in Refs. [32, 56].

The PESs and the corresponding symmetric and asymmetric LAPs computed with
UNEDF1${}_{\textrm{HFB}}$ for 258Rf, 262Sg, and 262Hs are shown in Fig.
4(a-c). For D1S, the asymmetric mode is not relevant for 262Sg. Consequently,
the remaining discussion pertains to SkM∗ and UNEDF1${}_{\textrm{HFB}}$ only.

Table 2: Similar as Table 1 except for 258Rf, 262Sg and 262Hs. |  258Rf |  262Sg |  262Hs
---|---|---|---
UNEDF1${}_{\textrm{HFB}}$ | $0.48$ | $0.85$ | $0.52$
SkM∗ |  $\sim$0 | $0.04$ |  $\sim$0

Table 2 shows $P_{s}$ for the superheavy nuclei shown in Fig. 4. SkM∗ favors
the asymmetric mode, while UNEDF1${}_{\textrm{HFB}}$ predicts competition
between the modes. The modes predicted by SkM∗ are consistent with those
identified in [32], and the reasons for small differences in $P_{s}$ were
discussed in Sec. III.1.

The mass and charge yields are shown in Fig. 4(d-i). Similar to 258Fm, the
yields computed using UNEDF1${}_{\textrm{HFB}}$ have a large symmetric peak
with asymmetric shoulders. Note that the fragment yield peak for each mode
agrees between EDFs, but contributes differently, according to its relative
probability. SPM [14] and GEF [20] predict asymmetric yields for 258Rf. BSM
[19] predicts asymmetric yields for 258Rf and 262Sg, and symmetric yields for
262Hs. Symmetric fission is predicted in [56] for the three nuclei, although
yields are not computed. The large spread between theoretical predictions
highlights the usefulness of multimodal fission for differentiating between
models.

##  IV Multimodal fission in 3D

In order to separate fission pathways that are unresolved in a lower-
dimensional collective space, it is advisable to consider additional
collective degrees of freedom such as the hexadecapole moment $Q_{40}$ that
controls the number of nucleons in the neck region. Indeed, studies
constraining $Q_{40}$ have identified fission modes not found in the reduced
$(Q_{20},Q_{30})$ collective space [42, 89, 30, 32, 90, 57, 91, 92]. In this
section, we study fission in the three-dimensional collective space
$(Q_{20},Q_{30},Q_{40})$.

Figure 6: The fragment mass (left) and charge (right) yields for 258Fm using
SkM∗ (top) and UNEDF1${}_{\textrm{HFB}}$ (bottom) calculated in the 2D (green
solid lines) and 3D (blue dashed lines) collective spaces. Experimental data
from [77] is shown as red filled circles. Figure 7: Similar as in Fig. 5 but
for 306122\. The 2D PESs are at constant $Q_{40}=0$.

We first consider 258Fm as part of the benchmark Fm chain. The outer turning
surface, LAPs, and localizations at $\mathcal{C}$ are shown in Fig. 5(a)-(b)
for SkM∗ and UNEDF1${}_{\textrm{HFB}}$, respectively. The part of the path
from g.s. to FI is not shown for clarity. Two symmetric modes are identified:
the compact (CS) and elongated (ES) modes. $\mathcal{C}_{\textrm{ES}}$ has
larger $Q_{40}$ than $\mathcal{C}_{\textrm{CS}}$, and hence the neck region
for $\mathcal{C}_{\textrm{ES}}$ is thicker. We also found two asymmetric
fission pathways: the weak (WA) and strong (SA) modes.
$\mathcal{C}_{\textrm{WA}}$ has a small ($Q_{30}<10$ b3/2) octupole moment.
The ES mode is present due to the $Q_{40}$ constraint; the other modes are
consistent with the discussion of Sec. III.1. The modes are also largely
consistent with the pathways identified in Refs. [32, 58].

Table 3: Relative probabilities for the modes identified in the 3D collective space. Mode |  258Fm |  306122
---|---|---
| SkM∗ | UNEDF1${}_{\textrm{HFB}}$ | SkM∗ | UNEDF1${}_{\textrm{HFB}}$
CS | $0.79$ | $0.47$ |  $\sim$0 | $-$
ES |  $\sim$0 |  $\sim$0 | $-$ | $-$
WA | $0.21$ | $0.53$ | $1.0$ | $1.0$
SA |  $\sim$0 | $-$ |  $\sim$0 | $-$

Table 3 lists the relative probability of each mode. The symmetric
contribution is now larger (smaller) than in 2D for SkM∗
(UNEDF1${}_{\textrm{HFB}}$). Figure 6 shows the corresponding fragment yields
compared to the yields calculated in the 2D collective space. For
UNEDF1${}_{\textrm{HFB}}$, $\mathcal{C}_{\textrm{WA}}$ is found closer to the
$Q_{30}=0$ axis than in 2D, resulting in a less asymmetric distribution. This
worsens the agreement with the tails of the experimental yields. For SkM∗,
agreement with experiment is improved due to the increase in the relative
contribution of the symmetric mode. This highlights the importance of adding
additional collective coordinates for mode identification.

