# Superheavy Elements and Ultradense Matter

Evan LaForge<EMAIL_ADDRESS>Department of Physics, The University of
Arizona, Tucson, AZ, 85721, USA  Will Price<EMAIL_ADDRESS>Department of
Physics, The University of Arizona, Tucson, AZ, 85721, USA  Johann Rafelski
<EMAIL_ADDRESS>Department of Physics, The University of Arizona, Tucson,
AZ, 85721, USA

(June 20, 2023)

###### Abstract

In order to characterize the mass density of superheavy elements, we solve
numerically the relativistic Thomas-Fermi model of an atom. To obtain a range
of mass densities for superheavy matter, this model is supplemented with an
estimation of the number of electrons shared between individual atoms. Based
on our computation, we expect that elements in the island of nuclear stability
around $Z=164$ will populate a mass density range of $36.0-68.4$ g/cm3. We
then extend our method to the study of macroscopic alpha particle nuclear
matter condensate drops.

##  I Introduction

We explore properties of superheavy elements with nuclear charge $Z\geq 104$
in a simplified model approach. Our primary objective is to improve the
systematic understanding of the mass density of such elements. Prior work
relied on the Hartree-Fock method Fricke1971 , which provides a detailed
insight into the microscopic atomic structure. We instead use the relativistic
Thomas-Fermi model (RTFM) Muller:1974fh , which allows us to significantly
enlarge the range of atomic charge $Z$ and provide a more comprehensive view
of superheavy elemental properties.

###  I.1 Ultradense Matter

Our study of mass density is motivated by the possibility of Compact
Ultradense Objects (CUDOs) Rafelski:2013 ; Dietl:2012 , which we define here
as objects, typically astronomical bodies, which have a mass density greater
than that of Osmium (22.59 g/cm3 Lide:2005 ), the element with the largest
experimentally measured mass density. In particular, some observed asteroids
surpass this mass density threshold. Especially noteworthy is the asteroid 33
Polyhymnia. Based on mass and size measurements obtained from independent
sources, its mass density is computed as CARRY201298

$\rho=75.3\pm 9.6\,\text{g/cm${}^{3}$}\,.$ (1)

Since the mass density of asteroid 33 Polyhymnia is far greater than the
maximum mass density of familiar atomic matter, it can be classified as a CUDO
with an unknown composition.

In this paper we suggest and explore the notion that CUDO matter could be
composed of superheavy elements beyond the known periodic table. Islands of
nuclear stability are thought to exist for high-$Z$ elements, particularly
around $Z=164$ Grumann:1969 . We hypothesize that these elements could exist
in the cores of dense asteroids like 33 Polyhymnia. We refer to Ref.
CARRY201298  for a full listing of asteroid mass densities. There are many
more entries with anomalous CUDO density values, albeit with larger margins of
error and thus reduced statistical significance for individual objects.
However, considered as a group, one could argue that there is strong evidence
that the case of asteroid 33 Polyhymnia is not singular.

The mass density of any element is dependent on atomic mass and the spacing
between atoms. The atomic mass is mostly localized in the atomic nucleus; the
electrons and their binding energy have a negligible effect on the atomic
mass. Regarding nuclear mass we note that as $Z$ increases, the number of
nucleons increases at a faster than linear rate: light elements have roughly
the same number of neutrons and protons, but heavy elements have roughly 1.5
times as many neutrons as protons. This increase originates in the need to
reduce the mutual Coulomb repulsion between $Z$ protons in the atomic nucleus.
On the other hand, as $Z$ increases, the strength of the Coulomb force on the
electrons, as well as the increased degeneracy in high angular momentum
states, prevents the distance between atoms from growing nearly as fast as the
mass. Therefore, on average, the density of elements grows with $Z$. Thus, if
matter is stable for particularly large $Z$ values, it is plausible that such
elements are dense enough to explain the composition of ultradense asteroids.

In this paper, we explore two examples of superheavy matter: we consider
superheavy (high-$Z$) elements, and alpha matter, in which nuclear matter is
composed of a gas of interacting alpha particles Clark:2023sin .

