11institutetext:  E. M. Holmbeck 22institutetext: The Observatories of the
Carnegie Institution for Science, Pasadena, CA 91101, USA
NHFP Hubble Fellow
22email<EMAIL_ADDRESS>33institutetext: T. M. Sprouse
44institutetext: Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA
Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los
Alamos, NM, 87545, USA
44email<EMAIL_ADDRESS>55institutetext: M. R. Mumpower 66institutetext:
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA
Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los
Alamos, NM, 87545, USA
66email<EMAIL_ADDRESS>
# Nucleosynthesis and observation of the heaviest elements

E. M. Holmbeck  T. M. Sprouse  M. R. Mumpower

(Received: 29 July 2022 / Accepted: 19 January 2023 / Published online: 15
February 2023)

###### Abstract

The rapid neutron capture or ‘ _r_ process’ of nucleosynthesis is believed to
be responsible for the production of approximately half the natural abundance
of heavy elements found on the periodic table above iron (with proton number
$Z=26$) and all of the heavy elements above bismuth ($Z=83$). In the course of
creating the actinides and potentially superheavies, the _r_ process must
necessarily synthesize superheavy nuclei (those with extreme proton numbers,
neutron numbers or both) far from isotopes accessible in the laboratory. Many
questions about this process remain unanswered, such as ‘where in nature may
this process occur?’ and ‘what are the heaviest species created by this
process?’ In this review, we survey at a high level the nuclear properties
relevant for the heaviest elements thought to be created in the _r_ process.
We provide a synopsis of the production and destruction mechanisms of these
heavy species, in particular the actinides and superheavies, and discuss these
heavy elements in relation to the astrophysical _r_ process. We review the
observational evidence of actinides found in the Solar system and in metal-
poor stars and comment on the prospective of observing heavy-element
production in explosive astrophysical events. Finally, we discuss the
possibility that future observations and laboratory experiments will provide
new information in understanding the production of the heaviest elements.

###### Keywords:

_r_ process nucleosynthesis actinides superheavy elements

###### Contents

  1. 1 Introduction
    1. 1.1 From atoms to nuclei
    2. 1.2 Nuclear transmutations and nucleosynthesis
    3. 1.3 Hints of heavy element nucleosynthesis
  2. 2 Nuclear physics of the heaviest nuclei
    1. 2.1 Nuclear binding energy and mass
    2. 2.2 Neutron capture
    3. 2.3 Nuclear decay
    4. 2.4 Nuclear fission
  3. 3 Astrophysical alchemy
    1. 3.1 The basic _r_ process
    2. 3.2 Production of the heaviest elements
    3. 3.3 Sensitivity to nuclear physics
    4. 3.4 Sensitivity to astrophysics
  4. 4 Observation of the heaviest elements
    1. 4.1 Actinides in the Solar system
    2. 4.2 Actinides in stars
    3. 4.3 Fission fragments
    4. 4.4 Beyond uranium
  5. 5 Summary & future outlook

##  1 Introduction

In the universe, we know of at least 118 chemical elements that can exist. To
visualize and sort these elements by their chemical properties, they are
placed on the periodic table and grouped according to their chemical structure
(see Fig. 1). The heaviest elements on the periodic table are those with
charge numbers in excess of $Z=83$, including the actinide series ($89\leq
Z\leq 103$) and the transactinide—often called ‘superheavy’—elements with
$Z>103$. Experimental efforts over the past several decades have resulted in
the discovery of new elements, including, at the present time, the heaviest
known element oganesson, Og, with $Z=118$ Karol2016 ; Karol2016b . While all
but two elements with $Z\leq 83$ have at least one state in which they live
longer than a Hubble time ($\gtrsim$ 14 Gyr), all of the known elements above
$Z=83$ are radioactive; that is, they only exist only for a finite amount of
time. However, an ‘Island of Stability’ Myers1966 ; Nilsson1969 ; Moller1986
has been proposed for elements near $Z\sim 114$, where some of these elements
are thought to be less radioactive and exist for longer than their superheavy
neighbors.

Because heavy elements tend to decay quite quickly when compared to the time
that has elapsed since their astrophysical production, most of the heavy
elements—especially the transactinides—cannot be found naturally; instead,
these may be created in laboratory settings. There are a handful of actinide
elements that live sufficiently long to be found naturally in significant
amounts: thorium (Th), protactinium (Pa), uranium (U), neptunium (Np), and
plutonium (Pu), with $Z=90$–94, respectively. Figure 1 highlights the actinide
and transactinide elements on the periodic table and colors them by their
abundance on Earth. In addition, all of the elements between Bi and Th
($83<Z<90$) occur naturally in the Earth’s crust as decay products of the
longer-lived actinides, though only in trace amounts. Beyond Earth, Th and U
are the only $Z>83$ elements that can also be identified in the light of
quiescent stars beyond our Solar system, notably in some of the oldest stars
in the Galaxy.

Figure 1: Periodic table of the elements showing the Earth abundances of the
heaviest elements above $Z>83$, including the superheavy elements ($Z>103$).
Elements in dark grey are not found on Earth, but instead have been detected
in laboratory settings.

The presence of the heavy elements both inside and outside the Solar system
attest to a natural production mechanism for these elements Kajino2019 .
However, no _trans_ -actinide element has yet been found in nature, neither in
the Solar system nor outside of it. Can they be created naturally at all? Can
the process that makes the actinides also create the superheavies, and is
there any chance of finding or observing these elements in their natal
production sites? To address these questions, this review first introduces in
Sec. 2 the nuclear physics involved in the production (and destruction) of the
heaviest elements. Next, Sec. 3 reviews where and how these nuclear properties
are active in the astrophysical context that creates the heavy elements. We
follow up with the observational consequences of heavy-element nucleosynthesis
in Sec. 4, discussing the possibilities of observing the heaviest elements and
their natural production. Lastly, Sec. 5 offers encouraging perspectives as we
move into the future of this topic at the intersection of physics and
astronomy.

###  1.1 From atoms to nuclei

We have, up until this point in our review, focused on the periodic table as
the taxonomy within which the rich variety of atomic nuclei may be assigned.
Nuclei may be grouped according to their overall number of protons into
specific elements. With relatively few exceptions, atomic nuclei belonging to
the same element will share a common electronic structure and, consequently,
share a common set of chemical and material properties, such that their shared
proton number presents an excellent basis for their classification throughout
much of chemistry, material science, engineering, and even our daily
interactions with the world.

Beginning around 100 years ago, it became increasingly apparent that the
classification of atoms according to their chemical properties alone was
insufficient for uniquely identifying atoms. The gold foil experiments by
Ernest Rutherford, Hans Geiger, and Ernest Marsden between 1908 and 1913 set
the initial groundwork by developing an atomic model comprised of a dense,
positively charged core (what we now know to be the nucleus) surrounded by a
relatively dispersed volume occupied by oppositely charged electrons
Rutherford1911 ; Geiger1913 . Subsequent experiments over the following few
decades—in particular, the discovery of the neutron—eventually developed the
understanding that the atomic core was comprised of neutrons in addition to
protons, giving rise to a further distinction between atoms of the same
element Rutherford1913 .

The many years that followed these pioneering efforts in nuclear physics
ultimately culminated in the modern understanding of the atomic nucleus: the
atomic nucleus is a bound system of neutrons and protons, each of which are
composite systems of elementary particles known as quarks. These constituent
quarks are bound together by the strong force and mediated by gluons. The
protons and neutrons themselves are similarly bound by the strong force but
mediated by the exchange of mesons, a quark-antiquark composite particle. The
total number of protons ($Z$) in a nucleus is still broadly responsible for
setting its chemical properties, but the widely variable number of neutrons
($N$) that may be combined with those protons to form bound nuclei of various
_isotopes_ calls for an independent axis along which they may be sorted,
namely neutron number. Similarly, a nucleus can be referred to by its ‘atomic
mass number,’ $A$: the total combined number of neutrons and protons in the
nucleus. In this scheme, a nuclide refers to a unique combination of some
number of protons and neutrons in a bound system and is the nuclear analogue
to the chemical ‘element.’ As the elements are arranged on the periodic table
according to their increasing proton number and electron shell structure,
nuclides are similarly arranged on a chart of nuclides, with axes not only
describing the number of protons in the nucleus, but also the number of
neutrons, indicating all the isotopes that may exist of a particular element.

We define superheavy nuclei to refer to those nuclei which possess an extreme
number of protons, neutrons or combination thereof; broadly categorizing
nuclei with $Z>82$. Superheavy elements refer to those species with $Z>103$,
forming a subset of superheavy nuclei. The definition of a superheavy element
depends on the atomic or nuclear context; here we take the nuclear
perspective.

An element’s proton number and electron shell structure (visualized by its
location on the periodic table) are the basic properties that describe how an
element reacts chemically. Similarly, in addition to its proton and neutron
number identity, there are a variety of properties of the nucleus that
determine how individual isotopes react and decay. These range from the
relatively straightforward, such as their mass or binding energy, to the more
complex, such as the precise description of how a nucleus may interact with
other nuclei. While any attempt to enumerate these properties in their
entirety requires a devoted, exhaustive review, we dedicate Sec. 2 to a more
focused discussion of the small subset of these properties which are
especially important for understanding the astrophysical production of the
heaviest elements.

###  1.2 Nuclear transmutations and nucleosynthesis

Insofar as all nuclei are comprised of the same basic building blocks, protons
and neutrons, it immediately follows that there may be some prospect that
nuclei may be able to reconfigure themselves into new combinations of the same
inherent building blocks. Indeed, this general phenomenon has been observed
since the infancy of nuclear physics—the observation of radioactivity being
perhaps the earliest example—even if it would not be understood as a
transmutation of nuclei per se for many years later.

Nuclear transmutations may be categorized in any number of ways. One of the
more natural ways to approach this task is to consider the circumstances under
which a transmutation takes place. For example, some nuclei transmute
spontaneously, in the sense that they proceed on their own without interaction
with any particles external to the nucleus itself. Spontaneous fission and
$\alpha$ decay are clear examples of such transmutations. These spontaneous
decays differ from induced transmutations, which only occur when a nucleus
interacts with some external particle, such as a baryon, another nucleus, or
perhaps a lepton.

Another commonly used basis for describing nuclear transmutations concerns the
types of interactions that mediate the transmutation. As it is ultimately the
strong interaction that binds nuclei together, a broad swath of nuclear
transmutations are principally driven in some complex way by the same category
of interactions. To highlight how this might proceed, consider the situation
in which an incident (‘projectile’) nucleus bombards another (‘target’)
nucleus. In this scenario, it is possible for some protons and/or neutrons be
exchanged between the two. After such a transmutation takes place, the overall
numbers of protons and neutrons are independently conserved, but they are
nevertheless rearranged to convert the initial target and projectile nuclei
into some other nuclei. It is not, however, the case that all nuclear
transmutations conserve the numbers of protons and neutrons, as weak
interactions may convert protons to neutrons and vice-versa, the prototypical
example of which is that of nuclear $\beta^{\pm}$ decay, which changes the
element of a nucleus but leaves the combined number of neutrons and protons
unchanged. Similarly, not all reactions that preserve proton and neutron
number are necessarily mediated by the strong force, e.g.,
photodistintegration is mediated by the electromagnetic force.

While the distinction between strong and weak transmutations is useful for
discussing such topics at a high level, the fundamental forces are always
active, and thus the distinction is not always clean-cut. For example, nuclear
$\beta$ decay (a weak process) leaves a nucleus in an excited state which may
emit a proton or neutron (via properties of the strong interaction) before
finding its way to its final energy state. Another example is $\alpha$ decay
which exhibits an interplay between strong and electromagnetic interactions.
In general, a thorough understanding of all transmutations in which a nucleus
might participate will require detailed consideration of all types of
interactions as a collective whole.

Given the rich and diverse types of transmutations in which nuclei may
participate, the question is raised: do these transmutations actually occur in
any significant way in physical environments? This is perhaps most evident
when we consider that the immediate aftermath of the Big Bang is understood to
have produced primordial nuclei comprised almost entirely of hydrogen, helium,
and lithium, in stark contrast with what we now observe to exist today, both
in the Solar system and throughout the universe (as shown, e.g., in Fig. 1).
In contrast, early studies of Big Bang nucleosynthesis predicted the _entire_
range of elements were created Alpher1948 . Given the modern understanding
that the Big Bang only produced the lightest elements, the logical explanation
for the observation of the rest of the elements found presently would be that
nuclear transmutations have been taking place throughout the universe since
its very beginning, converting the light nuclei into successively heavier
ones, up to and including the naturally occurring thorium, uranium, and
plutonium observed, e.g., here on Earth. ‘Nucleosynthesis’ is the generic term
that refers to any process which produces new nuclei in physical systems, with
‘astrophysical nucleosynthesis’ referring specifically to nucleosynthesis that
takes place in astrophysical environments.

Figure 2: Abundance of each isobar (constant proton number plus neutron
number, $A$) in the Solar system from data in Ref. Lodders2009 . The nuclei
with primary origins in neutron-capture nucleosynthesis are highlighted in
blue.

The elements we find in our Solar system, then, are a product of roughly nine
billion years of distinct and incredibly diverse astrophysical nucleosynthesis
processes that created the heavier elements out of the building blocks left by
the Big Bang. Figure 2 shows the abundance of each isobar (those nuclei
sharing the same total number of protons and neutrons in their nucleus, $A$)
found in the Solar system on a logarithmic scale. While hydrogen and helium
are still the most abundant elements—reflective of their primordial origin—the
rest of the periodic table ($Z\geq 3$) accounts for only about a total of 2%
of the Solar system material Lodders2003 ; Lodders2009 .