Figure 8: The fragment mass (a) and charge (b) yields for 306122 calculated
with UNEDF1${}_{\textrm{HFB}}$ (black solid lines) and SkM∗ (magenta dashed
lines).

Finally, in Fig. 7 we consider 306122, a superheavy nucleus that may exhibit
multimodal fission [86, 56, 93, 19, 94, 14, 95]. A pronounced asymmetric
valley forms beyond the outer turning surface leading to the cluster decay
channel, similar to 294Og [96]. The relative probabilities for both EDFs are
listed in Table 3, and the SkM∗ results are consistent with [86, 56]. The
yields, shown in Fig. 8, are in qualitative agreement between the EDFs. As
with 294Og, the heavy fragment is close to the doubly magic nucleus 208Pb. The
yields disagree with the symmetric yields calculated in the BSM model [19].

##  V Conclusions

In this study, we have explored the competition between SF modes for
254,256,258,260Fm and several superheavy nuclei within the axial quadrupole-
octupole and quadrupole-octupole-hexadecapole collective spaces. We used the
nudged elastic band method [34] to find SF modes and their relative
probabilities, and computed fragment yields using the hybrid approach from
Refs. [35, 36].

In general, identified fission modes are consistently predicted by the EDFs
used. However, the relative probabilities of the modes were found to be fairly
sensitive to the choice of EDF, leading to different overall fragment yields.
In the 2D collective space, UNEDF1${}_{\textrm{HFB}}$ was found to agree best
with experiment. The differences in the fragment yields highlights the utility
of multimodal fission when differentiating between models.

For 258Fm, an elongated symmetric fission mode has been identified in the 3D
collective space, and the yields were found to be more symmetric than in the
2D space. Introducing the hexadecapole moment brought theoretical results
closer to experiment for SkM∗. The nucleus ${}^{306}122$ is predicted to decay
via cluster emission, similar to 294Og [96] and 284Cn shown in Fig. 1.
Identification of a cluster decay of superheavy nuclei remains an important
experimental challenge.

Coexistence of fission pathways is expected for many $r$-process nuclei, and
has a strong impact on the fragment yields. We demonstrated in this work that
the nudged elastic band method [34] is capable of efficiently identifying such
competing pathways, enabling future studies of multimodal fission in several
collective coordinates.

Furthermore, the above highlights the importance of including more collective
coordinates when studying SF. The triaxial quadrupole moment, for instance, is
important when describing the collective motion from the ground state to the
fission isomer [97, 58, 98, 44]. One can also consider a particle number
dispersion term, which controls dynamic pairing fluctuations [99, 100, 44,
51]. However, increasing the dimension of the collective space beyond the
presented 3D calculations is computationally expensive. In this regard, DFT
emulators will help by reducing computational costs [101, 102, 103, 104].

One interesting alternative to the presented description of SF is dynamic
reaction processes such as quasifission, which exhibits similar fragment
production characteristics as fission and has been identified as a promising
experimental avenue for studying the fission properties of hard-to-produce
nuclei [105, 106, 107]. Time-dependent reaction methods do not constrain the
system to a set of collective coordinates, allowing the most probable modes to
be determined efficiently [108]. This efficiency enables systematic
calculations across the nuclear chart, in addition to uncertainty
quantification when describing real-time large amplitude collective motion
beyond existing fusion studies [109]. Regardless of the method of exploration,
comprehensive studies of multimodal fission are needed across the chart of
nuclides. These systematic studies of fragment properties for spontaneous,
neutron-induced, and compound-nucleus fission all contain rich information
that will inform $r$-process nucleosynthesis and provide key insights about
the underlying structure and dynamics of exotic nuclei.

$\bullet$

###### Acknowledgements.

This work was supported by the U.S. Department of Energy under Award Numbers
DOE-DE-NA0004074 (NNSA, the Stewardship Science Academic Alliances program),
DE-SC0013365 (Office of Science), and DE-SC0023175 (Office of Science, NUCLEI
SciDAC-5 collaboration); by the National Science Foundation CSSI program under
award number 2004601 (BAND collaboration); and by the Spanish Agencia Estatal
de Investigación (AEI) of the Ministry of Science and Innovation (MCIN) under
grant agreements No. PID2021-127890NB-I00 and No. RYC2021-031880-I funded by
MCIN/AEI/10.13039/501100011033 and the "European Union NextGenerationEU/PRTR".

D.L. and E.F. contributed equally to this work.

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