###  I.2 Overview

The RTFM, which is discussed in section II, approximates the quantum structure
of an atom as a fixed nuclear charge distribution surrounded by a Fermi
electron cloud. The ionic size is naturally defined at the radius at which the
RTFM electron density vanishes. At this radius, the nuclear charge will not be
entirely neutralized by the electron cloud, resulting in a slightly ionized
atom (an ion) with a net positive charge. This ion can be thought of as an
atomic “core,” with all remaining unscreened electrons appearing in the
conduction band.

Defining the radius of such an atomic core as $R_{\mathrm{core}}$, we know
that the potential outside the core takes the form of a Coulomb potential

$eV(r)=\frac{-\gamma\alpha\hbar c}{r},\;r\geq R_{\mathrm{core}}\,,$ (2)

where $e$ is the charge of an electron, $V$ is the combined electric potential
of the nuclear charge and electron cloud, $\alpha$ is the fine structure
constant, and $\gamma$ is the the net charge of the core ion.

Based on experimentally measured mass densities, we show numerically that
within a predefined class of metals, $\gamma$ scales approximately linearly
with $Z$. We consider two classes of metals: typical metals, defined as having
similar mass densities to other metals near them on the periodic table, and
heavy metals, defined as having the highest mass density of any element in
their respective period. It should be noted that most metals are at least as
dense as typical metals and thus have a mass density in between that of
typical and heavy metals with similar atomic number. There are some
exceptions, such as Europium ($Z=63$) and Ytterbium ($Z=70$) Lide:2005 , which
are less dense than typical metals, though these exceptions are rare.

Figure 1 depicts the relationship between $\gamma$ and $Z$ for both considered
classes of metals. For each value of $Z$ marked on the plot, the value of
$\gamma$ was chosen by determining what value yielded the experimentally
measured density when used as an input for the RTFM. We can then extrapolate
$\gamma$ to higher $Z$ to be used as an input to the RTFM equations.

It should be noted that $\gamma$ was not restricted to integer values. This
allows for $\gamma$ to become a fudge parameter, compensating for the
systematic error, such as the use of spherical ions as a proxy for crystalline
structure elements. In section III, we will use this extrapolation to
determine the range of mass densities that superheavy elements are likely to
fall within for a few values of $Z$, including $Z=164$.

Figure 1: The number of electrons shared in the conduction band, $\gamma$, as
function of the atomic number, $Z$, of a given element obtained according to
the relativistic Thomas-Fermi model: Nearly all metals will have a $\gamma$
value within the shaded area, see text for more explanation.

We will also explore alpha matter using the RTFM in section IV. Unlike normal
matter, significant screening of the Coulomb repulsion within a very large
nucleus of alpha particles can occur by the electrons “diving” into the
nucleus and potentially forming more alpha particles. This would allow for
CUDO systems with much higher values of $Z$ to remain stable. We will use as
an example $Z=10^{5}$. The relative simplicity of the RTFM makes this value of
$Z$ only slightly more computationally intensive than that of standard matter.
However, using the more precise Hartree-Fock method would require calculating
on the order of $10^{5}$ self-consistent orbitals, making it computationally
challenging even today.

##  II The Relativistic Thomas-Fermi Model of the Atom

The RTFM atom consists of a fixed, spherically symmetric nuclear charge
distribution surrounded by an electron cloud. The density of that cloud is
determined by prescribing a relativistic Fermi energy $E_{f}$. The
relativistic Fermi momentum is given by

$p_{f}^{2}(r)c^{2}=[(E_{f}-eV(r))^{2}-(m_{e}c^{2})^{2}]\Theta(E_{f}-eV(r)-m_{e}c^{2})\,,$
(3)

where $\Theta$ is the Heaviside function. Note that at a certain radius, the
right side of Eq. (3) vanishes. This defines the edge of the ion,
$R_{\mathrm{core}}$, from the following condition

$eV(R_{\mathrm{core}})\equiv E_{f}-m_{e}c^{2}\;,\qquad E_{f}-m_{e}c^{2}\equiv
E_{f,\mathrm{NR}}\;,$ (4)

where $E_{f,\mathrm{NR}}$ is the non-relativistic Fermi energy.