As can be seen in Fig. 2, there is a steady decline in abundance with
increasing atomic mass number, where the heaviest elements are also the least
abundant. A surprisingly robust theory for explaining this pattern as a
consequence of nucleosynthesis was first proposed in the landmark 1957 result
of Burbidge, Burbidge, Fowler, and Hoyle Burbidge1957  and contemporaneous
works, e.g., by Al Cameron Cameron1957 . These works, which gave birth to the
field now known as ‘nuclear astrophysics,’ interpreted the distinct peaks in
the Solar system abundance pattern as a clear signature of nuclear physics
structure and stellar evolution. The peaks of locally high abundances,
decreasing through the series of C, O, Ne, Si, and S, correspond to the
$\alpha$-nuclei preferentially formed in the fusion stages of progressively
more massive stars, terminating in the iron peak around $A=56$ produced by
core-collapse supernovae that mark the fusion end-point of very massive stars’
lives. Beyond this point, i.e., moving towards the heavier nuclei populating
the Solar system (indicated by the shaded region of Fig. 2), the Coulomb
barrier increases for heavier elements, making charged particle reactions
increasingly unsustainable in stellar environments. In other words, the fusion
of like-charged nuclei (e.g., Fe and Fe), which for elements lighter than Fe
produced energy that once supported the star, now become energetically
expensive for elements heavier than Fe, leading to stellar collapse. It
follows, then, that the elements beyond Fe cannot be produced astrophysically
by stellar _fusion_. Rather, the capture of free neutrons onto nuclei is
responsible for the vast majority of the abundances of elements with $Z\gtrsim
30$ in the Solar system, though charged-particle captures are still
responsible for the production of some Solar-system nuclei. However, because
free neutrons decay with a half-life of around ten minutes, the
nucleosynthesis of these heaviest nuclei requires specific and relatively rare
astrophysical environments: that is, an environment which also produces the
neutrons to be subsequently captured. The capture of light charged nuclei
(e.g., a proton) is a possible route to synthesizing heavier neutron-deficient
elements. The rapid capture of protons (_r_ p process) Wallace1981  is thought
to produce light _p_ nuclei up to mass number $A\sim 110$, see Schatz2001  and
references therein. The astrophysical conditions (temperature and density)
must be appropriate to favor incident protons overcoming the Coulomb barrier
of heavier nuclei. In environments with sufficient neutrino fluxes, anti-
neutrino capture on protons, or the $\nu$p process, may ensue. This process
can explain the production of 92,94Mo and 96,98Ru in the Solar system
Frolich2006 . As far as the Solar system is concerned, the heavy elements owe
the majority of their production to two generic types of neutron capture: slow
and rapid Meyer1994 . Any prospect for the production of the (super)heavy
elements would necessarily proceed via _rapid_ neutron-capture nucleosynthesis
(or the ‘ _r_ process’), a concept which we explore in much deeper detail in
Sec. 3.

###  1.3 Hints of heavy element nucleosynthesis

In parallel to breakthroughs in the understanding of atoms and nuclei were
landmark identifications of heavy elements on Earth. Today we know that trace
amounts of several heavy elements produced by the _r_ process can be found on
Earth (see Fig. 1). Uranium was first discovered in 1789 from mineral samples
Klaproth1789  and 244Pu more recently in deep-sea sediments Hoffman1971 ;
Wallner2015 ; Wallner2021 . (For an exhaustive review of the history of
actinide discoveries, see Fry2013  and Kragh2013 .) Even before the direct
detections of the actinide elements, the existence of Th, U, and Pu was
inferred from measurements of Xe and Pb as the radioactive ‘mother’ nuclei
that have undergone nuclear transmuation by decaying into lighter ‘daughter’
species (e.g., Chackett1950 ; Dalton1953 ; Kuroda1960 ). As early as the
1910’s, going one nuclear generation further by requiring the origin of
uranium as itself a decay daughter product of a still-heavier element was both
hypothesized and criticized Haas1912 ; Kragh2013 . As we will see in this
review, this hypothesis was ironically not completely incorrect. The actinides
can indeed be decay products of the superheavy elements, and observations of
the actinides may be an indirect route to inferring production of the
superheavies Oganessian2006 .

The definition of ‘observation’ is not limited to direct sample collection and
measurement such as that conducted on Earth through, e.g., meteorites or deep-
sea sediments; most of the usable and available information about the universe
comes from starlight. A star is essentially a cool layer of material
surrounding a hotter core that radiates light over a continuum of energies
(i.e., wavelengths). Elements in the stellar photosphere can ‘absorb’ photons
radiated by deeper layers of the star at certain energies. Specifically, if a
photon has enough energy to excite an electron to a higher atomic energy level
(or ionization state), the photon may be ‘absorbed’ by the electron. This
exchange of energy manifests in the electromagnetic spectrum of the relatively
cool stellar photosphere as an atomic ‘absorption feature’ in which the
brightness of specific wavelengths of light is less than its neighboring
wavelengths that are not absorbed (or not absorbed as strongly). In a
quiescent star—such as the Sun—the hot stellar core produces a continuum of
light at different wavelengths, and elements in the cooler atmosphere of the
star may absorb light emitted by the hot core, producing absorption features
unique to the elemental composition of the stellar photosphere. Following this
principle, many elements can be identified in the light from the Sun. Indeed,
helium was first discovered in 1868 from sunlight during a solar eclipse and
thus owes its name to the Greek word for ‘sun,’ _helios_ Haig1868 ; Rayet1868
. Somewhat more recently—and more than one hundred year after its terrestrial
discovery in 1828—thorium was first identified in the Solar spectrum Moore1943
, and we emphasize that this identification would not have been possible
without dedicated ‘laboratory astrophysics’ efforts to characterize the atomic
transitions of thorium in order to guide where they can be found in starlight
McNally1942 . Outside the Solar system, the energy spectrum of starlight can
also reveal the elements present in other stars. In Sec. 4, we will discuss
stellar observations of the heaviest elements in more detail.

Observations both on Earth and from the Sun’s spectrum of light are able to
capture the elemental composition of the Solar system to produce diagrams like
Fig. 2, showing how much of each element/isobar exists in the Solar system
Arnould2020 . Observations from light and those from physical samples are
fundamentally different, however. Atomic absorption features visible in
stellar spectra are set by the proton number in the nucleus, and only in rare
cases do the absorption features show sensitivity to the total number of
nucleons in the nucleus. In other words, abundances derived from starlight
only indicate the total elemental abundance (i.e., summed over all isotopes
that may be present) Roederer2017 ; Hansen2018 . On the other hand, laboratory
techniques are able to separate elements not only by their charge, but also by
their mass number, providing another axis along which to characterize the
elemental abundances and allowing abundances as a function of atomic mass
number to be made (such as in Fig. 2). Because the Solar system is the only
system for which isotopic measurements can be made across the whole observable
periodic table, it remains the only system for which we have an _isotopic_
abundance pattern Cowan2021 . Stardust from other systems arriving to the
Solar system in the form of pre-Solar grains also allows some isotopic
information to be gained, but these grains are uncorrelated with stars that
are presently observed with stellar spectroscopy. The abundance patterns of
quiescent stars other than our Sun are strictly by element, save for the rare
handful of elements for which isotopes can be identified in stellar spectra
Wenyuan2018 ; Roederer2022 .

Similarly, atomic features (absorption or emission) can be detected in non-
quiescent events as well. For example, William Huggins made the first
spectroscopic observations of a novae in 1866 Huggins1866 , noting that the
hydrogen lines are notably different from those found in stars Hoffleit1986 .
Today, classes of novae and supernovae are categorized based on what elements
are detectable in their spectra. In Sec. 4, we will address if, when, and how
evidence for the heaviest elements can be found in spectra of explosive
astrophysical events.

##  2 Nuclear physics of the heaviest nuclei

The production of the heaviest elements in nature critically depends on the
properties of very short-lived, unstable nuclei. To date, many of these
properties remain unmeasured, and therefore theoretical model calculations of
nuclear properties must be used to supplement nuclear data when simulating
nucleosynthesis taking place in extreme regions of the chart of nuclides. In
this section, we review the nuclear properties that are of particular interest
to the formation of the heaviest elements.

###  2.1 Nuclear binding energy and mass

One of the basic properties of a nucleus is its binding energy, or the minimal
energy required to separate the nucleus into its constituent nucleons
(neutrons and protons). From the Einstein energy-mass relation, this can be
equivalently stated in terms of mass. In principle this quantity can be
derived from the lowest energy state of the many-body Hamiltonian; however,
this calculation is a tremendous task for light nuclei Carlson1998  and such
complexities only increase for heavy species Koszorus2021 ; Mougeot2021 .

The first estimates of nuclear binding energy were based on the liquid drop
model of a nucleus, and it was shown that this energy saturates per nucleon
Gamow1930 . Within a few years after this idea, a phenomenological formula was
developed by Weizsäcker Weizsacker1935 . This semi-empirical formula consists
of additive terms, each describing a unique aspect of the physics of nuclear
binding energies:

$\begin{split}E_{B}(Z,N)=a_{V}A-a_{S}A^{2/3}-a_{C}\frac{Z^{2}}{A^{1/3}}\\\
-a_{A}\frac{(N-Z)^{2}}{A}+\delta(Z,N)\ ,\end{split}$ (1)

where the first term describes the increase in binding energy as more nucleons
are added to the nucleus, the second term describes the surface contribution
of nucleons near the edge of the nucleus, the third (Coulomb) term describes
the electrostatic repulsion of protons, the fourth term measures the asymmetry
(Pauli exclusion), and the final term controls pairwise spin-coupling. The
coefficients ($a_{i}$) for each of these terms are fit to the measured nuclear
masses of nuclei, such as those compiled and summarized in the Atomic Mass
Evaluation (AME) Audi2003 ; Wang2017 ; Wang2021 . Note that going from
measured _atomic_ masses and _nuclear_ masses requires an additional term to
account for the binding energy of the electrons. The latest (2020) release of
the AME dataset contained the atomic masses of roughly 2,500 species. For the
three latest releases of the AME, the average reported uncertainty among all
nuclei was 40, 30, and 25 keV respectively. From this progression it is clear
that the precision of measurements has continually improved over time,
particularly with the replacement of $\beta$ endpoint measurements with
measurements from Penning traps Block2016 .

Perhaps the most immediately apparent shortcoming of the Weizsäcker mass
formula is its inability to reproduce the so-called ‘closed shell’ features
appearing in nuclear masses. These closed shells correspond to nucleons
completely occupying discrete energy levels of the nucleus and are a direct
parallel to the closed valence electron shells of the noble gas elements
(i.e., the last column of Fig. 1). By observing the nuclear masses as a
function of proton and neutron number, one notes the relatively stronger
binding energies of nuclei with specific numbers of neutrons and/or protons
compared to their immediate neighbors on the chart of nuclides. The canonical
closed shells occur at nucleon ‘magic’ numbers of 2, 8, 20, 28, 50, 82, and
126. For heavy or deformed nuclei, the situation becomes somewhat more
complicated, with the closed shells appearing at different numbers of protons
and neutrons. For example, the heaviest confirmed closed shell for the proton
occurs at $Z=82$, with the next accessible closed shell suggested at 114; for
the neutron, the heaviest closed shell is at $N=126$, and it is suggested that
$N=184$ may also be a closed shell Sobiczewski1966 ; Myers1966 . The shell
behavior of heavy, neutron rich nuclei at $N=126$ and $N=184$ has a uniquely
important role in mediating the production of the heaviest elements in
astrophysical environments, which we discuss in more detail in Sec. 3. Nuclei
with both closed neutron and proton shells tend to be spherical and are
considered ‘doubly magic.’ The ‘Island of Stability’—a region of the nuclear
chart in which superheavy elements are theorized to be long-lived—is
conjectured to occur around the postulated doubly magic closed shell at
$Z=114$ and $N=184$, though there is debate on where exactly on the $N$-$Z$
plane the Island of Stability would exist Mosel1969 ; Lalazissis1996 ;
Cwiok1996 ; Gambhir2005 ; Santhosh2017b ; Adri2021 .

While Eqn. 1 can reproduce basic patterns arising in the masses of measured
nuclei, the growing expansion of our knowledge of the nuclear landscape
demands the need for more complex theoretical descriptions of the atomic
nucleus, both to capture the aforementioned nuclear shell effects and other,
more subtle features of nuclear structure. Below, we survey some of the more
common approaches that have been applied to the heaviest nuclei in recent
years.

The semi-classical approach—sometimes called the macroscopic-microscopic
method—breaks the description of nuclear binding down into two parts: (1) the
energies related to the macroscopic shape, a more refined counterpart to the
Weizsäcker formula, and (2) the energies related to quantum phenomena. The
motivation for this approach is empirical in nature and stems from the mean-
field picture of atomic nuclei Negele1982 . Popular semi-classical models
include the widely used Finite-Range Droplet Model (FRDM) Moller1995 ;
Moller2012 ; Moller2016 , Dulflo-Zuker (DZ) Duflo1995 , KTUY Koura2005 , and
Weizsäcker-Skyrme (WS) Wang2010 ; Liu2011 ; Wang2014 . Much of the modern day
understanding of neutron-rich nuclei relevant to the _r_ process comes from
the fundamental insights provided by semi-classical models Nix1969 ;
Sobiczewski1969 ; Moller1974 ; Krappe1979 ; Moller1981 ; Nix1989 ;
Ichikawa2019 . To date, macroscopic-microscopic approaches provide a match to
mass evaluations on the order of a few hundred keV Mumpower2016r .

A number of powerful and predictive methods seek to model the atomic nucleus
based on the nucleon-nucleon interactions between its constituent protons and
neutrons, e.g., no-core shell models Navratil2000 ; Navratil2016 , Green’s
Function Monte Carlo (GFMC) Pudliner1995 ; Carlson2015 , or Configuration-
Interaction (CI) methods Sherrill1999 . However, as a result of the many-body
problem in physics (which observes the simple fact that the complexity of
describing nuclear systems grows combinatorically with atomic mass number), it
becomes quickly apparent that these and/or similar methods cannot be
reasonably applied to even modestly heavy ($A\gtrsim 100$) nuclei, much less
so for the heaviest/superheavy nuclei with $A\sim 300$; the computational
resources that would be needed far exceed that which could be envisioned any
time in the near or distant future.

