# Decay of superheavy nuclei based on the random forest algorithm

Boshuai Cai  Sino-French Institute of Nuclear Engineering and Technology, Sun
Yat-sen University, Zhuhai, 519082  Cenxi Yuan<EMAIL_ADDRESS>Sino-
French Institute of Nuclear Engineering and Technology, Sun Yat-sen
University, Zhuhai, 519082

###### Abstract

How nuclides decay in the superheavy region is key information for
investigating new elements beyond oganesson and the island of stability. The
Random Forest algorithm is applied to study the competition between different
decay modes in the superheavy region, including $\alpha$ decay, $\beta^{-}$
decay, $\beta^{+}$ decay, electron capture and spontaneous fission. The
observed half-lives and dominant decay mode are well reproduced. The dominant
decay mode of 96.9 % nuclei beyond 212Po is correctly described. $\alpha$
decay is predicted to be the dominant decay mode for isotopes in new elements
$Z=119-122$, except for spontaneous fission in some even-even ones because of
the odd-even staggering effect. The predicted half-lives show the existence of
a long-lived spontaneous fission island at the southwest of 298Fl caused by
the competition of nuclear deformation and Coulomb repulsion. More
understanding of spontaneous fission, especially beyond 286Fl, is crucial to
search for new elements and the island of stability.

##  I Introduction

The limit of nuclear landscape [1, 2] is always an intriguing subject. Exotic
properties of nuclei are found around the boundary of nuclear limits, e.g.,
the shell evolution [3, 4, 5, 6], the 4$n$ resonant state [7, 8], the 4$p$
unbound state [9], etc. The discovery of new elements (nuclides) generally
faces three problems: production, separation and identification [10]. Because
the nucleus is unstable, generally with a rather short half-life, one has to
utilize some probes. One of the most direct is the decay mode [10, 11], using
the decay products as the signal of existence. Thus, it is important to
investigate and predict the dominant decay mode of those unknown nuclides. The
nuclear binding energy and the half-life are key data for understanding the
decay mode of an atomic nucleus. The former measures the stability of nuclides
through energy criteria, and the latter describes the possibility of decay.

Both microscopic and macroscopic methods have been used to study the nuclear
binding energy and the partial half-life of each decay channel. The
microscopic theory starts from the nucleon-nucleon interaction, either
realistic or phenomenological. The macroscopic theory uses selected variables
with physical considerations to construct semi-empirical formulas and fit the
experimental data, with a risk of overfitting and inappropriate parameters.
Besides, the exotic nuclei may deviate far from the general fitting as
outliers. Decreasing the deviation between theoretical predictions and
observables is always a critical issue.

With computing and storage power advancement, machine learning algorithms have
become available and helpful in many fields with various successes [12]. As
summarized in a recent colloquium, estimating the residuals of nuclear
properties through the machine learning algorithms is a powerful strategy
[13]. The neural network was used to compensate for the residuals of nuclear
masses [14, 15, 16] and nuclear charge radii [17, 18, 19] with structure
optimization and careful choice of the inputted parameters with definite
physical meanings. The applicability of the Decision Tree (DT) was verified by
training and testing with residuals of binding energy in 2020 [20]. However,
the Random Forest (RF) [21], developing from the DT algorithm, has been tested
for neither nuclear mass nor partial half-life of a specific decay channel, of
which the semi-empirical formulas have suggested several major components but
with residuals. The machine learning algorithms can include the possible
features to make a training for the residuals, while the RF, with bootstrap
sampling, can not only avoid the overfitting but also take into account the
correlation between data combinations and several features, which increases
the robustness and is conducive to the extrapolation.

Until now, none of work has investigated the competition between different
decay modes by the partial half-lives estimated by the machine learning
algorithm. The present work applies the RF machine learning algorithm to study
the major decay mode of heavy and superheavy nuclei. The competition of
$\alpha$ decay, $\beta$ decay and spontaneous fission (SF) of new elements and
the possible long-lived island are discussed in the superheavy region.

##  II Method

The present work concentrates on the region of $Z\geqslant 84$ and $N\geqslant
128$. The partial half-lives of $\alpha$ decay, $\beta^{-}$ decay, $\beta^{+}$
decay, electron capture (EC), and SF are calculated by the semi-empirical
formulas and then the residuals of each formula are trained by the RF
algorithm respectively. The minimum partial half-life of one mode corresponds
to the dominant decay mode.