The usual relation between the number density of electrons and the Fermi
momentum also applies in the relativistic model as follows Landau:1980 :

$\displaystyle\rho_{e}(r)$
$\displaystyle=\frac{p_{f}^{3}(r)c^{3}}{3\pi^{2}\hbar^{3}c^{3}}\,,$ (5)
$\displaystyle=\frac{[(E_{f}-eV(r))^{2}-(m_{e}c^{2})^{2}]^{3/2}}{3\pi^{2}\hbar^{3}c^{3}}\,.$
(6)

Note that writing $E_{f}$ in terms of $E_{f,\mathrm{NR}}$ in Eq. (6), when
$|eV|<<m_{e}c^{2}$ and $E_{f,\mathrm{NR}}<<m_{e}c^{2}$, the conventional non-
relativistic Thomas-Fermi model arises.

The edge of the ion defined in Eq. (4) provides a natural method for
calculation of mass density, which only requires determination of the self-
consistent potential associated with the prescribed nuclear structure and
Fermi energy. The size of the atom is obtained from the radius
$R_{\mathrm{core}}$, while the mass will be sourced in the atomic nuclei.

The potential of the atom is determined by solving the Poisson equation for
the electric potential $V(r)$ with the density of electrons in the Fermi
surface Eq. (6) as the source term. In the following equation, the proton
density $\rho_{P}(r)$ and the total density $\rho_{T}(r)$ are number
densities, not charge densities. The Poisson equation then reads

$-\Delta
V(r)=\frac{|e|\rho_{T}(r)}{\epsilon_{0}}=\frac{-e\rho_{P}(r)+e\rho_{e}(r)}{\epsilon_{0}}\,.$
(7)

The proton density $\rho_{P}(r)$ will be established in the following
subsection.

###  II.1 Nuclear Charge Distribution

In order to determine the potential, the nuclear charge density (i.e. proton
density) must be specified. This was chosen to be a Fermi distribution
normalized to have $Z$ charges, as is standard. This also allows for our
computation to be directly compared to and validated by earlier studies which
used the RTFM Rafelski:1976ts . The proton density is

$\displaystyle\rho_{P}(r)$
$\displaystyle=\frac{C}{1+\text{exp}\left[\frac{4\log(3)(r-R_{N})}{t}\right]}$
(8) $\displaystyle t$ $\displaystyle=2.5\,\mathrm{fm}$ (9) $\displaystyle
R_{N}$ $\displaystyle=1.2A^{1/3}$ (10)

where $t$ is the shell thickness, $R_{N}$ is the nuclear radius, and $C$ is a
normalization constant defined by the integral

$\int_{0}^{\infty}4\pi r^{2}\rho_{P}(r)\text{d}r=Z\,,$ (11)

which is calculated numerically. The number of nucleons $A$ in the nucleus is
assumed to be

$A=2.5Z\,.$ (12)

This assumption provides a good approximation for superheavy elements. It will
be altered when we consider alpha matter. With these assumptions in place, the
nuclear distribution is entirely determined by the parameter $Z$. Note that
the choice of $R_{N}$ Eq. (10) fixes the density at the center of the nucleus
to be roughly one nucleon for every $\sim 7.2$ fm3.

###  II.2 Fermi Energy

The key RTFM parameter required is the Fermi energy $E_{f}$, which is defined
as the energy up to which electron states are filled. The value of $E_{f}$
determines the ion-electron density. As determined by Müller and Rafelski
Muller:1974fh , in the domain where the spontaneous pair production is
energetically prohibited and stable atomic structure arises, the Fermi energy
$E_{f}$ obeys the following inequality

$-m_{e}c^{2}\leq E_{f}\leq m_{e}c^{2}\,.$ (13)

The case $E_{f}=m_{e}c^{2}$ corresponds to a neutral atom while
$E_{f}<m_{e}c^{2}$ corresponds to a cation. If the Fermi energy is greater
than $m_{e}c^{2}$, then the binding energy must be positive, which is
impossible. If the Fermi energy is less than $-m_{e}c^{2}$, then the binding
energy of the least bound electron will be greater in magnitude than
$2m_{e}c^{2}$, meaning there is enough energy in the system to produce
electron-positron pairs; when pair production occurs, additional electrons
will be spontaneously added to the system and their associated positrons will
be ejected.