Alternatively, energy density functional (EDF) theoretic methods in nuclear
modeling intend to capture, in an approximate way, some of the rich
microscopic physics present in atomic nuclei while avoiding the combinatorical
explosion of complexity otherwise introduced in heavier nuclear systems
Dobaczewski1984 ; Beiner1975 ; Bender2003 ; Stone2007 ; Goriely2009b ;
Schunck2019 . These approaches focus on describing the nuclear potential
energy in terms of local, one-body particle number densities and related
quantities (kinetic energy density, spin densities, and spin-orbit densities,
among a number of others). Application of the variational principle
immediately offers a path to identifying the ground-state configuration by
minimizing the energy density functional with respect to all possible
configurations, from which nuclear mass (among other nuclear structure
observables) may be inferred. Nuclear pairing effects prove something of a
challenge for these methods, so model predictions in the nuclear EDF approach
are generally restricted to even-even nuclei, although methods have been
developed to also treat nuclear systems with unpaired protons and/or neutrons,
e.g., via a blocking procedure Schunck2010 . With phenomenological corrections
applied on top of ‘bare’ model predictions, these models can generally match
known data on the order of a few hundred keV Goriely2009c ; Kortelainen2010 ;
Kortelainen2012 ; Erler2012 ; Goriely2013c ; Agbemava2014 ; Kortelainen2014 ;
Goriely2015c ; Goriely2016 . This situation is largely comparable to that of
macroscopic-microscopic approaches; note, however, that in either approach,
experimental measurements have characteristic uncertainties around an order of
magnitude (or more) lower than any theory currently provides.

Figure 3: Mass model predictions along the U ($Z=92$) isotopic chain relative
to FRDM2012. The plus markers and dark grey band show mass measurements and
uncertainties, respectively, from each $Z=92$ entry in AME2020. (For clarity,
the extent of current mass measurements along this isotopic chain is also
indicated.) The light grey horizontal band indicates an overall 500 keV
uncertainty, which is readily achieved by theoretical models within the region
where mass measurements exists. Outside measurements, model predictions
diverge rapidly.

In any of these approaches, theoretical mass models use experimentally known
nuclei to constrain the underlying parameters within the model. In some cases,
other information can also be used in this parameter optimization step,
including properties of extended nuclear matter, fission barrier heights,
nuclear deformation, nuclear saturation density, the nucleon-nucleon
interaction, and neutron skin depths.

Of chief concern for studying the production of the heaviest nuclei is how
well models perform when extrapolating far from known, measured nuclei
Pearson2006 ; Sobiczewski2014a ; Sobiczewski2014b ; Goriely2014 ; open
questions still persist around the evolution of deformation and collective
effects Casten1987 ; Casten1999 ; Barbero2012 , as well as the strength of
shell closures Strutinsky1967 ; Tondeur1978 ; Retamosa1997 ; Hirsch2010 .
Figure 3 shows the behavior of several mass models for the uranium isotopic
chain along with measurements from the AME2020 compilation where available.
All sets of masses are compared to a baseline mass model, FRDM2012, which
allows for the opportunity to gauge extrapolations. The difference between the
mass model predictions and AME are within the standard uncertainty (500 keV)
for theory mass calculations where there is measured data. However, outside of
the range of measured masses, the divergence between mass models quickly
exceeds 500 keV, especially on the neutron-rich side. This divergence of model
predictions with increasing neutron excess is a general trend with current
modeling efforts.

Accurate mass measurements of neutron-rich nuclei (those on the order of
$\sim$25 keV uncertainty or less) are the primary way to resolve discrepancies
among mass predictions Dilling2018 . Measurements at modern facilities have
provided a wealth of knowledge in recent years Knobel2016 ; Izzo2018 ;
Ascher2019 ; Reiter2020 ; Smith2020 ; Li2022 ; Orford2022 . Insights include
the behavior of highly deformed nuclei Orford2018 ; Vilen2020 ; Vilen2020errat
, the behavior of nuclear near shell closures Baruah2008 ; Meisel2015 ;
Lascar2017 ; Tang2020 , the resolution of isomeric states Yordanov2016 ;
Ayet2019 ; Hornung2020  and the first direct mass measurement of a superheavy
nuclide Schury2021 .

As we will see in the following subsections, the masses of nuclei play a
critical role in nuclear reaction and decay rates, and remains one of the
largest sources of uncertainty for the properties of neutron-rich nuclei.
Section 3 will discuss in detail how different theory predictions for nuclear
masses affect the production of the superheavy elements, helping or hindering
the potential to observe them.

###  2.2 Neutron capture

Neutron capture is a transmutation process in which a nucleus absorbs a
neutron, increasing the mass number by one unit. This process is typically
followed by the emission of $\gamma$-rays in the excited compound nucleus. For
the incident neutron energies ($E_{\rm n}\sim$ 1–100 keV) relevant to
astrophysical processes that creates the heaviest elements, neutron capture
can be described by Hauser-Feshbach theory Koning2005 ; Herman2007 ;
Kawano2016 . This statistical approach is based on the formation of a compound
nucleus and is valid for nuclei that have sufficiently high level densities
Kawano2006 ; Koning2008 . For nuclei with low level density (e.g., near closed
shells) or for nuclei near the neutron dripline, the direct capture mechanism
can also contribute in a significant way Mathews1983 ; Rauscher1998 ;
Goriely1997 ; Otsuki2010 . Pre-compound reaction mechanisms such as semi-
direct capture are less likely to play a role at the relatively low neutron
energies found in astrophysical environments Xu2014 .

Statistical model calculations require inputs from nuclear structure models
including the optical model potential Koning2003 ; Kunieda2007 , nuclear level
density Demetriou2001 ; Goriely2008 ; Minato2011 , and the $\gamma$-strength
function Bartholomew1973 ; Kopecky1990 ; Uhl1994 ; Liddick2016 . The optical
model—together with the associated optical model potential—is used to describe
the overall cross section for the absorption of a neutron into a compound
nucleus. The nuclear level density and $\gamma$ strength functions are then
used to infer the relative probability of the compound nucleus to de-excite.
When the excitation energy is well above the neutron separation energy, as in
the case with $\beta$-decay of neutron-rich nuclei, neutron emission will
dominate $\gamma$ emission in the statistical approach. In contrast, for
neutron capture in astrophysical environments, the total excitation energy in
the compound nucleus is generally below the particle emission threshold,
thereby limiting the reaction channels primarily to elastic, inelastic or
radiative ($\gamma$) emission. In radiative neutron capture, denoted
(n,$\gamma$), the nucleus de-excites by the emission of one or more photons
populating either the ground state or an isomeric state of the compound
nucleus Mumpower2017 . For some light nuclei the (n,$\alpha$) channel and its
inverse may become energetically favorable Bliss2017 ; Psaltis2022  while
neutron-induced reactions on nuclei near the neutron dripline Hansen2001  may
also exhibit the (n,2n) channel due to the low separation energies Goriely2011
. In both of these cases the channels will compete with (n,$\gamma$). Besides
the competition between neutron and $\gamma$ emission of the compound nucleus,
for very heavy nuclei, the compound nucleus can be found in a state in which
nuclear fission may also compete; this is discussed in much more detail in
Section 2.4.2.

Closely related to neutron capture is its inverse process, photodissociation,
in which a photon has enough energy to liberate a neutron from the nucleus.
The energy required for such photons to cause photodissociation must naturally
exceed that of a nucleus’s one-neutron separation energy, without which the
combined photon-nucleus system would have insufficient energy to strip a free
neutron from the nucleus. Unlike neutron-capture cross sections, which must
generally be modeled directly, the rate for the inverse process may quickly be
arrived at by applying arguments based on statistical mechanics—in particular,
the principle of detailed balance. In this way, the photodissociation rate can
be expressed via the equation:

$\lambda_{(\gamma,n)}=\frac{2}{n_{n}}\frac{G^{\prime}(T)}{G(T)}\left(\frac{\mu
k_{B}T}{2\pi\hbar^{2}}\right)^{3/2}\lambda^{\prime}_{(n,\gamma)}\mathrm{e}^{-S_{n}/k_{B}T},$
(2)

where $n_{n}$ is the neutron density; $G(T)$ is the nuclear partition
function; $\mu$ is the reduced mass of the target and projectile nuclei; $T$
is the temperature; $S_{n}$ is the neutron separation energy;
$\lambda_{(\gamma,n)}$ and $\lambda_{(n,\gamma)}$ are the photodissociation
rate and neutron capture rate, respectively; and primed quantities correspond
to those of the nucleus with one fewer neutron relative to those of the
unprimed quantities. For any fixed temperature, the separation energy controls
the photodissociation rate insofar as it appears inside the exponential
factor.

Overall, for a fixed neutron density, the specific point along an isotopic
chain at which equilibrium is reached between the two processes, i.e., where
the flow of nuclear abundance ‘stalls,’ requires a detailed understanding of
both the astrophysical conditions (which sets the temperature of the
nucleosynthetic environment) as well as nuclear structure for neutron-rich
nuclei across the chart of nuclides (which sets the one-neutron separation
energies). The specific implications of this relationship for heavy-element
nucleosynthesis are discussed further in Sec. 3.

###  2.3 Nuclear decay

A basic property of a nuclear species is how long it may persist, on average,
before succumbing to radioactive decay. The radioactive decay of a species
involves changing the identity of one or more nucleons (as in the case of
nuclear $\beta$ decay, Sec. 2.3.1) or breaking into multiple, smaller nuclei
($\alpha$ decay and fission, Sec. 2.3.2 and 2.4.2, respectively). The elapsed
time for a collection of identically prepared species to reduce to half of its
initial population via radioactive decay is known as a half-life, $T_{1/2}$.
It is important to recognize that this definition is statistical in nature,
and thus describes the average behavior of an ensemble of identically prepared
states.

Figure 4: Half-lives of the heaviest measured isotopes. Note the extremely
short half-lives encountered by species after 208Pb ($Z=82$) and before 223Ra
($Z=88$). Closed spherical shells that are confirmed (solid lines) and
proposed (dashed lines) are shown for reference.

Half-lives of the ground state of nuclei may range from fractions of a second
to longer than the age of the universe. Figure 4 shows the total half-lives
for the heaviest measured isotopes as reported in the NuBase (2020) evaluation
Kondev2020 . Stable isotopes are indicated by black squares while radioactive
nuclei are colored according to (the $\log_{10}$ of) their half-life. Many
elements in the actinide region, as well as all but two elements with a lower
charge than bismuth ($Z<83$), have at least one long-lived isotope. The
exceptions of lighter elements with no stable isotopes are technetium ($Z=43$)
and promethium ($Z=61$). A gap of very short-lived nuclei exists between these
two regions where $\alpha$-decay is the dominant decay mechanism. Outside the
region of measured masses, nuclei become evermore short-lived and, for lighter
nuclei below $Z\sim 90$, are predicted to primarily undergo $\beta$-decay.

When two or more nuclear decay processes proceed in tandem, the total half-
life is the sum of the inverse partial half-lives, which are defined as if
each of the decay processes happened independently. The total half-life is
particularly relevant for the heaviest nuclear species where $\beta$ decay,
$\alpha$ decay, and (spontaneous) fission may all compete. Figure 5 plots the
primary measured decay mode across the chart of nuclides. The majority of
nuclei far from stability (including those involved in the creation of the
heaviest elements) will undergo $\beta$ decay. However, in the heaviest
region, there is competition between these channels, and any individual
nucleus may have significant chance to decay by $\beta$, $\alpha$, or fission.

Figure 5: Primary measured decay mode across the chart of nuclides. Evaluated
data taken from NuBase (2020) Kondev2020 . Closed shells that are confirmed
(solid lines) and proposed (dashed lines) are shown for reference above
$N/Z=20$.

Although the specific half-life of a nucleus depends on the combined
probabilities of all its decay channels (which typically involve $Q$-value
differences), the location of the _most_ $\beta$-stable nucleus along an
isobaric chain can nonetheless be approximated from the binding energy.
Returning to the semi-empirical mass formula, one can estimate the minimum
binding energy per nucleon (and therefore which combination of $Z+N=A$ is the
most stable):

$N/Z\sim 1+\frac{a_{C}}{a_{A}}(Z+N)^{2/3},$ (3)

where $a_{C}\sim 0.7$ MeV and $a_{A}\sim 23$ MeV are the Coulomb and asymmetry
coefficients. The small, positive factor implies that heavier stable nuclei
have slightly more neutrons than protons. This fact can be observed in Figs. 4
and 5 as the known stable nuclei bend progressively away from $N=Z$.

Below, we discuss the principle modes of spontaneous decay for heavy, neutron-
rich nuclei that involve particle emission. Spontaneous fission and
$\beta$-delayed fission, which are also important in determining the overall
(in)stability of the heaviest nuclei, are reserved specifically for our
broader discussion of fission in Sec. 2.4.2.

####  2.3.1 Beta ($\beta^{-}$) decay

Nuclear $\beta^{-}$ decay is the primary decay mode for the majority of nuclei
involved in the neutron-capture processes responsible for creating the
heaviest elements, the _r_ process. This nuclear transmutation process is
governed by the weak interaction that converts a neutron into a proton in the
nucleus. The nucleus simultaneously emits an electron and an anti-neutrino to
conserve charge and lepton number. In a somewhat humorous letter by Pauli
Brown1978 , the neutrino itself was postulated in order to explain the
continuous energy distribution of the electron that was observed decades
earlier Sime1996 . Later, Fermi provided a theoretical framework for
performing calculations Fermi1934  (see also Wilson’s English translation
Wilson1968 ). For completeness—though not especially relevant during the
astrophysical production of heavy nuclei via rapid neutron capture—$\beta^{+}$
decay occurs on the neutron-deficient side of nuclear stability when a proton
in the nucleus transmutes into a neutron via the emission of a neutrino and a
positron. The decay of nuclei via $\beta^{+}$ (or $e^{-}$-capture) may
contribute non-negligibly in the slow neutron-capture process, especially at
branch points. Unless otherwise specified, any mention of ‘$\beta$’ decay here
refers to $\beta^{-}$ decay.