###  II.1 Decay Half-life Formulas

The universal decay law (UDL) [22, 23],

$\displaystyle\text{log}_{10}T_{1/2,\alpha}=$ $\displaystyle
aZ_{\alpha}(Z-Z_{\alpha})\sqrt{\mu/Q_{\alpha}}$ (1) $\displaystyle+b\sqrt{\mu
Z_{\alpha}(Z-Z_{\alpha})(A_{\alpha}^{1/3}+(A-A_{\alpha})^{1/3})}$
$\displaystyle+c,$

is used to fit the $\alpha$ decay half-life. $Z_{\alpha}$, $A_{\alpha}$,
$Q_{\alpha}$ and $\mu=A_{\alpha}(A-A_{\alpha})/A$ denote the proton number,
the mass number of $\alpha$ particle, the $\alpha$ decay energy and the
reduced mass, respectively. The channel is supposed to be from the ground
state to the ground state.

Table 1: The coefficients and the corresponding RMSE of UDL, SF3 and Eqs. (3, 4, 5, 6, 7) when fitted to the nuclei with $Z\geqslant 84$ & $N\geqslant 128$. The RMSEs of RF trained UDL, SF3 and Eq. (7) are listed in the last two columns. WS4 and UNEDF0 in the subscript denotes that the source of the unmeasured energies. |  $a$ (log${}_{10}B_{\text{GT}}$) | $b$ | $c$ | $d$ | $e$ | $f$ | RMSE | RMSERF,WS4 | RMSERF,UNEDF0
---|---|---|---|---|---|---|---|---|---
UDL | 0.407 | -0.382 | -23.896 | - | - | - | 0.883 | 0.598 | 0.669
SF3 ($Z<104$) | -1.129 | -6997.113 | 79.803 | - | - | - | 3.070 | 1.195 | 1.195
SF3 ($Z\geqslant 104$) | -1.363 | -13272.729 | 113.415 | - | - | - | 1.267 | 0.825 | 0.825
SF${}_{\text{ren}}$ | -1002.490 | 35.340 | -0.332 | -0.747 | 31.565 | - | 2.875 | - |
SF${}_{\text{xu}}$ | -1.031 | 0.145 | 6.28E-07 | -0.00986 | -1.135 | 572.558 | 3.313 | - |
SF${}_{\text{santhosh}}$ | -37.871 | 0.432 | 3325.452 | -8270.979 | 496.073 | - | 2.857 | - |
SF${}_{\text{soylu}}$ | -19.408 | 180.953 | -0.491 | -0.807 | 3.07E-06 | -1851.543 | 3.181 | - |
Eq. (7)${}_{\beta^{+}}$ | 1.378 | - | - | - | - | - | 1.957 | 0.439 | 0.437
Eq. (7)${}_{\beta^{-}}$ | -1.819 | - | - | - | - | - | 1.451 | 0.656 | 0.667
Eq. (7)${}_{\text{EC}}$ | -2.112 | - | - | - | - | - | 2.360 | 0.971 | 0.996

As to SF, a three-parameter formula (noted as SF3),

$\log_{10}T_{\mathrm{SF}}=a\frac{(Z-\nu)^{2}}{(1-\kappa
I^{2})A}+\frac{b}{A}+c,$ (2)

is proposed based on several existing formulas [24, 25, 26, 27, 28, 29], where
$\nu$ presents the blocking effect from unpaired nucleons, takes 0 for even-
even nuclei and 2 for other nuclei [24], $\kappa$ takes 2.6 [30, 28],
$I=\frac{N-Z}{A}$, and $a$, $b$, and $c$ are fitting coefficients. Eq. (2) is
particularly fitted to nuclei with $Z<104$ and the rest because of a
systematic difference as shown in TABLE 1. $T_{1/2,\text{SF}}$ of nuclei with
$Z<104$ increases largely with the decrease of $Z$ since the Coulomb repulsion
decreases. The rather long $T_{\text{SF}}$ ($>10^{8}$ s) of some nuclei in
this region cannot be universally described at present and are not taken to
fit Eq.(2) because the competition of such SF is very weak comparing to other
decay modes. This formula avoids the divergence of other SF formulas in Refs.
[24, 25, 26, 27] during extrapolation, which write as:

$\displaystyle\text{log}_{10}T_{1/2,\text{SF},\text{ren}}=$ $\displaystyle
a\frac{Z-90-\nu}{A}+b\frac{(Z-90-\nu)^{2}}{A}$ (3)
$\displaystyle+c\frac{(Z-90-\nu)^{3}}{A}$
$\displaystyle+d\frac{(Z-90-\nu)(N-Z-52)^{2}}{A}+e,$
$\displaystyle\text{log}_{10}T_{1/2,\text{SF},\text{xu}}=$ $\displaystyle
aA+bZ^{2}+cZ^{4}+d(N-Z)^{2}$ (4) $\displaystyle+eZ^{2}A^{-1/3}+f,$
$\displaystyle\text{log}_{10}T_{1/2,\text{SF},\text{santhosh}}=$
$\displaystyle a\frac{Z^{2}}{A}+b(\frac{Z^{2}}{A})^{2}$ (5)
$\displaystyle+c\frac{N-Z}{A}+d(\frac{N-Z}{A})^{2}+e,$
$\displaystyle\text{log}_{10}T_{1/2,\text{SF},\text{soylu}}$
$\displaystyle=aA+bA^{2/3}+cZ(Z-1)A^{-1/3}$ (6)
$\displaystyle+d(N-Z)^{2}/A+eZ^{4}+f.$

When the higher order terms in these four formulas enhance their
interpolation, the divergence is introduced into the extrapolation, which will
be discussed in the last part of Sec. III.

The $\beta$ decay half-life is estimated by the formula in Refs. [31, 32].
Assuming that the ground state $\beta$ decay is one effective Gamow-Teller
(GT) transition, the partial half-life is expressed as

$\displaystyle\text{log}_{10}T_{1/2,\beta}=\text{log}_{10}\kappa_{1}-\text{log}_{10}f_{0}-\text{log}_{10}B_{\text{GT}},$
(7)

where $\kappa_{1}=\frac{2\pi^{3}\hbar^{7}\ln
2}{m_{e}^{5}c^{4}G_{\text{F}}^{2}}=6147$ s, $f_{0}$ is the phase-space factor
and $B_{\text{GT}}$ is the GT reduced transition probability [31]. For EC, the
phase-space factor is deduced as

$\displaystyle f_{0}^{\text{EC}}\approx
2\pi(\frac{Z}{137})^{3}(1-\frac{1}{2}(\frac{Z}{137})^{2}+E_{0})^{2},$ (8)

while for $\beta^{\pm}$ decay, it is

$\displaystyle
f_{0}^{\beta^{\pm}}\approx\frac{\mp(E_{0}^{5}-10E_{0}^{2}+15E_{0}-6)2\pi(Z\mp
1)/137}{30(1-\exp(\pm 2\pi(Z\mp 1)/137))},$ (9)

where $E_{0}$ is the renormalized $\beta$ decay energy. Because $Q_{\beta}$
provided by AME2020 [33] is the difference of atomic mass, the electron mass
should be reconsidered:

$\displaystyle E_{0,\beta^{+}}$
$\displaystyle=\frac{Q_{\beta^{+}}+2m_{e}c^{2}}{m_{e}c^{2}}$ (10)
$\displaystyle E_{0,\beta^{-}}$
$\displaystyle=\frac{Q_{\beta^{-}}+m_{e}c^{2}}{m_{e}c^{2}}$ $\displaystyle
E_{0,\text{EC}}$ $\displaystyle=\frac{Q_{\text{EC}}-m_{e}c^{2}}{m_{e}c^{2}}.$

Finally, the log${}_{10}B_{\text{GT}}$ is estimated as the average of
log${}_{10}(f_{0}T_{1/2,\beta}/\kappa_{1})$. The fitting results are listed in
TABLE. 1

###  II.2 Random Forest Method

RF is an integration of the DT and bootstrap algorithm. DT is a non-parametric
supervised learning algorithm. For a dataset consisting of $S$ samples of $I$
features (variables) $\\{(\theta_{1},...,\theta_{I})_{s},s\in[1,S]\\}$ and
object (observable) $\\{y_{s},s\in[1,S]\\}$, it establishes a binary tree
structure which divides the dataset into $L$ subsets based on the values of
features, each subset is called as a leaf. Such partition aims at the minimum
root-mean-square error (RMSE)

$\mathrm{RMSE}=\sqrt{\frac{1}{S}\sum_{s=1}^{S}(y_{s}-f(\theta_{1},...,\theta_{I}))^{2}}$
(11)

of the whole dataset, assigning each leaf a value.