We are most concerned with the densest of superheavy elements, so we will
consider metals with a compact, crystalline structure. Within this structure,
atoms are tightly packed together and share a few electrons in the conduction
band. The nucleus, as well as the unshared electrons outside of the conduction
band, can be thought of as an atomic core. Effectively, the core is an ion
that is neutralized by the electrons in the conduction band.

We will approximate the volume of a single atom within a metal as a sphere
with radius $R_{\mathrm{core}}$, which will enable us to calculate the mass
density.

###  II.3 Boundary Conditions

We will need to supplement the RTFM differential equation Eq. (7) with
boundary conditions in order to specify a unique solution. In particular, we
must know the potential at the center of the nucleus $V(r=0)$ as well as its
derivative. At the origin the electric field vanishes, so that the derivative
of the potential is

$\frac{\partial V}{\partial r}\bigg{|}_{r=0}=0\,.$ (14)

The value of $V(0)$ must be determined in a way that is self-consistent with
the two parameters $Z$ and $E_{f}$. In the absence of an analytic formula
giving $V(0)$ as a function of these parameters, we will instead use the known
asymptotic form of the potential for large $r$; outside the core, there is no
charge so the solution takes the form of a Coulomb potential with the
unscreened charge, $\gamma$, as shown in Eq. (2). Recall that $\gamma$ refers
to the net charge of the core, which is dependent on $Z$ and $E_{f}$. When
considering a metal in a crystalline structure, $\gamma$ can be thought of as
the number of electrons in the conduction band.

We use a shooting method Press:2007  to numerically determine $V(0)$. There is
a single value of $V(0)$ that makes the solution converge for large $r$, while
all other values will result in a divergent solution. When the magnitude of
$V(0)$ is too high, we find that the solution for $V$ will diverge above the
expected $1/r$ form and approach infinity. When the magnitude of $V(0)$ is too
low, the solution will change signs, which is impossible in a bound system.
Thus, for any values of $E_{f}$ and $Z$, $V(0)$ can be found through
bisection.

With this procedure in mind, we can determine the mass of the atom using
$A\,m_{N}$, where $m_{N}$ is the mass of a bound nucleon and $A$ is the
nucleon number, for simplicity we set $m_{N}\simeq m_{p}$ where $m_{p}$ is the
proton mass. Other effects of nuclear and electron binding, such as the
difference in mass between protons and neutronsthe or the mass of electrons,
have been neglected. The volume of the atom is obtained using
$R_{\mathrm{core}}$ as its radius, which is dependent on both $Z$ and $E_{f}$.
Because $\gamma$ is dependent on $Z$ and $E_{f}$, we can instead fix $Z$ and
$\gamma$ as our two parameters. This allows the calculation of the mass
density of superheavy elements based solely on the nuclear charge and the
number of shared electrons in the conduction band.

###  II.4 Applying the Method

We developed a C-program that solves the relevant differential equation Eq.
(7) using a fifth order predictor corrector method for prescribed values of
the parameters $Z$ and $\gamma$. In order to test the method, we considered in
Figure 2 the results of this method applied to Radon ($Z=86$), which were
obtained using $\gamma=2$. Radon was chosen because it is a noble gas with
relatively high $Z$, so the Fermi surface electron cloud approximation of the
RTFM is reasonable.

Figure 2: The potential energy of an electron, $eV$, as a function of the
distance from the origin, $r$, for an RTFM atomic cation with $Z=86$ and
$\gamma=2$. The nuclear and RTFM core radii have been marked with vertical
lines.

To visualize the asymptotic behavior of the solution, Figure 3 depicts the
same numerical solution with $-\frac{reV}{\alpha\hbar c}$ instead plotted on
the vertical axis. All values were made positive to enable log scaling of the
vertical axis. In order to illustrate the convergence of this solution at the
edge of the atomic core, the graph depicts the potential outside the atomic
core $eV(r)=\frac{-\gamma\alpha\hbar c}{r}$, which, due to the scaling,
appears as a flat line at $+\gamma$.

Figure 3: This figure depicts the same data as Figure 2, with the y-axis non-
dimensionalized to $-reV/\alpha\hbar c$ and log-scaled. The asymptotic
potential outside the atomic core is also depicted and appears as a flat line
at $+\gamma$.

To verify the accuracy of the numerical method, we fed the RTFM potential into
a numerical Schroedinger equation solver to determine if the potential for
Radon ($Z=86$) yielded the expected behavior. More specifically, we verified
that the various electron shells were filled in the proper order according to
NIST_ASD .