The $\beta$-decay process may leave the nucleus in an energetically excited
state. In this excited state, the nucleus can either emit energy in the form
of photons to decay to its ground state, or, if energetically allowed, emit
($\beta$-delayed) particles, possibly changing its identify yet again (a
possibility we will discuss later in this section and in Sec. 2.4.2). The
total energy available in $\beta$ decay is measured by $Q_{\beta}$, the energy
(or mass) difference between the initial and final ground states:
$Q_{\beta}=m_{i}-m_{f}-m_{e}$. Whether the daughter nucleus decays to a ground
state or into a different nucleus depends on how much of the total available
energy ($Q_{\beta}$) is allocated to the daughter nucleus. Theoretical models
of nuclear $\beta$ decay require both the relative energy between the parent
and daughter states as well as a prediction of the transition strength (or
intensity) between these participating levels Menendez2016 . This requirement
amounts to applying Fermi’s Golden Rule to specific nuclear operators
Engel2015 ; Engel2017 ; Nanni2018 . The pairing of this information may be
summarized in a $\beta$-decay strength function of a nucleus Hansen1973 , from
which the overall $\beta$ decay rate may be immediately calculated Moller1990
. When, following $\beta$ decay into an excited state, the population of
excited states in the daughter nucleus exceeds particle threshold, delayed
particle emission may ensue Pappas1972 ; Gjotterud1978 . For neutron-rich
nuclei, delayed particle emission may occur when the neutron separation energy
in the daughter nucleus is less than the parent’s $Q_{\beta}$, i.e.,
$S_{n}(Z+1,N-1)<Q_{\beta}(Z,N)$, leading to so-called $\beta$-delayed neutron
emission Borzov2014 ; Dimitriou2021 .

Overall, the excited nucleus will favor particle emission over electromagnetic
decay whenever it is energetically allowed Spyrou2016 . However, a more
refined portrait may be obtained by employing principles of statistical decay,
in very much the same way as de-excitation is modeled following the formation
of a compound nucleus bohr1935  following neutron capture (Sec. 2.2). In this
way, the branching ratios and energy spectra of emitted particles may be
calculated Kawano2008 ; Mumpower2016b . Because the overall energy of the
compound nucleus is determined by $Q_{\beta}$, which can be quite large (ten
MeV or more) for the very neutron-rich nuclei populated during an _r_ process,
it is energetically favorable for multiple neutrons to be emitted before a
$\beta$-unstable nucleus finds its way to a stable nucleus. For nuclei with
large $Q_{\beta}$, i.e., as one approaches the neutron dripline, modern models
predict approximately five delayed neutrons emitted per $\beta$ decay
Minato2021 . In the most extreme cases, as many as eight or more neutrons may
be emitted following $\beta$ decay in _r_ process nuclei.

Figure 6: Isotopic $\alpha$-decay $Q$-values from the 2020 release of the
Atomic Mass Evaluation (AME2020) (points) Wang2021  compared with those from
the FRDM2012 model (solid lines) Moller2016  along constant proton number.
Thick black lines indicate proton-number intervals of ten.

For nuclei in a localized region of the chart of nuclides, the nuclear shell
model may be used to calculate $\beta$-decay rates Langanke2000 ;
MartinezPinedo1999 ; MartinezPinedo2000 ; Zhi2013 . However, currently, to
compute all relevant nuclei, theories with a global reach must be employed
Minato2013 ; Minato2022 . Global calculations of $\beta$-decay properties have
been calculated within the macroscopic-microscopic method based on the Quasi-
particle Random Phase Approximation (QRPA) Moller1997  with phenomenological
treatment of first-forbidden transitions Moller2003 . This approach generally
yields $\beta$-decay properties dominated by Gamow-Teller transitions
Moller2019 . In contrast, microscopic approaches suggest that first-forbidden
transitions play a strong role, especially above the $N=126$ shell closure
Marketin2016 . More recently, microscopic approaches have utilized the finite-
amplitude method to speed up the calculation of $\beta$-decay properties
Mustonen2016 ; Ney2020 . It is important to keep in mind that the current
large discrepancies in $\beta$-decay properties will have substantial impact
on the creation of the elements in astrophysical environments. We return to
this point in Section 3.

####  2.3.2 Alpha ($\alpha$) Decay

Alpha decay is a nuclear transmutation in which an $\alpha$ particle (a helium
nucleus consisting of two protons and two neutrons) is emitted from the
nucleus. Alpha decay is technically possible when the $Q$-value
($Q_{\alpha}=m_{i}-m_{f}-m_{\alpha}$) is positive, and it is one of the
primary decay modes of the heaviest nuclides. However, because of a relatively
strong Coulomb barrier established by the parent nucleus, there are many
nuclei which exhibit a positive $Q_{\alpha}$ but which undergo $\alpha$ decay
with effectively zero probability, and those nuclei with significant $\alpha$
decay rates are those with comparatively larger $Q_{\alpha}$ values.

Figure 7: Fission barrier heights across the heaviest region of the chart of
nuclides, as calculated using the FRLDM model Moller2015 . Note the region of
particularly low barrier heights around $Z=100$ and $N=190$, which boosts
nuclei’s tendency towards fission and substantially inhibiting their capacity
to produce the superheavy elements.

Figure 6 shows the behavior of $Q_{\alpha}$ along isotopic chains for heavy
nuclei. Measurements reported in the AME2020 are shown in points while lines
indicate $Q_{\alpha}$ predictions from FRDM2012. Black solid (dotted) lines
specify proton numbers divisible by ten (five). This figure reflects the
general qualitative agreement between theory and experimental values when used
to determine relevant $Q$-values for nuclear reaction and decay calculations.
In reference to the _r_ process specifically, however, it should be noted that
those nuclei with significant $Q_{\alpha}$ lie much closer to those with known
masses, so the difficult and persistent complication of uncertain nuclear
masses in relation to, e.g., theory calculations of neutron capture or
$\beta$-decay of very neutron-rich nuclei is not nearly so severe for
analogous predictions of $\alpha$-decay properties.

The partial half-life of $\alpha$-decay may be calculated via an assumption of
quantum tunneling through a barrier, which results in the well-known Viola-
Seaborg relation Viola1966 . This kind of phenomenology was crucial in
anticipating the possible $N=184$ shell closure. Following this work,
subsequent predictions of $\alpha$-decay half-lives have been made using
various methods Poenaru1983 ; Samanta2007 ; Kiren2013 ; Sahu2016 ;
Santhosh2017 ; Santhosh2018 . While measurements have recently been made of
lighter $\alpha$-decaying nuclei Braun2017 , it is the $\alpha$-decay chains
of the heaviest nuclei that impact the actinide and superheavy elements.

###  2.4 Nuclear fission

The heaviest nuclear species may undergo fission, which is the general term
for processes that splits the nucleus into two or more lighter ‘fission
fragments.’ The splitting of a nucleus into two main fragments along with,
possibly, some small handful of neutrons is the most common modality, as first
reported by Meitner, Strassman, and Hahn Meitner1938 .

Fission rates are the most important fission property in controlling whether
and how many heavy and superheavy nuclei might be produced following an _r_
process; if the rates for certain nuclei are too fast, for example, nuclei
will have a small ‘survival probability’ during their decay towards less
neutron rich nuclei that occurs at the end of any _r_ process. At the same
time, electromagnetic and particle radiation spectra, as well as fission
fragment yields, can be important observational tools for examining the
possibility of heavy and superheavy element production in _r_ -process
environments and their associated remnants.

Overall, a complete theoretical framework for calculating the numerous fission
properties of interest to the _r_ process has proven to be a formidable task,
and a number of approaches have been developed to make progress towards this
persistent challenge in nuclear theory. Broadly speaking, the theoretical
approaches mirror those applied in nuclear structure (see discussion of Sec.
2.1), with fully microscopic, macroscopic-microscopic, and phenomenological
models all providing important developments in our understanding of, and
predictions for, fission properties across the nuclear chart (see Ref.
Schunck2022  for a recent and thorough review). Microscopic approaches, in
particular, are at a uniquely promising point in their history, as recent
advances in computational hardware are beginning to provide the considerable
resources needed to fully apply such methods to the problem of fission
Jach2017 ; Hessberger2017 ; Regnier2017 ; RodriguezGuzman2017 ; Sekizawa2017 ;
Tao2017 ; Verriere2017 ; Giuliani2018 ; Navarro2018 ; Lemaitre2019 ; Zhao2019
; Bulgac2020 ; RodriguezGuzman2020 ; Bulgac2021 ; Jin2021 ; Marevic2021 ;
Verriere2021b ; Bulgac2022  (also see Ref. Schunck2019  and references
therein). At the same time, macroscopic-microscopic approaches have continued
to fill a gap insofar as their comparatively less demanding computational
methods enable studies that predict fission properties across the entire
nuclear chart Abe1996 ; Moller2004 ; Moller2015b ; Randrup2011a ; Randrup2011b
; Sierk2017 ; Mumpower2020 ; Verriere2021 . Still too, systematics fill
important gaps in our ability to make reasonable predictions for certain types
of fission observables, particularly with regards to the calculation of
fission rates Hill1953 ; Bjornholm1980 ; Xu2005 ; Karpov2012 ; Panov2013  and
approximate fragment yields Kodama1975 ; Wahl2002 . Modern treatments now also
include fragment energy partitioning, deexcitation, and corresponding neutron
and gamma emission spectra Schmidt2016 ; Jaffke2018 ; Okumura2018 ; Lovell2021
; Talou2021 ; Vogt2021 .

We explore the specific implications of fission for nucleosynthesis in Sec. 3
while focusing the remaining portions of the present discussion on the modern
understanding of fission in and of itself.

####  2.4.1 Fission barriers

In any prediction of fission rates—regardless of fission process or
nucleus—the fission barrier is one of the strongest predictors of a nucleus’s
propensity to fission. Because the fission properties of so few nuclei have
been measured experimentally (and fewer still, when only nuclei of interest to
the _r_ process are considered), _r_ process simulations rely on global models
for predicting fission barriers which are, in turn, used to calculate actual
fission rates for the relevant fission processes. Such global predictions are
available in the BCPM Giuliani2013 , ETFSI Aboussir1992 ; Aboussir1995 , FRLDM
Moller2015 , HFB-14 Arnould2006 , and Thomas-Fermi Myers1999  models. In each
of these models, the nuclear potential energy of each nucleus is computed
across a domain of collective coordinates, and the resulting geometry of the
potential energy surface is analyzed to identify the fission barrier in
reference to the ground state and splitting (scission) configurations. The
difference between the ground state and fission barrier energies is taken to
be the ‘fission barrier height,’ which is the quantity ultimately used to
predict the rates for the different fission processes Nix1972 . When a nucleus
is initially in an excited state (e.g., as in the compound nucleus formed in
neutron-capture or the excited energy states nuclei $\beta$ decay into), the
excitation energy may be subtracted from the barrier height to obtain an
effective value for further use in such calculations.

In Fig. 7, we show the global behavior of barrier heights across the
fissioning region of the nuclear chart for the FRLDM model Moller2015 .
Particularly noteworthy is the transition to a broad, diffuse region of low
barrier heights for very neutron-rich nuclei located around $Z=100$ and
$N=190$. In the 1960s, it was already expected that this general region would
have relatively lower barrier heights that would enhance the propensity to
fission as _r_ process attempt to traverse this region on a path towards the
superheavy elements, ultimately inhibiting or altogether preventing the
production of these elements in nature Moller2010 . Modern calculations
involving the aforementioned models generally agree with each other on the
order of $5$ MeV (but diverge more significantly for the most neutron-rich
nuclei), but in any case generally demonstrate this region of lower barrier
heights, along with the corresponding implications for superheavy element
production Thielemann1983 ; Giuliani2013 ; Mumpower2018 ; Vassh2019 ;
Minato2021 . Nevertheless, these calculations are subject to the same level of
uncertainties characteristic of nuclear theory in general; future work that
aims to improve on these methods is surely warranted. We will explore in Sec.
3.2 how this region of low fission barrier heights affects superheavy-element
formation in modern _r_ -process simulations.

####  2.4.2 Fission Processes

In the most general sense, fission occurs when a nucleus proceeds through a
complex configuration space, temporarily passing through one or more regions
associated with a relatively high potential energy, until it arrives at one of
many possible highly deformed ‘scission configurations’ with much lower
potential energy than it began with. Once a scission configuration is realized
by the nucleus, the nucleus is energetically driven to split into two or more
fragments. The fission barrier measures the minimum excitation energy
(relative to the ground state) that the nucleus must possess in its initial
state to undergo this process in the classical sense, i.e., such that the
total energy never drops below zero throughout its dynamical trajectory to
fission. Of course, the nucleus is clearly a quantum mechanical system, so it
is often the case that the nucleus proceeds through the barrier by tunneling
on occasion, and the fission barrier is therefore used as a proxy for
predicting fission rates in phenomenological approaches.

The different fission processes observed in nature, then, are categorized in
reference to the different ways that the initial state of a fissioning nucleus
might be prepared. Perhaps the simplest among these is the case of spontaneous
fission, which can occur in ground or metastable states of especially heavy
nuclei, in which fission proceeds in situ by tunneling through the fission
barrier Nilsson1968 . In this case, the rate of spontaneous fission is a
constant property of the nucleus (i.e., not determined by extrinsic factors
such as temperature, density, or other nuclei) Randrup1973 ; Randrup1974 ;
Randrup1976 . Even though spontaneous fission may technically be possible for
a large number of unstable heavy nuclei, it is frequently the case that it is
negligible when compared with other available decay modes that proceed much
more quickly. For nuclei produced during the _r_ process, this is generally
the case. However, on longer timescales after the _r_ process has concluded,
it is likely to contribute significantly once other decay and/or fission modes
have become inaccessible (e.g., after the exhaustion of free neutrons
effectively closes off the neutron-induced fission channel, discussed below).
While a small number of spontaneous fission rates have been experimentally
measured Kondev2020 , the intersection of these instances with _r_ process
nuclei is quite small. Theoretical calculations may be based on barrier
penetration arguments Sadhukhan2020a  or empirical phenomenology Xu2005 , some
of which broadly capture the dependency of the rate on the fission barrier
Zagrebaev2011 ; Karpov2012 .

In contrast with spontaneous fission, there are any number of conceivable ways
that a nucleus can be prepared with a wide range of excitation energies. When
this occurs, a nucleus’s total energy will be brought much closer to the
fission barrier, which can dramatically increase the probability of fission.
Of particular interest to _r_ process applications are the neutron-induced and
$\beta$-delayed fission processes, both of which may provide a nucleus with
such excitation.

In neutron-induced fission, a compound nucleus forms in much the same way that
of neutron capture. The total energy of a nucleus immediately after it
captures a neutron is equal to the sum of the neutron energy and the one-
neutron separation energy ($S_{n}$) of the product nucleus. In ordinary
neutron capture, the compound nucleus will undergo a cascade of
electromagnetic decays to the ground state. However, for the heavier nuclei,
where the fission barriers are sufficiently low, fission competes with the
electromagnetic decays, such that some (possibly significant) fraction of
neutron-capture events terminate with fission, rather than with the ground
state associated with the compound nucleus. The total neutron-induced fission
and neutron-capture rates can be evaluated by considering both channels in
statistical Hauser-Feshbach approaches Kawano2021 .

Figure 8: Maxwellian averaged cross sections (MACS) for neutron capture (red)
and neutron-induced fission (blue) across the Np ($Z=93$) isotopic chain. The
cross sections are evaluated at $T_{9}=1$ GK, a typical temperature that an
_r_ process might traverse through.

In Fig. 8, we plot both of the neutron capture and neutron-induced fission
cross sections as a function of increasing neutron number for the Np ($Z=93$)
isotopic chain and for thermalized environments at a temperature broadly
representative of the _r_ process, $T=1$ GK. Recalling the fission barriers
plotted in Fig. 7, there is a diffuse region of considerably lower fission
barriers that the $Z=93$ isotopic chain passes through for isotopes with
neutron numbers roughly between $N=165$ and $N=175$. This trend in fission
barrier height is clearly reflected when comparing the relative cross sections
for fission vs. neutron capture, where the sharp peak in the fission cross
section precisely coincides with the nuclei with the lowest barriers, with
neutron-induced fission clearly dominating over neutron capture by around
three orders of magnitude in the most extreme instances.

Importantly, the total neutron-induced fission rate for each nucleus,
$\lambda_{(n,f)}$, depends directly on free neutron number density, $n_{n},$
such that $\lambda_{(n,f)}\propto n_{n}$. Correspondingly, in the extremely
neutron-rich environments characteristic of the _r_ process, neutron-induced
fission can be quite significant, especially during the earlier phase of the
_r_ process where the total number of neutrons exceed all other nuclei in the
environment. Indeed, network calculations suggest that neutron-induced fission
can, for a brief but critical period of time during nucleosynthesis, be the
fastest fission process Petermann2012 .

The third and final fission process typically considered in _r_ -process
studies is $\beta$-delayed fission. In this scenario, a nucleus undergoes an
initial $\beta$ decay which leaves it in any number of excited nuclear states
of the final nucleus, ranging from the ground state up to the $Q$-value for
the decay, with the remaining energy being carried away by the electron and
anti-electron-neutrino that are immediately produced by the decay. As in
neutron-induced fission, the excited nucleus formed immediately after $\beta$
decay has some probability to fission, the extent of which is determined
largely by the relative magnitude of the excitation energy and the fission
barrier. The total $\beta$-delayed fission rate is then taken to be product of
the total $\beta$-decay rate and the probability to subsequently fission, and
the effective $\beta$ decay rate is the difference of the total rate with the
$\beta$-delayed fission rate.

An interesting variation on $\beta$-delayed fission happens when the most
neutron-rich nuclei of the _r_ process are considered. As one moves across an
isotopic chain towards increasingly neutron-rich nuclei, the neutron
separation energy decreases while the $Q$-value for $\beta$ decay increases.
As discussed in Sec. 2.3.1, this provides ideal conditions for the emission of
one or more delayed neutrons. By coupling a probability to fission at each
stage of this decay sequence, and provided that the fission barriers are
sufficiently low that fission becomes an appreciable possibility, this
introduces the prospect for multi-chance $\beta$-delayed fission, a process
which comprises a spectrum of possibilities for the emission of some number of
delayed neutrons followed by fission (as opposed to electromagnetic decay to a
ground or metastable state). The overall process of $\beta$ decay is then a
complex convolution of electromagnetic decays, neutron emission, and fission.
The cumulative effects of the competition between these decay modes can be
carefully resolved in a statistical Hauser-Feshbach framework Mumpower2018 ;
Minato2021  in much the same way that the competition between different exit
channels are treated to resolve the probability of neutron-induced fission
relative to neutron capture.

While neutron-induced fission can be the fastest fission process during the
first second(s) of an _r_ process, when free neutrons are still extremely
abundant, $\beta$-delayed fission and multi-chance $\beta$-delayed fission
rates, which are constant and necessarily do not depend on neutron abundance,
can become the primary fission process once the neutron abundance drops to
sufficiently low levels, thereby eliminating any significant possibility for
neutron-induced fission to occur. Because this generally occurs when nuclei
are decaying towards longer-lived and less neutron-rich nuclei, accurate
predictions for $\beta$-delayed fission rates across the nuclear chart are
especially important for understanding the formation of heavy and superheavy
elements in the aftermath of the _r_ process Kodama1975 ; Thielemann1983 ;
Petermann2012 ; Goriely2015b ; Ghys2015 ; Mumpower2018 ; Holmbeck2019a .

##  3 Astrophysical alchemy

###  3.1 The basic _r_ process

Figure 9: Schematic description of the _r_ -process pathway and the dominant
reaction channels active in the heavy-element region. Black boxes indicate
stable nuclei, outlined boxes measured nuclei, and grey-filled a theoretical
extent for which nuclei could possibly exist. Note that for these heavy-
element regions, the _r_ -process pathway is very far from the extent of
measured nuclear data (i.e., far from the black-outlined boxes denoting the
extent of the AME2020), indicating that the nuclear data in these regions rely
on theory and become very uncertain.

As we alluded to in Sec. 1.2, the natural abundance of elements heavier than
iron are expected to be mostly produced by neutron capture nucleosynthesis,
with the heaviest among these (those heavier than lead) in particular being
produced entirely by a subset of neutron-capture nucleosynthesis known as the
rapid neutron capture process, or the _r_ process Horowitz2019 . Up until now,
we have made many references to the _r_ process while describing the
properties of the heaviest nuclei that are created—even if momentarily—by this
nucleosynthetic mechanism. Now, equipped with knowledge of the relevant
nuclear physics in the heavy-element region, we may turn to a comprehensive
description of the _r_ process itself.

Schematically, the _r_ process proceeds as follows: given a system with a
small abundance of ‘seed’ nuclei but subjected to an extraordinary flux of
free neutrons, the neutrons will radiatively capture onto the seed nuclei,
transmuting them along isotopic chains to nuclei with heavier mass numbers but
the same atomic number. In this way, the nuclei may be pictured as moving
horizontally on the chart of nuclides (when plotted in $N$ versus $Z$). As
neutron number increases for a given isotopic chain, the $\beta$-decay rates
tend to become much faster Shafer2016 . At the same time, the neutron
separation energies tend to decrease as nuclei become more neutron-rich, such
that the rate of neutron capture will become slower overall, with the inverse
process, photodissociation of neutrons from nuclei, further reducing the
overall capacity of nuclei to net-capture neutrons. The combined effect of the
increasing $\beta$-decay rate and decreasing neutron-capture rate is that the
seed nuclei will continue to capture neutrons (forming heavier nuclei) until
$\beta$ decay rates become fast enough to compete with neutron capture, which
will then increment the nuclei to the next successive element. This process
repeats for the next isotopic chain, with neutron capture increasing mass
number and $\beta$ decay transmuting nuclei into heavier elements.

Figure 9 shows a schematic for how the heaviest nuclei are populated during
the _r_ process in terms of their primary reaction channels: captures, decays,
and fission; recall discussion in Sec. 2. The _r_ -process ‘path’ shown by the
lines with arrows approximately denotes a neutron-rich pathway near the
neutron dripline traversed in creation of the heaviest elements. Along the red
line, there are more captures per decay, as indicated by its shallow slope.
The blue line indicates that the process of moving up a closed neutron
shell—such as that predicted at $N=184$—is moderated by decays. Once the
predicted shell closure is surmounted, large uncertainties make the definition
of the _r_ -process path (and the dominant channel) difficult to determine.

The _r_ -process path denoted in Fig. 9 is purely schematic. It is more
rigorously defined as the set of the most abundant nuclei along each isotopic
chain at a given time in the _r_ process. In the early stages of the _r_
process at high temperatures and densities, the path can be defined by an
equilibrium between (n,$\gamma$) and ($\gamma$,n) along any given isotopic
chain. Once the bulk of these reactions are no longer in equilibrium, a point
known as (n,$\gamma$) freezeout, the path may be defined by a subset of nuclei
for which the $\beta$-decay rates have become competitive with the neutron-
capture rates Cowan1991 .

Figure 10: Snapshots of nuclei produced in an _r_ process network calculation
using astrophysical conditions from the neutron-star merger simulations of
Rosswog2014  and nuclear data from Mumpower2018  for three key times: (a)
before (n,$\gamma$) freezeout, (b) decay back to stability, and (c) well after
the _r_ -process event. Colored boxes show the abundances of the nuclei at
each moment in the calculation. Stable nuclei are shown as black squares, and
the limit of mass measurements from AME2020 Wang2021  is shown by the black
outline. The grey shaded region shows the theoretical extent of bound nuclei
that may exist according to the FRDM2012 mass model Moller2012 , and the
vertical and horizontal bars at $N=126$ and $Z=82$ denote closed nuclear
shells.

After this trade-off between neutron-induced reactions and $\beta$ decay has
continued for some time (generally within a second or so), the neutrons will
be mostly or entirely exhausted, after which $\beta$ decay overtakes neutron
capture as the dominant transmutation mechanism. Figure 10 shows the
abundances of each nucleus at freezeout (panel ‘a’) and other key times in an
_r_ process calculation using neutron-rich conditions (trajectory from
Rosswog2014  with nuclear physics as in Mumpower2018 ). After freezeout, the
nuclei begin their long series of decays, as show in Fig. 10b, eventually
terminating at the stable and extremely long-lived nuclei and beginning an
isochronic phase of the _r_ process (Fig. 10c) Sprouse2022 . Save for a
handful of late-time $\alpha$ decays, spontaneous fission events, and latent
neutron capture that can adjust the final abundances Zhu2018 ; Wu2019 , the
abundances in Fig. 10c show the stable and long-lived nuclei created by the
_r_ process that, e.g., are now observed in the Solar system and throughout
the universe.

Both during the _r_ process itself as well as after freshly synthesized nuclei
have decayed to their final, stable/long-lived configurations, the closed
neutron shells at $N=50$; 82; 126; and, possibly, 184 manifest as relatively
over-abundant ‘peaks’ in the abundance pattern. For example, the peaks in the
Solar system isotopic pattern in Fig. 2 at $A\sim 80$, 130, and 195 are a
direct consequence of the closed neutron shells. (The sharper peaks at higher
mass numbers in Fig. 2 are also from the same closed neutron shells, but occur
from the _slow_ neutron-capture process, which populates those shells at
higher charge numbers, and hence, higher mass numbers overall.) These peaks
are explained simultaneously by an increase in half-lives along the shell
closure and the sudden drop in neutron separation energies that occurs just
after the shell closures are crossed Mumpower2015a . The net effect of which
is to strongly impede the nuclear flow across such isotones, leading to the
retention of nuclei at a neutron richness considerably less than that found
elsewhere throughout the chart of nuclides Mumpower2015b . Because the
$\beta$-decay rates for these closed-shell nuclei are slower than surrounding
counterparts, the closed shells act to restrict the flow of nuclear abundance
to heavier nuclei, and nuclei will tend to accumulate around these discrete
neutron numbers. Notice in Fig. 10a that nuclei at the closed neutron shell
$N=126$ indeed show the largest abundances in the calculation. After
freezeout—and depending on the extent of latent neutron capture—the nuclei
once populated at the closed $N=126$ shell maintain their high abundances
along the same _isobars_ ($A\sim 195$), but the high abundance peak shifts to
lower neutron numbers. This shift of the abundance peak can just be begun to
be seen in Fig. 10b by the slight shift to lower neutron numbers of the high-
abundance region at $Z=70$. Finally, at late times this high-abundance feature
(produced pre- and mid-freezeout as an effect of the neutron shell closure at
$N=126$) now populates $N\sim 118$, as can be seen in Fig. 10c.

The proposed closed neutron shell at $N=184$ is particularly relevant for the
superheavy elements, as it is around the $N=184$ and $Z=114$ doubly magic
region that the Island of Stability is speculated to occur Chapman2020 . We
discuss next the region beyond $N=126$, where the abundances of nuclei during
a standard _r_ process depend on the complicated interplay of reactions beyond
just neutron capture and $\beta$ decay.

###  3.2 Production of the heaviest elements

As described in the previous section, a basic canonical picture of the _r_
process involves neutron capture, its inverse process of photodissociation,
and $\beta$ decay. However, as soon as an _r_ process produces the heaviest
elements, particularly those well beyond Pb, the situation becomes
substantially more complicated, and the effects of a more diverse set of
processes—such as $\alpha$ decay and fission—may significantly contribute to
the overall dynamics of the ensuing nucleosynthesis. The competition between
inflowing and outflowing material at high charge and mass ultimately
determines abundance of heavy elements produced by the _r_ process, including,
possibly, the superheavies.