Additionally, we verified that there was a proportionately large energy gap
between the least bound filled state and the most bound unfilled state,
corresponding to high chemical stability. The least bound filled state was the
6p state with a binding energy of $7-8$ eV. The most bound unfilled state was
the 7s state with a binding energy of $3-4$ eV. This is roughly half of the
binding energy of the least bound filled state.

The experimentally measured value of the lowest ionization energy is 10.7 eV
NIST_ASD . This is larger than the numerical result by about $25$%, indicating
the relative accuracy of the RTFM to describe properties of the noble gas
radon: the qualitative features of the atomic structure is properly described
by the self consistent RTFM potential.

##  III Mass density of superheavy elements

Given $Z$, we will determine what unique value of $\gamma$ yields the correct
mass density for a few elements with similar chemical properties. From these
values of $\gamma$, we will linearly extrapolate a range of values of $\gamma$
for metals with a given value of $Z$. More specifically, we will consider
typical metals and heavy metals. Typical metals are defined by having similar
mass density to elements with similar atomic number, while heavy metals are
defined as being the densest elements of their respective periods within the
periodic table.

For typical metals, we chose Lanthanum ($Z=57$), Erbium ($Z=68$), and
Californium ($Z=98$). For heavy metals, we chose Copper ($Z=29$), Rhodium
($Z=45$), and Osmium ($Z=76$). All of these elements are marked on Figure 4.
Note the humps in the graph produced by the varying chemical properties
throughout individual periods on the periodic table.

Figure 4: This figure depicts the density in g/cm3 for all elements up through
$Z=99$, the limit of what has been experimentally measured.Lide:2005  Labeled
on the graph are the typical metals, marked with blue squares, and the heavy
metals, marked with red triangles, that were used for finding the baseline
$\gamma$ values. The density values for all gasses were recorded at standard
temperature and pressure.

With the elements of known mass density chosen, the next step is determining
the appropriate values of $\gamma$ that yield these mass densities from the
RTFM. This was done via bisection similar to the determination of $V(0)$. For
each value of $Z$, a test value of $\gamma$ was chosen. For each test value,
the potential was calculated and then used to determine $R_{\mathrm{core}}$.
The mass density was then calculated using the equation below

$\rho_{\mathrm{mass}}=\frac{Am_{p}}{\frac{4}{3}\pi R_{\mathrm{core}}^{3}}\,,$
(15)

where $m_{p}=1.67\times 10^{-24}$ g is the mass of the proton. This assumes
that the volume occupied by one atom is equal to a sphere with the size of
$R_{\mathrm{core}}$. Deviations from this assumption are compensated for by
allowing $\gamma$ to take non-integer values.

Once the mass density was calculated, the test value of $\gamma$ was increased
or decreased in order to better match the mass density until precision up to
1% was achieved. Table 1 and Figure 5 depict the values of $\gamma$ for all
six considered input elements used in the linear extrapolation of $\gamma$ up
to $Z=180$. $\gamma$ ranges roughly from $2$ to $9$ in the range of $Z$ for
which the RTFM is valid ($Z\gtrsim 25$).

$Z$ | 57 | 68 | 98
---|---|---|---
$\rho_{\mathrm{mass}}$(g/cm3) | 6.15 | 9.07 | 15.1
$\gamma$ | 2.48 | 3.06 | 3.70

$Z$ | 29 | 45 | 76
---|---|---|---
$\rho_{\mathrm{mass}}$(g/cm3) | 8.96 | 12.4 | 22.59
$\gamma$ | 3.54 | 4.04 | 5.28

Table 1: For three input elements with value of $Z$ we show: The experimental
mass densities Lide:2005  $\rho_{\mathrm{mass}}$(g/cm3) and our calculated
$\gamma$-values. The top table section corresponds to typical metals, and the
bottom table section corresponds to the heavy metals, see text. Figure 5: The
number of electrons shared in the conduction band, $\gamma$, as function of
the atomic number, $Z$, of a given element obtained according to the
relativistic Thomas-Fermi model: nearly all metals will have a $\gamma$-value
within the shaded area, compare figure 1, here showing at $Z=164$ the
intercepts with our linear extrapolation marked to highlight the range of
$\gamma$ for that value of $Z$.