Recalling Fig. 9, neutron capture and $\beta$ decay bring material to the
actinide region. Once occupying the actinides, nuclei have a chance to undergo
fission through neutron-induced, $\beta$-delayed, and/or spontaneous fission
channels Mumpower2018 ; Hao2022 . As described in Sec. 2.4.2, nuclei with
proton number larger than $Z\sim 90$ may be driven to fission by an incoming
neutron Panov2003 ; Martinez-Pinedo2007 . Similarly, fission can follow the
$\beta$ decay of a nucleus if that decay places it in a sufficiently excited
state Thielemann1983 ; Ghys2015 ; Mumpower2018 . For some theoretical models,
$\beta$-delayed and neutron-induced fission can ultimately terminate the _r_
process and prevent very high-$A$ nuclei from being synthesized. This effect
is apparent in Fig. 7 by the extremely low fission barrier heights near
$Z=100$ and just beyond $N=184$.

Figure 11: Most likely $\beta$-decay paths (dashed) of some neutron-rich
nuclei that can populate the superheavy elements as they traverse through
regions where there is a probability to fission (colors), calculated using
decay and reaction data based on the FRDM2012 mass model Moller2012 ;
Moller2016  and FRLDM fission barrier heights Moller2015 . Only
$\beta$-delayed fission and spontaneous fission are included in panel (a), and
panel (b) additionally includes the neutron-induced fission channel. The
black-outlined region shows the extent of current nuclear mass measurements
from the AME2020 compilation Wang2021 , and the grey boxes show a theoretical
extent for what nuclei could possibly exist. Superheavy elements that have
been observed in laboratory settings are marked with red, and the black circle
indicates the region of the theoretical superheavy ‘Island of Stability.’

Modern simulations of the _r_ process suggest a final fission termination
point between $A\sim 275$ and $A\sim 300$ Eichler2015 ; Mumpower2018 ;
Goriely2013 . The nuclei populated in this region of the chart of nuclides are
certainly superheavy in the sense that they have an extreme neutron excess.
Whether superheavy elements are created this far from stability early on in
the _r_ process as well as their possible survival to more stable species
remains an open question. These questions are related to the maximum
termination point of the _r_ process, which is also uncertain. Sufficiently
high fission barriers for these extremely neutron-rich species would raise the
terminus of the _r_ process to even higher mass numbers Shibagaki2016 .

Once the neutron-flux drops and the $\beta$-decay timescale dominates over
neutron-capture, the newly synthesized neutron-rich nuclei can decay to even
heavier elements, whereby they have another chance to populate the
superheavies. Figure 11 shows paths along which some neutron-rich nuclei that
could be made in an _r_ process may $\beta$ decay. The most neutron-rich
nuclei are more likely to emit at least one neutron after $\beta$ decay
($\beta$-delayed neutron-emission), which further decreases the mass of the
nuclei that are ultimately formed at the end of those $\beta$-decay chains.
However, these nuclei are not completely free to $\beta$-decay undisturbed;
during these $\beta$-decays, fission processes can continue to evacuate
material from the heavy-element region. Sufficiently low fission barrier
heights in this region can cause nuclei to prefer fission over $\beta$ decay
and therefore throttle production of the actinides and superheavies.

The green and blue colors in Fig. 11a show the probability that a nucleus will
survive fission and $\beta$ decay in the absence of neutron-induced fission.
These probabilities were calculated by taking the ratio of the $\beta$-decay
rate to the sum of the $\beta$-decay, $\beta$-delayed, and spontaneous fission
rates (from the FRDM2012 mass model Moller2012 ; Moller2016  and FRLDM fission
barrier heights Moller2015 ) on a per-nucleus basis. By taking the product of
the probabilities along a particular $\beta$-decay path (e.g., one of the
black dotted paths), we can obtain an upper limit for how likely it is that a
particular nucleus at freezeout will transmute into a superheavy species. To
create, e.g, 290Ds ($Z=110$), a progenitor 304Cm ($Z=96$) nucleus would have
to traverse through several regions where the probability to fission exceeds
the probability to $\beta$ decay. In total, the cumulative probability that
304Cm and the daughter nuclei that are created by subsequent $\beta$ decay
will survive each opportunity to fission and create 290Ds is found to be about
0.1% in the absence of neutron-induced fission.

When the neutron-induced fission channel is included (i.e., by adding an
additional term to the rate ratio), even at low temperatures and neutron
fluxes, neutron-induced fission at late times when the nuclei are closer to
stability can thwart superheavy-element production, as shown in Fig. 11b. In
the case of 290Ds, even a low neutron flux decreases the total probability of
production by more than an order of magnitude. On the lower-mass side, there
is a small window with lower probabilities to fission where nuclei can decay
into the superheavy series. For example, in the absence of neutron-induced
fission 294Pu ($Z=94$) has about a 0.8% chance to survive every fission event
along its most-probable decay chain and create 283Hs ($Z=108$). (Neutron-
induced fission changes this endpoint nucleus to 283Db ($Z=105$) with a
combined survival probability of 0.04%.) In the next section, we will explore
how these predictions vary for different assumptions of theoretical nuclear
physics properties.

###  3.3 Sensitivity to nuclear physics

In the extremely neutron-rich regions of the chart of nuclides, all properties
of nuclei entering into _r_ process simulations depend on theory calculations,
and predictions by theory may vary. For example, the strength of the closed
neutron shells affects how long nuclei remain at that nucleus, eventually
determining how much mass occupies the closed shells. Neutron shells that are
too strong effectively stockpile abundance at the closed shells, which may
prevent material from reaching the actinide region and have important
consequences for superheavy elements production Schatz2002 . In particular,
the $N=126$ is the highest known spherical neutron shell closure. Though
theories predict the next spherical neutron shell closure at $N=184$, its
existence has not yet been confirmed. In modern, dynamic simulations, neutron-
rich _r_ processes that proceed beyond the $N=126$ shell closure are typically
terminated around $N=184$ by fission processes, as discussed in Sec. 2.4.2 and
3.2. The strength of this closed shell plays an additional role in the
production of heavy and superheavy nuclei moderating late-time neutrons.
Recent microscopic fission models Giuliani2020  show that material accumulated
around closed shells like $N=184$—resistant to neutron capture—can allow more
late-time free neutrons, which may in turn induce fission on the heavy nuclei
decaying to stability, for example as is the case in Fig. 11b. On the other
hand, if the $N=184$ shell closure does not exist, or is weakened as a
function of neutron excess, it is possible that the _r_ process terminates at
higher mass numbers, providing an easier pathway for the population of
superheavy nuclei. Such considerations are sensitive to the behavior of
nuclear reactions at large neutron excess Patra2009 .

Similarly, $\beta$ decay determines how quickly material traverses between
(and beyond) the closed neutron shells. This is particularly important for the
$N=126$ closed shell as it acts as a gatekeeper to the production of the
heaviest elements Tang2020 . Decay rates that include First-Forbidden
transitions for nuclei above the $N=126$ shell closure increase the
$\beta$-decay rates by over an order of magnitude on average compared to model
predictions that do not fully include these types of transitions (e.g.,
Marketin2016  versus Moller2019 ). As a consequence, heavy nuclei spend very
little time in the $N>126$ region before reaching the fission region. In the
end, overall less material populates the actinide region by the time $\beta$
freezeout occurs, resulting in a decreased likelihood that the superheavy
elements will be robustly populated. Other studies of $\beta$-decay involve
experimentally characterizing $\beta$-delayed neutron emission for neutron-
rich nuclei Dimitriou2021 ; Hall2021 . Not only can late-time neutron emission
increase the probability of heavy nuclei to fission, but understanding the
competition the between $\beta$-delayed neutron-emission and $\beta$-delayed
fission channels will also inform how much material is mapped to lower mass
numbers in the _r_ process due to $\beta$-delayed fission.

If the theoretical (and experimental) mass and $\beta$-decay predictions are
generous enough to abundantly populate the actinide region during the _r_
process, the heavy nuclei will then have to contend with fission, another
source of uncertainty in nuclear theory. As shown in Fig. 7, there are
significant regions with low barriers to fission, which is intrinsically
related to the fission rate. Such low barrier heights cause fission to be the
dominant transmutation channel, decreasing the probability that a heavy
nucleus will survive as it undergoes $\beta$ decay though the low fission
barrier region Petermann2012 ; Goriely2015b . Low fission barrier heights
beyond $N=184$ inhibit the production of superheavy elements, particularly as
nuclei pass through a region where the probability to undergo $\beta$-delayed
fission is nearly 100% Mumpower2018 . Fission barrier heights—and therefore
the probability to undergo fission—differ between theory predictions. Theories
that predict higher fission barriers and a weaker (or no) $N=184$ shell
closure have an increased probability of populating heavier masses that may
decay to superheavy elements, provided that the barrier heights are
sufficiently high that $\beta$-delayed fission is also suppressed.

Another interesting possibility that would impact the population of actinides
and superheavies is the emergence of an alternative shell closure in this
region of superheavy nuclei. A deformed or spherical shell closure below
$N=184$ would hold material up at lower mass number, likely leading to a
relatively larger population of the lightest actinides. Conversely, the
absence or weakening of the proposed $N=184$ closure paired with another
higher closure could lead to a larger production of the heaviest acitnides or
superheavy elements, assuming fission does not substantially interfere during
the decay back to stability.

Figure 12 shows how differing mass model and fission barrier height
combinations can produce different elemental abundances in an _r_ -process
simulation. To explore the current variation in model predictions, HFB14
barriers were used in combination with UNEDF0 and HFB27 masses, FRLDM barriers
were used for FRDM2012 and WS4 masses and KTUY barriers were used for DZ
masses. At approximately seven hours after an _r_ -process event, a relatively
robust pattern emerges for elements below $Z\sim 98$, whereas above Cf, the
predictions vary substantially. The heaviest elements are produced with the
DZ/KTUY dataset, resulting in a termination around $Z\sim 108$. The HFB model
produces the highest yield of superheavies while the masses with FRLDM
barriers represents a more pessimistic case. According to their primarily
$\beta$-decay population mechanism, the superheavies that would be synthesized
by the _r_ process would naturally be on the neutron-rich side of the chart of
nuclides and hence likely to be more neutron-rich than what has currently been
created in the laboratory.

The select models used here suggest that thorium ($Z=90$), uranium ($Z=92$),
and plutonium ($Z=94$) are among the most populated actinide species around
7.5 hours after an _r_ -process event. Subsequent $\alpha$-decay from heavier
actinides can continue to to contribute to their production on much longer
timescales than that of the neutron capture phase of the _r_ process. Above
californium ($Z=98$) the models begin to substantially diverge in their
production of the heaviest elements. The KTUY fission barrier model, with its
high fission barriers, consistently affords the ability to produce the nuclei
with highest proton number, even with variation in the assumed masses (not
shown). It is also found that HFB models consistently produce the highest
population of superheavies; in the case of Fig. 12, the bulk of the population
resides in rutherfordium ($Z=104$). The diversity of species populated in Fig.
12 will result in a significant variation in the final decay pathways of the
heaviest elements, which in turn influences the potential for observations
that occur on longer timescales.

Figure 12: Total elemental abundances at about seven hours in an _r_ -process
calculation using different nuclear mass models and fission barrier heights.
For the UNEDF0 and HFB27 models, barrier heights from HFB14 were used. Both
WS4 and FRDM2012 use FRLDM fission barrier heights Moller2015 , and DZ uses
KTUY barriers Koura2014 .

Revisiting the values in Fig. 11 with a more optimistic choice like HFB27, the
cumulative probability that 304Cm survives fission to populate 290Ds is about
7.6% without neutron-induced fission, and 1.8% with it active. For a slightly
lower-$Z$ superheavy, say 283Hs, the decay path from ${299}$Pu goes between
regions of low barrier heights, leading to a cumulative survival probability
of 63%. Masses and fission barrier heights thus have a significant effect on
the superheavy elements that can be produced by the _r_ process.

It is important to recognize several points here regarding the simulations
shown in Fig. 12; even in optimistic scenarios (e.g. DZ/KTUY) with significant
superheavy elements created, they are likely to be found in trace amounts
relative to the abundances of lighter nuclei which can be populated in many
more, less extreme, astrophysical conditions. Second, simulations presented
here do not fully estimate all the nuclear uncertainties involved in the
production of the heaviest nuclei. Additional uncertainties, particularly with
$\beta$ half-lives will serve to increase the variations observed in Fig. 12.

Given the orders of magnitude variation found in the production of superheavy
elements with current modeling, extensive work involving measurements and
theoretical calculations are required to resolve this situation. Continued
efforts to understand the fission of heavy neutron-rich nuclei will clarify
the likelihood of heavy nuclei to survive their long $\beta$-decay chains and
eventually populate the superheavy elements Erler2012 ; Ishizuka2022 .
Observational implications of fission will be discussed in Sec. 4.

###  3.4 Sensitivity to astrophysics

Not only is the production of superheavy elements sensitive to the as-of-yet
unknown nuclear physics of neutron-rich nuclei, but it is also sensitive to
the astrophysical conditions responsible for an _r_ -process event.
Temperature, entropy, density, and neutron-richness shape the initial
elemental composition—which nuclei are present when the _r_ process begins—as
well as the reaction rates between nuclei during the _r_ process, in part
determining how far in mass number the _r_ process progresses. For example, at
the same temperature and density, astrophysical conditions with a higher
neutron-richness (more free neutrons available for capture) will generally
produce a stronger _r_ -process, and thus a higher probability of producing
the heaviest elements. The dependence of isotopic production on _r_ -process
dynamics is not unique to the discussion of superheavy elements. For example,
the formation of the rare-earth nuclei has been shown to depend on the
conditions of _r_ process freezeout Surman1997 ; Otsuki2010 ; Arcones2011 ;
Goriely2013 ; Mumpower2016a ; Vassh2019 , and the production of the heaviest
elements has also been thoroughly explored in previous work (e.g., Kratz1993 ;
Qian2007 ; Eichler2016 ; Eichler2019 ).