Using the extrapolation of $\gamma$, we determine the range of $\gamma$ values
that should appear near $Z=164$. For each $\gamma$ value, the RTFM is used to
obtain the radius of the atomic cores, which enables calculation of the mass
density of the metal. Because mass density increases monotonically with
$\gamma$ (because the core size decreases as $\gamma$ increases), the actual
mass density of superheavy element $Z=164$ is found between the mass densities
calculated using these two extrapolated $\gamma$ values.

The extrapolations for typical and heavy metals respectively yield the
equations

$\displaystyle\gamma_{\mathrm{typical}}$ $\displaystyle=0.0279Z+1.01$ (16)
$\displaystyle\gamma_{\mathrm{heavy}}$ $\displaystyle=0.0373Z+2.42$ (17)

as shown in Figure 5. We are encouraged that for the lower range of $Z$ values
for which the RTFM is valid ($Z\gtrsim 25$), $1.7<\gamma<3.4$, which is a
reasonable range for the number of electrons in the conduction band for these
metals.

We see that $\gamma_{\mathrm{typical}}=5.6$ and $\gamma_{\mathrm{heavy}}=8.5$
for $Z=164$. The RTFM solution then gives us the typical and heavy metal core
radii $R_{\mathrm{core}}=1.656,\,1.338$ Å, respectively. From Eq. (15) we then
predict that $Z=164$ will have a density between $36.0$ and $68.4$ g/cm3, see
Figure 6. These predictions align with results of Fricke Fricke1971 , who
predicted that $Z=164$ would have a density of $46$ g/cm3.

Figure 6: The predicted mass density bounds for superheavy elements in domains
of atomic number $Z=$ 114, 140, and 164 (round (green) dots; input (triangles
(red) and squares (blue); shown as function of atomic number $Z$. The vertical
line indicates the predicted ‘island of stability’ at/near to $Z=164$; the
region $Z=114$ is already discovered. It is unlikely that $Z=140$ yields a
stable nucleus, but this value of $Z$ makes the shape of our result more clear
compared to dashed lines presenting a naive linear extrapolation.

Note that our predicted mass density range will work not only for $Z=164$ but
for elements with similar atomic number as well. This is because increasing or
decreasing $Z$ by small amounts will make very little difference on the mass
density range. Thus, all elements near $Z=164$ will have similar mass density
ranges, and if there is a broad enough island of nuclear stability, then atoms
will likely populate the range of mass densities from $36.0$ to $68.4$ g/cm3.

We applied our method to determine a rough mass density range for any metals
with $Z$ between $110$ and $118$, which elements have been discovered, though
their mass densities are yet to be experimentally measured. A prior study of
elements ranging from $Z=103$ to $Z=111$ using density functional methods
predicts a range of mass densities of $14.4-27.3$ g/cm3 Gyanchandani:2011 .
The mass densities of elements $113$ to $116$ were predicted by Fricke to be
between $13.6$ and $15.1$ g/cm3 Fricke1971 .

Because $Z=114$ is the midpoint of the range between $Z=110$ and $Z=118$,
these elements should have mass densities within the bounds obtained for
element $Z=114$. Our extrapolation yields the range $4.2<\gamma<6.7$ for
$Z=114$. From these values, the lower bound on mass density is $19.3$ g/cm3,
and the upper bound is $39.9$ g/cm3, see Figure 6. Our result is near to the
prior predictions quoted above but allows a slightly larger upper-range. It is
clear that these relatively recently discovered elements could have
surprisingly high mass density. However, they neither have a high enough mass
density to explain CUDOs like asteroid-33 Polyhymnia, nor the stability that
is required.

It is worth considering whether we could consider a naive linear fit of mass
density and avoid numerical calculations, see dashed lines in Figure 6.
However, because mass density is the product of both nuclear and atomic
properties, it is unlikely that its form will be as simple as a linear $Z$
dependence originating primarily in the change of nuclear mass. We believe
that an understanding of the number of shared electrons, $\gamma$, and
evaluation of the cationic size which incorporates an understanding of atomic
structure is necessary to obtain mass density of unknown superheavy elements.