Figure 13 highlights an example of the dependence of heavy-element production
on two astrophysical parameters in a standard _r_ process: specific entropy
($s$) and initial composition ($Y_{e}$). The trajectories used for the
construction of Fig. 13 derive from the parameterization used in Lippuner2015
, where we have fixed the timescale parameter $\tau=10\ \textrm{ms}$.
Generally, lower $Y_{e}$—and thus more neutron-rich conditions—yield a higher
actinide abundance which is indicated by bright yellow. Similarly, a higher
entropy generally enables even moderately low-$Y_{e}$ conditions to produce
significant actinide material since high-entropy conditions allow more
photodissociation of neutrons from nuclei. Noting the trends in Fig. 13, the
production of the heaviest elements is clearly not straightforwardly
quantifiable as a simple function of astrophysical parameters. In any case,
the greatest chance of observing the superheavy elements would likely result
from an astrophysical environment in which the abundance of the heavy elements
is maximized.

Figure 13: Actinide production as a function of entropy (vertical axis) and
electron fraction (horizontal axis) for parameterized astrophysical conditions
with fixed expansion timescale $\tau=10\ \textrm{ms}.$

Currently, the conditions that favor heavy and superheavy element production
are low-entropy, low-$Y_{e}$ astrophysical sites, generally agreeing with
early studies of superheavy element production in _r_ processes that use
parameterized calculations Ohnishi1972 ; Schramm1973 ; Berlovich1974 ;
Mathews1976 . Modern simulations of binary neutron star mergers robustly
predict low-entropy, low-$Y_{e}$ conditions Goriely2011 ; Korobkin2012 , which
could prove suitable for superheavy-element production. Neutron star mergers
contain several interesting sites possible for _r_ -process nucleosynthesis
including a post-merger accretion disk Surman2006 ; Surman2008  and tidally
stripped ejecta Just2015 . Complex simulations with Monte-Carlo radiation
transport of the accretion disk Miller2019 ; Foucart2020  currently do not
produce significant quantities of the heaviest elements, but this conclusion
depends strongly on uncertainties in neutrino physics which controls the
morphology of the disk McLaughlin1996 ; Surman2005 ; Patton2012 ; Zhu2016 .
Tidal ejecta may be a more favorable location for the production of the
heaviest elements owing to generally lower initial electron fractions and less
time for neutrinos to interact with the material Radice2016 ; Bovard2017 ;
Radice2018a ; Radice2018b ; Hotokezaka2015 . Complete simulations from the
inspiral phase until creation of the post-merger disk will be very valuable in
assessing the capacity of heaviest element synthesis in neutron star mergers.

Other compact object mergers of interest include black-hole neutron-star
(BHNS) mergers which also have the potential to produce the heaviest elements
Nishimura2016 ; Roberts2016 . The study of BHNS binaries offers for
exploration a wide range of physical processes beyond nucleosynthesis,
including the neutron star equation of state, stellar evolution, high-energy
astrophysics, and the expansion of the universe Foucart2020b . However, black-
hole neutron-star mergers may be unlikely multi-messenger sources due to the
requirement of high-spin for an electromagnetic signal which is further
compounded by limitations of current observing technologies Fragione2021 .

Finally, it remains possible for magnetorotational supernovae or collapsars to
reach suitable conditions for the formation of the heaviest elements
MacFadyen1999 ; MacFadyen2001 ; Surman2011 ; Nakamura2015 ; Siegel2019 ;
Miller2020 . Further investigation is warranted on this front, particularly
with the starting configurations of the disk which strongly influences the
resultant nucleosynthesis.

##  4 Observation of the heaviest elements

We have discussed the properties of the heaviest nuclei from a nuclear
perspective and how those properties play into the potential astrophysical
production of these elements. If superheavy elements are made in the universe,
where and how can we observe them, if at all? The definition of ‘observation’
in this context extends from direct measurements through, e.g., atomic mass
spectrometry and stellar spectroscopy, to indirect ones through, e.g., heavy-
element decay products and lightcurves following explosive astrophysical
events. Here we review a brief history of translead observations both in the
Solar system and outside of it and discuss potential observations regarding
the heaviest elements that may be formed in the _r_ process.

###  4.1 Actinides in the Solar system

The existence of heavy elements on Earth and in meteorites gives some
indication of the astrophysical events that produced those elements before
Solar system formation. Studies that seek to explain the last enrichment event
by the _r_ process before the Solar system formed frequently employ not only
the heavy elements, but also their short-lived radioactive decay products
Lugaro2018 ; Wallner2021 ; WangX2021 . Understanding the production of heavy
elements and their decay products can constrain galactic chemical evolution
studies by elucidating which _r_ -process sites dominate at which times in
galactic history Wanajo2006 ; Wehmeyer2015 ; Cote2018 ; Ojima2018 . In any
case, the measurements of translead nuclei created before Solar system
formation is constrained by the half-lives of those elements. Recall from Fig.
4 that all known translead nuclei are unstable, and the half-lives of
superheavy elements generally become shorter with increasing atomic number.
Even if superheavy elements were present at Solar system formation, they
likely would have decayed to less than trace abundances over the 4.6-Gyr age
of the Solar system.

Starting in the early 1960’s, fossil tracks were found in naturally occurring
mica, believed to be made from the fragment from the spontaneous fission of
238U Price1962 . These observations followed a study on artificially
irradiated mica that discovered uranium fission fragments can leave such a
record Silk1959 . Extending this logic, it was believed that fossil tracks
from cosmic rays could be found in cosmically irradiated meteorites, and such
tracks were identified for medium nuclei and heavy nuclei, including 244Pu,
within the decade Maurette1964 ; Fleischer1965 ; Fleischer1967 ;
Fleischer1967b . Extrapolating further still, experimental undertaking were
designed in the specific search for superheavy nuclei in meteorites
Ginzburg2005 ; Aleksandrov2008 . Results from these experiments claim
detections from superheavy nuclei as well as lighter, transuranic species that
could possibly be decay products from the same long-lived superheavies
Bagulya2013 ; Alexeev2016 ; Alexandrov2019 ; Alexandrov2022 .

Figure 14: Observed stellar spectrum (points) in wavelengths centered on the
absorption features of Th (left) and U (right) compared to models (colours)
used to derive the abundance of the element in the star. Dotted lines show
model spectra with no Th and U present. Absorption features of notable
elements and molecules are also labelled.

Inferring the existence of short-lived mother nuclei by measuring the
abundances of their longer-lived daughter nuclei has been applied to other
actinides elements as well. For example, after a series of $\beta$ and
$\alpha$ decays, 247Cm ($T_{1/2}=15.6$ Myr) eventually transmutes into the
much longer-lived 235U ($T_{1/2}=708$ Myr) Stirling2005 . The ratios of long-
lived actinide species like 235U, 238U ($T_{1/2}=4.47$ Gyr), and 232Th
($T_{1/2}=14$ Gyr) are often used to calculate the time between the _r_
-process event that produced them and the time they are observed (i.e.,
present day) through a method known as ‘cosmochronometry’ Dicke1969 ;
Truran1998 ; Cowan1999 ; Goriely1999 ; Schatz2002 ; Wanajo2002 ; Otsuki2003 ;
Brennecka2010 . In the Solar system, ages calculated from the ratio of 235U to
238U have shown some disagreement from those calculated from other ratios,
e.g., 232Th to 238U. Since all actinides are necessarily _r_ -process-only
nuclei, they must all have been made in the same event(s), and variations
between the ages inferred from the ratio point to an inconsistency in
understanding their production rather than variations in the age of the
material itself. To solve this inconsistency, it has been suggested that the
decay of 247Cm, which would have been present in the early Solar system,
causes a residual enhancement to 235U Tatsumoto1980 , thus explaining the
different in ages calculated with different actinide ratios. Indeed, 247Cm
found in meteorites is consistent with a single _r_ -process event that is
responsible for the bulk of actinide material currently found in the Solar
system Francois2016 . A similar conclusion is drawn from comparing the
abundances of actinides to lighter elements, i.e., through the 129I/247Cm
ratio Cote2021 . Further observational evidence of a consistency in the
129I/247Cm ratio would strongly suggest that both lighter _r_ -process nuclei
and the heaviest _r_ -process nuclei are all created in a single event. This
ratio is also able to estimate that the time between the last _r_ -process
event that enriched the pre-Solar gas and condensation of that gas into solids
was about 100–200 Myr Cote2021 . Observing _r_ -process-only nuclei like the
actinides in the Solar system thus provides powerful ways of understanding
their astrophysical production.

###  4.2 Actinides in stars

As mentioned in Sec. 1, atomic absorption features in stellar spectra are
correlated with elements present in the star’s photosphere. Through the
identification and measurement of these absorption features, the elemental
composition may be characterized and quantified in stars. For example, Fig. 14
shows absorption features identified in a stellar spectrum over two narrow
wavelength regions. The continuum of light is normalized to unity, and
significant deviations from unity correspond to atomic transitions of elements
present in the stellar photosphere. Of measured translead nuclei, only three
actinide nuclei—and no superheavies—have half-lives sufficiently long for the
species to be observable today in starlight: 232Th, 235U, and 238U. The most
prominent atomic transitions of Th and U in optical wavelengths are shown in
Fig. 14.

It is important to note that observations through stellar spectroscopy are in
most cases naturally limited to deriving elemental—not isotopic—abundances.
Unless the hyperfine structure of a nucleus alters the absorption feature
within measurable limits (e.g., the detector’s resolving power), the shape and
central wavelengths of atomic absorption features are degenerate with isotopic
mass. Only in rare cases (or extreme conditions) can isotopic abundances be
inferred from stellar spectra Cowley1989 ; Mashonkina1999 ; Roederer2008 . In
the case of the actinides, it is safe to assume that the only isotope of Th
present in stars is 232Th, since the half-life of the next longest-lived Th
nucleus is less than 10,000 years and would have decayed away possibly even
before the star was born. Similarly, for stars sufficiently old, 238U
dominates over the faster-decaying 235U, and the abundance of U derived from
stars can be almost entirely attributed to 238U. As mentioned in the previous
section, this assumption does not hold for young stars like the Sun. Beyond
238U, no heavier element has been identified in stellar spectra, and there are
currently only five stars for which even the abundance of U has been derived
Cowan2002 ; Honda2004 ; Frebel2007 ; Placco2017 ; Holmbeck2018 .

The identification of actinides in stellar spectra (primarily through Th)
ranges over cosmic time and is not limited to the Sun or our Solar system.
Thorium has also been identified in stars similar to the Sun and could play a
role in planet formation and habitability around Sun-like stars Botelho2019 ;
Nimmo2020 . In these ‘Solar twins,’ the abundance of Th roughly agrees with
the Sun. Since the actinides are necessarily produced by _r_ process
conditions, the consistency of finding Th across several stars could be
considered evidence that either (1) one _r_ process site dominates Th
production, or (2) the _r_ process behaves universally.

Figure 15: Thorium abundances ([Th/Fe]) for stars across a variety of
metallicities ([Fe/H]) and stellar populations: very metal-poor (VMP) Galactic
halo stars ($[\rm{Fe/H}]<-2.0$) from the JINAbase compilation and references
therein Abohalima2018 , extra-Galactic systems Aoki2007 ; Ji2018 , the M5
globular cluster Lai2011 , Milky Way thin-disk delPeloso2005 , and Galactic
solar twins Unterborn2015 ; Botelho2019 . The five metal-poor stars that have
a derived abundance of uranium are from Cowan2002 ; Frebel2007 ; Siqueira-
Mello2013 ; Placco2017 ; Holmbeck2018 . Zero on both axes defines the Solar
system value.

The Sun and stars similar to it are young by cosmic standards. The gas that
made the Sun and its planets is the result of the preceding nine billion years
of chemical evolution. The universe started with H, He, trace amounts of Li,
and negligible amounts of anything heavier. The most massive stars—such as the
first generation of stars in the universe was thought to be—continue
subsequent stages of fusion in their cores up to iron. Over time, the nearly
pure H-He composition of the universe became enriched with Fe and other
elements ejected by stars as they evolved. The oldest stars, therefore, are
reasonably assumed to be those with the lowest amount of Fe (low
‘metallicity’), since they were made at a time when the universe was still
chemically primitive.

Not only can actinides be measured in the atmosphere of both our Sun and stars
similar to it, but Th (and sometimes U) can also be found in stars much more
chemically primitive than the Sun. These ‘metal-poor’ stars were made from gas
that had less than 1% of elements with $Z>=3$ than the gas from which the Sun
was made. Given that the universe evolved from a metal-free state to its
current metal content, the low metallicity of these stars indicates that they
are much older than the Sun, possibly even belonging to the second generation
of stars ever made in the Galaxy. Galactic chemical evolution modeling
supports this basic assumption, though the correlation of metallicity with age
becomes less straightforward at early times (e.g., Wise2012, ).

As can be seen by Fig. 2, elements made by the _r_ process are a small
fraction of the total composition of the Solar system. In metal-poor stars,
the abundances of heavy elements ($Z\geq 26$) are reduced even further by a
factor of ten or more in comparison. However, even though their absolute
elemental abundances are low, heavy elements can still be identified in metal-
poor stars Roederer2013 . Figure 15 shows the abundance of Th relative to Fe
(scaled to the Sun) in stars inside and outside the Galaxy spanning a range of
metallicities ([Fe/H]), with the most metal-poor (lowest [Fe/H]) likely being
the oldest stars. Stars with $[\rm{Fe/H}]<-2.0$ have less than 1% of the iron
that the Sun has and are called ‘very metal-poor (VMP).’ The ‘[X/Y]’ notation
indicates the logarithm of the number abundance ratio of element X per Y, then
scaled to the Solar ratio. For example, stars with $[\rm{Th/Fe}]=1.0$ have ten
times the number of thorium atoms per iron atom than the Solar system. The
(currently only) five stars in which U—in addition to Th—is detected Cowan2002
; Frebel2007 ; Siqueira-Mello2013 ; Placco2017 ; Holmbeck2018  are indicated
in Fig. 15 along with two cases in which Th has been identified in stars
outside of the Milky Way (i.e., in the dwarf spheroidal Galaxy, Ursa Minor
Aoki2007 , and in the ultra-faint dwarf galaxy, Reticulum II Ji2018 ). The
presence of the _r_ process elements, including Th, in metal-poor stars is
evidence that the _r_ process has been active even since early times, both in
this galaxy and in others.