##  IV Alpha matter

Much of the work in the previous section was built on the assumption of
nuclear stability of superheavy elements and there is obvious interest in
$Z>>164$ to obtain CUDOs of greater mass density. However, given that the
Coulomb energy increases rapidly with $Z$, islands of nuclear stability are
unlikely to occur beyond $Z=164$, and it is unclear whether $Z=164$ will be
stable enough to explain the mass density of asteroid 33 Polyhymnia. However,
a different high-density nuclear structure could lead to stable ultradense
matter.

As an example, we explore alpha matter: nuclear matter composed of alpha
particles in a Bose-Einstein condensate-like configuration. This configuration
could be much less tightly packed than standard nuclear matter. For a recent
review of this nuclear structure, see Clark and Krotscheck Clark:2023sin . For
large enough $Z$, the proton density can be much lower while still maintaining
nuclear attraction while the electron cloud is able to nearly overlap with the
proton distribution to roughly counteract the Coulomb force responsible for
nuclear instability. While this section focuses on condensates of alpha
particles for simplicity and conciseness, similar matter could form using
carbon-12 nuclei, oxygen-16 nuclei, or any other particularly stable bosonic
nuclei.

To model the large, loosely packed cloud of alphas, we will use the Gaussian
charge distribution

$\rho_{P}(r)=\rho_{P}(0)e^{\frac{-r^{2}}{2R_{N}^{2}}}\;,$ (18)

where $R_{N}$ is some arbitrary nuclear radius and $\rho_{P}(0)$ is defined
from the constraint Eq. (11). This radius is more of a characteristic size
than an actual approximation of the size of the nucleus. A better description
of the nuclear charge range is about $3R_{N}$, because about 97% of the
nuclear charge is contained within that radius. As example we consider case
$Z=10^{5}$. We will choose $\gamma=250$ because this large value of $Z=10^{5}$
is well beyond the range of the extrapolation from the previous section. This
approach should provide a reasonable lower bound for the mass density of these
forms of alpha matter.

To explore the atomic properties of this form of nuclear matter broadly, we
will consider a wide variety of characteristic nuclear radii $R_{N}$. In Table
2 the considered characteristic radii are depicted and central charge density
(left two columns) is obtained for $Z=10^{5}$, assuming $\gamma=250$. Table 2
further lists the central potential energy $eV(0)$, the Fermi momentum at the
nuclear radius $p_{f}{R_{N}}$, and the mass density $\rho_{\mathrm{m}}$ of the
corresponding alpha matter for the considered values of $R_{N}$. We note that
at the smallest nuclear radius, given that the potential at the origin exceeds
twice the muon mass $2m_{\mu}=212$ MeV/$c^{2}$, an additional charge
distribution must be considered to accommodate for a few muons. However, this
effect is not significant here.

$R_{N}$[fm] |  $\rho_{P}(0)$[fm-3] |  $eV(0)$[MeV] |  $p_{f}|_{R_{N}}$[MeV/c] |  $\rho_{\mathrm{m}}$[g/cm3]
---|---|---|---|---
50 | $5.08\times 10^{-2}$ | -218 | 183 | 2240000
100 | $6.35\times 10^{-3}$ | -109 | 91.6 | 1820000
1000 | $6.35\times 10^{-6}$ | -10.4 | 9.17 | 712000
10000 | $6.35\times 10^{-9}$ | -0.702 | 0.924 | 101000
20000 | $7.94\times 10^{-10}$ | -0.244 | 0.466 | 36000
50000 | $5.08\times 10^{-11}$ | -0.0480 | 0.189 | 5350
Table 2: For $Z=10^{5}$ and chosen characteristic Gaussian nuclear radius
$R_{N}$ we show from left to right: potential energy at the origin $eV(0)$,
Fermi momentum at the nuclear radius $p_{f}|_{R_{N}}$, and mass density
$\rho_{\mathrm{m}}$.

We note in Table 2 that between the three largest radii $R_{N}$, the Fermi
momentum dips below 0.511 MeV/$c$, marking the onset of the transition to a
non-relativistic electron distribution: In the non-relativistic limit, every
alpha has two electrons bound to it, forming an atomic condensate that will
not be considered in this work. However, in systems where the alpha particles
are nearer to standard nuclear density, the electrons can be shared across all
of the alphas, which we do consider in this study.