Figure 16: Top: elemental abundance patterns of three metal-poor stars with
varying degrees of actinide enhancement (DES J0355$-$5404 Ji2018 , CS
22892-052 Sneden2003 , and 2MASS J0954+5246 Holmbeck2018 ) compared to the
Solar _r_ -process component scaled by a constant. Bottom: differences between
the stellar abundances and the Solar _r_ -process component by element,
normalized to the average difference among the lanthanide elements.

The range spanned by [Th/Fe] in Fig. 15 indicates that either these elements
are not produced in the same _r_ -process site, or not always with the same
ratio every time. If they were, the Th/Fe ratio would be constant over time
(and over metallicities). Instead, the stochasticity of thorium—especially the
high abundances at low metallicities—is evidence that rare, prolific _r_
-process sites were active at early times in the Galaxy. The convergence of
[Th/Fe] towards the Solar value ($[\rm{Th/Fe}]=0$) by Solar metallicity
($[\rm{Th/Fe}]=0$) could be explained by the actinide-producing _r_ -process
site becoming more frequent and less prolific (i.e., a constant production
over time), or simply that the actinide abundances ‘average out’ due to, e.g.,
gas mixing as the Galaxy ages. In any case, sites that produce an abundance of
actinides could possibly have produced non-trivial amounts of the superheavy
elements as well. However, due to the very old ages of stars showing the
highest actinide abundances ($>$10 Gyr), any superheavy elements that were
produced would have all but decayed by the time of their modern observation.
Although these stars in and of themselves offer little hope of observing the
superheavy elements naturally through their starlight, their existence raises
questions about the _r_ process site: are prolific producers of the actinide
elements still active in the Galaxy today? If so, then is there a change of
catching superheavy-element production in the act?

Many other _r_ -process elements in stars can be measured through stellar
spectroscopy. As long as a strong atomic transition occurs within visible
wavelengths, it is possible to identify the corresponding absorption feature
with ground-based telescopes. For more energetic transitions with wavelengths
in the ultraviolet (UV) range, space telescopes with UV capabilities must be
used. Fortunately, about half ($\sim$30) of the _r_ -process elements have
atomic transitions in the visible range which are strong enough to allow
ground-based telescopes to obtain nearly complete elemental abundance patterns
of stars, similar to Fig. 2. Another dozen or so _r_ -process elements have
strong transitions mainly in the UV regime, requiring space-based telescopes
for even more complete abundance patterns. Recently, 42 _r_ -process elements
were measured in the spectrum of a metal-poor star by combining space and
ground-based observational facilities, representing the most complete
abundance pattern for a star outside the Solar system Roederer2022 . Figure 16
shows three metal-poor stars in which their elemental abundances appear in an
_r_ -process pattern, similar to the _r_ -process fraction derived for the
Solar system. The Solar _r_ -process pattern, scaled down to roughly the level
of abundances in the other stars, is shown for comparison. Note again here
that their abundance patterns are necessarily elemental since most atomic
absorption features are degenerate to the isotopic number of the element. In
addition, there are no measured elements with $Z=43$ (Tc), $Z=61$ (Pm), and
$82<Z<90$ (everything between Pb and Th) since all known isotopes of these
elements are radioactive over stellar lifetimes. Elements with $47\lesssim
Z<56$, which includes most of the second _r_ -process peak, and around $Z=80$
do not have strong atomic transitions within the visible range, making their
detection in stellar spectra unfeasible with ground-based telescopes. Some
elements ($46\lesssim Z\leq 52$) can be obtained with UV-capable facilities
(such as the Hubble Space Telescope), which has been accomplished for a
handful of metal-poor stars Cowan2002 ; Cowan2005 ; Roederer2012 ;
Roederer2012b ; Siqueira-Mello2013 ; Peterson2020 .

Despite the tendency of elemental _r_ -process abundances observed in stars to
show a considerable similarities—appearing to suggest a sort of universality
to their production mechanism—they show significant deviations in their
actinide abundance compared to the rest of their _r_ -process elements. One
such deviation is the small subset of stars that demonstrates a strong
enhancement in their actinide abundance. These stars have been termed
actinide-boost stars due to their enhanced levels of Th relative to the
lanthanide elements (e.g., Eu), which otherwise do display remarkable
universality Lai2008 ; Mashonkina2014 ; Eichler2019 ; Holmbeck2019a . Of the
VMP stars only a couple dozen have thorium abundances derived, and about one-
third of those display the actinide-boost signature Hill2017 . Abundance
variations like the actinide boost challenge the notion of ‘ _r_ -process
universality,’ and some studies show that star-to-star actinide variations may
be attributed to different astrophysical conditions within a single
astrophysical _r_ -process site Holmbeck2019b . One such actinide-boost
star—2MASS J0954+5246, the most enhanced yet known Holmbeck2018 —is shown in
Fig. 16. Compare its Th abundance to a star considered ‘actinide normal’
(approximately agreeing with the Solar value after accounting for its age), CS
22892-052 Sneden2003 , and a star that is potentially actinide _deficient_
(much lower than Solar), DES J0335$-$5404 Ji2018 , belonging to the extra-
Galactic dwarf galaxy Reticiulum II.

Besides the important implications for understanding the production of
actinides throughout the universe, this overproduction of actinide phenomenon
may, in principle, be used to explore the possibility of superheavy elements,
as those astrophysical conditions producing the most actinide material could
suggest the possibility that superheavy progenitors along decay series may be
maximally produced as well. Alternatively, it may be that the production of
material of superheavy elements could inhibit the production of actinides. If
nuclei along superheavy decay series exhibit strong spontaneous fission
branchings, material that would otherwise be destined for long-lived actinides
would be effectively rerouted into much lighter elements. In this scenario,
then, the overproduction of the long-lived actinides may be correlated with
the production of fewer superheavy elements. In either case, advances in
nuclear theory and precision measurements will both be crucial for resolving
the decay modes and branching ratios that are critical to fully understand
this relationship.

If superheavy elements were ever measured in stars, they would have to have
half-lives comparably long to the age of the star in order to be detected
through stellar spectroscopy, as they would otherwise decay on observational
timescales. Elements occupying the Island of Stability provide the most
optimistic detection prospects among the superheavies, as they are projected
to be the longest lived. Unfortunately, current estimates of these half-lives
(seconds to years) Karpov2012  are still far too short to support their
observation over the relevant astrophysical timescales. At the present, the
only remaining feasible possibility is to attempt to catch element formation
‘in the act.’ This would involve unearthing unique features in atomic or
nuclear spectra for a nearby event Kasen2013 ; Korobkin2020 ; Misch2021 . For
example, a sufficiently strong $\gamma$-ray signal as part of a larger set of
multi-messenger observations (e.g., gravitational waves), could distinguish
between light and heavy _r_ -process components Timmes2019 . A complementary
approach may also be possible via indirect observation that provides a
distinct signature of the influence of long-lived decay products of the
heaviest elements Wanajo2014 ; Zhu2018 ; Wu2019 ; Zhu2021 ; Barnes2021 ;
Terada2022 .

###  4.3 Fission fragments

Calculations that produce superheavy elements in an _r_ process typically find
that the longest timescales for the populated nuclei are on the order of days
to years Petermann2012 ; Panov2018 . As discussed above and in Sec. 2, these
short half-lives and low fission barriers in the transactinide region reduces
the chances of directly detecting the superheavy elements and/or their lower-
charge progenitor nuclei. However, fission could be harnessed as a way to
infer the existence or production of the superheavy elements. The fission
products of heavy nuclei populate lower mass and charge numbers, and—after
finding their own way back to stability via $\beta$ decay—eventually lead to
an enhanced production of lighter nuclei (around the second _r_ -process peak
$A\sim 130$). Since all but two $Z\leq 84$ elements have at least one stable
isotope, such an overproduction from fission deposition can persist on
indefinite (i.e., observable) timescales. Substantial overproduction of key
elements in the fission product region could therefore be observed in, e.g.,
stars and thus would indicate that fission of heavy or superheavy elements
must have occurred.

The particular elements that would unequivocally be evidence of fission by
(super)heavy nuclei are uncertain. The landscape of fission barrier heights
determines where the bulk of heavy nuclei undergo fission, and this region is
highly uncertain and unconstrained, as discussed in Sec. 2.4.2 Vassh2019 ;
Giuliani2020 . Similarly, the fragment distribution of the extremely neutron-
rich nuclei undergoing fission in the _r_ process is also unknown Mumpower2020
. Theoretical descriptions of fission fragment deposition vary from producing
lower-mass nuclei around $A\sim 105$ to being responsible for the rare-earth
peak at $A\sim 165$ Goriely2013 ; Shibagaki2015 ; Vassh2021 . In any case, if
uncertainties on the fission rates (from the fission barrier heights) and
fission fragment distributions can be reduced, key elements observed in metal-
poor stars can be associated with and even quantify the extent of
overproduction via fission e.g., through network tracing Sprouse2021 .

###  4.4 Beyond uranium

Figure 17: Enhancement to energy released from the $\alpha$ decay and fission
of very heavy and superheavy elements. Standard _r_ process conditions (cold,
very neuron-rich) were run using three nuclear mass models and fission barrier
heights: HFB with HFB-14 barrier heights Arnould2006  (yellow), DZ Duflo1995
with KTUY Koura2014  (green) and FRDM2012 Moller1995 ; Moller2012 ; Moller2016
with FRLDM Moller2015  (pink). The late-time bump around 130 days that appears
in all three cases is attributed to the spontaneous fission of 254Cf.

Although the presence of actinide elements are an indication that the _r_
process can occur naturally, their existence alone is insufficient to
determine the full elemental extent of _r_ -process production. The most
promising avenue towards potentially confirming the production of superheavy
elements in nature is through direct observation of _r_ -process events,
especially in the light that is emitted by radioactive nuclei produced
therein. High-opacity lanthanides and actinides affect the bulk properties of
‘kilonova’ light (see Metzger2019  for a recent review) or by producing an
observable feature in their spectra Watson2019 . Following kilonova and gamma-
ray bursts on short timescales after an _r_ -process producing event could
reveal fingerprints of extremely radioactive nuclei with short half-lives. For
example, it has been found that the radioactive decay of 254Cf
($t_{1/2}\approx 60$ days) can leave a distinguishable feature in the light
curve following a nucleosynthetic event Zhu2018 . The energy released by the
(spontaneous) fission of 254Cf was initially thought to be responsible for
powering the light curves of Type Ia supernovae Burbidge1956 . However, as
supernovae have fallen out of favor as actinide-producing _r_ -process sites,
254Cf has resurfaced as theoretically contributing to kilonova light curves
following, e.g., the merging of a compact-object binary system Wanajo2014 ;
Zhu2018 . Zhu et al. found that the fission of 254Cf produces sufficient
energy to delay the otherwise power-law decay of kilonova lightcurves at about
60–100 days post-merger.

Figure 17 shows the impact of $\alpha$ decay and fission of the heavy elements
on the heating rate from material produced by the _r_ process. This ‘heat’
tracks the energy produced by nuclear reactions and decays (see Sec. 2) and,
through the opacities of atoms constituting the material, bears directly on
the observable lightcurve from such an event. A standard _r_ -process
simulation is used with three nuclear physics combinations that optimistically
and pessimistically produce superheavy elements, according to Fig. 12: DZ,
HFB27, and FRDM2012. As a baseline case, the total energy release as a
function of time is taken and any contribution by $\alpha$ decay and fission
from elements with $Z\geq 89$ is removed, which includes all the actinides and
superheavy elements. The total energy release is then recalculated, this time
including the contribution from all actinides, but no superheavy elements
(e.g., only $Z\leq 103$). Lastly, all elements are allowed to contribute to
the energy release.

To visualize the impact from all $Z\geq 89$ and $Z\geq 104$ elements, we
compare the two latter calculations that have $Z\geq 89$ contribution to the
first calculation that does not include these elements. In Fig. 17, the shaded
regions show the effect that simply including actinides (but no superheavy
elements) has on the total energy release. The solid line shows the effect
when superheavy elements (in addition to the actinides) are included. When any
of the $Z\geq 89$ elements are included, the energy released is increased
relative to the case without those elements, as a consequence of the energetic
$\alpha$ decay and fission of those elements now being allowed to add energy
(heat) to the _r_ -process ejecta. Note that the three mass models
qualitatively agree on when the most significant contribution occurs, but
disagree in their details of relative impact.

The most interesting comparison is between the solid lines and filled regions
in Fig. 17. For the most part, the solid and filled curves are
indistinguishable (especially for FRDM2012), meaning that superheavy elements
are not contributing significantly more than just the actinides themselves. A
revisit of Fig. 12 explains this result; the FRDM2012 mass model most
pessimistically produces the superheavy elements, so their low abundances have
little to no impact on the total energy release. On the other hand, HFB27
produces the highest total abundance of superheavy elements at that time,
though that abundance is concentrated at $Z=104$ (Rf). Notably, the excess
energy provided by the production of superheavies is on the order of a factor
of two. The DZ model shows a lower total superheavy abundance than HFB27, but
a greater extent in charge number, peaking at $Z=108$ (Hs). Consequently,
HFB27 and DZ do show a deviation between the cases with and without superheavy
elements (see around 0.1 and 1 day in Fig. 17), presenting a chance to witness
their natural, astrophysical production. These energy-excess features—whether
from solely the actinides or the superheavy elements—would reveal themselves
in both the bolometric light curves of _r_ -process events and the near-
infrared ($JHK$) bands Wu2019 . Observations of late-time infrared excess in
lightcurves could therefore indicate the presence of actinides and superheavy
elements Zhu2021 . Such observations may be possible with the NIRSpec detector
on the recently launched JWST, which has observational capabilities in the
near-infrared region Ferruit2022 . Long-term follow up of future kilonova
events can be a route to observationally confirming the extent of heavy-
element production by the _r_ process.

Not only would such observations directly confirm the synthesis of actinide
and possibly superheavy material by the astrophysical _r_ process, but they
also have the potential to reveal unknowns of the _r_ process itself. Recent
studies have explored the impact that nuclear physics far from stability has