It should be noted that the nuclear radius of 50 fm (densest option listed in
Table 2) still has a $\sim 30\%$ lower nuclear density at the origin than
standard nuclear matter. For $Z=10^{5}$ the size of the nuclear distribution
$3R_{N}=150$ fm is less than half of the Compton wavelength of an electron.
Therefore, the electron cloud is not capable of fully counteracting the
nuclear charge density and cannot fully eliminate the Coulomb repulsion
between alpha particles; our numerical results indicate that only $90\%$ of
the charge could be screened near the origin. Therefore the resulting
potential well near the origin is fairly deep.

Figure 7 characterizes this situation more fully: We show the proportion of
proton density at the origin that is not neutralized by electrons. For smaller
radii, the proportion of charge not neutralized by the electron distribution
is roughly 10%, which still corresponds to a significant amount of
cancellation. For higher radii, this proportion decreases significantly as the
two distributions overlap nearly perfectly. This results in the Coulomb
repulsion decreasing significantly as well, indicating the importance of
further studying the nuclear alpha matter’s stability with large $Z$.

Figure 7: The proportion of nuclear charge at the origin not neutralized by
electrons, defined as 1 - $\rho_{e}/\rho_{P}$, as a function of alpha-matter
Gaussian size $R_{N}$.

##  V Discussion

The purpose of this study was to determine whether CUDOs with extreme mass
density could be achieved without the need for the usually invoked strange or
dark matter. We have done this while exploring two different nuclear systems
using the relativistic Thomas-Fermi model. From the exploration of both
standard nuclei and alpha matter, it is clear that both types of nuclear
matter could explain the density seen in CUDOs such as asteroid-33 Polyhymnia.

Considering elements between $Z=110$ and $Z=118$, we predict that their mass
densities are less than $40$ g/cm3. It is therefore unlikely that these
elements have a large enough mass density to explain the extraordinary mass
density of asteroid-33 Polyhymnia; moreover, these elements are already known
to lack the required stability. However, elements in the other theoretical
island of nuclear stability near $Z=164$, which we predict to populate mass
density values between $36.0$ and $68.4$ g/cm3, are reasonable candidates. If
some significant part of the asteroid were made of these superheavy metals, it
is plausible that the higher mass density could be near the experimentally
measured value listed in Eq. (1). If there are other superheavy elements well
beyond $Z=164$, these could have an even higher mass density, further
increasing the chances that our stated hypothesis of CUDOs containing
superheavy elemental matter is correct.

We also applied the RTFM to alpha matter, which is another nuclear form of
matter capable of explaining the large mass density observed in CUDOs. The
forms of alpha matter described in Table 2 for $Z=10^{5}$ have mass densities
of thousands or even millions of g/cm3. This means that even a relatively
small amount of alpha matter present in CUDOs like 33 Polyhymnia could result
in the large observed mass density. These results, while preliminary, show the
wide array of exploration that can be done using the computational efficiency
of the RTFM. in such large alpha-nuclear matter drops there is nearly full
cancelations of nuclear Coulomb repulsion.

There are a number of ways in which this work could be expanded in the future.
The nuclear distribution was kept fixed, though electrons intersecting the
nucleus would have likely resulted in alterations to the nuclear distribution
and are capable of enhancing the stability of high $Z$ nuclei. Calculation of
the nuclear distribution in parallel with the electron distribution would be
required for this improvement, which could be accessible in the realm of the
Walecka model of high-$Z$ nuclei Walecka:1974qa , or using and extrapolating
methods addressing the alpha-matter nuclei Clark:2023sin .

Joint computation of atomic and nuclear structure is particularly needed for
alpha matter due to the significant overlap between the nuclear charge
distribution and the electron cloud. In addition, the electron mass in this
model was assumed to be fixed, but using an effective mass would make more
sense when such large binding energies are being considered. The exploration
of atomic structure of alpha matter could also be more in depth, with more
detailed models developed to describe their properties with more precision
than the calculations made in this paper.

Our study has demonstrated that the relativistic Thomas-Fermi model of the
atom is a useful tool. Because of its broad applicability and relatively small
computation time, in particular for high $Z$, it should not be overlooked in
applications to the atomic properties of CUDO matter.

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