Bootstrap is a statistical method with a basic idea of randomly resampling
with replacement, through which the possible combination and weighting of data
are automatically taken into account [34, 35]. Each time a new dataset is
obtained, a new DT is trained and used to predict the object of each sample in
the whole dataset. Repeating this $M$ times, one obtains a forest of $M$
trees. The final predicted value of the object for a sample is the average of
results calculated by all trees in the forest. Since each tree is trained by a
part of samples in the dataset, the value for each sample predicted by the
forest is an average of interpolation and extrapolation, which decreases the
divergence when the calculation is implemented to the unmeasured nuclei. The
open source scikit-learn [36] is used for machine learning. The forest is
assumed to be composed of $10^{5}$ trees to decrease the dispersion of RMSE in
the present work.

##  III Results and Discussion

The residuals of the decay formulas of $\alpha$ decay, $\beta^{-}$ decay,
$\beta^{+}$ decay, and EC are trained by RF with features $Z$, $N$, $A$, the
odevity of $Z$ and $N$, and decay energy. Because no decay energy can be
defined for SF, the fission barrier (FB) extracted from Ref. [37] is used to
replace the decay energy in the features set to consider the deformation
effect. The leaves number chooses 11, which is same as that determined for
training the binding energy in this region in our previous work [38]. FIG. 1
compares the residuals of these decay formulas before and after RF training.
Two conditions are assumed to determine outliers here. One is locating out of
the dash line with corresponding color, which means the scatters deviate two
times of RMSE from the experimental $\mathrm{log}_{10}T_{1/2}$. Another is
$|\mathrm{log}_{10}(T_{1/2,\mathrm{cal}}/T_{1/2,\mathrm{exp}})|$ value larger
than 3, which denotes that the calculation is three times of magnitude far
from the experimental value. In this way, one avoids missing (adding) outliers
due to the very large (small) RMSE. After training, the biases of outliers of
these decay formulas are well reduced, and the RMSE of formulas decreases
(TABLE 1) as expected. The condition of outlier is not too strict because our
aim is not to decrease the RMSE as small as possible but go to an appropriate
scale where the dominant decay mode can be described. The present work chooses
the same features and the same leaves number of RF to train the residuals of
different decay formulas, which avoids the overfitting in seeking an extreme
small RMSE.

Figure 1:  Comparison of residuals of UDL ($\alpha$ decay), SF3 (SF), Eq. (7)
($\beta^{-}$, $\beta^{+}$ and EC) before and after RF training. The dash lines
denote two times of RMSE of corresponding formulas.

Figure 2:  The dominant decay mode (left panels) and the minimum partial half-
lives (right panels) of $\alpha$ decay, $\beta^{-}$ decay, $\beta^{+}$ decay,
EC and SF. (a-b) The experimental data in NUBASE2020. (c-f) The energies are
calculated by WS4 and UNEDF0. Specifically, the FB is used to replace the
decay energy to learn SF. Nuclides, of which the predicted partial half-life
is longer than $10^{4}$ s, are marked by star.

Totally, 445 nuclides with measured partial half-lives and branch ratios of
these five decay modes are collected from NUBASE2020 [39]. The dominant decay
mode and the partial half-life of nuclides are drawn in FIG. 2(a-b). A long-
lived $\alpha$ decay valley from ${}^{226}_{88}$Ra138 to ${}^{251}_{98}$Cf153
lies between a narrow $\beta^{+}$/EC decay band and a neutron-rich $\beta^{-}$
region. Away from this valley, the half-life of nucleus decreases. The
southwest direction is dominated by $\alpha$ decay while the southeast
direction is occupied by $\beta^{-}$ decay. At the northwest direction,
$\beta^{+}$ decay and EC compete with $\alpha$ decay and lose after $Z$
increases. At the northeast direction, $\alpha$ decay and SF compete with each
other and a region extended from the $\alpha$ valley seems to be dominated by
SF. Though the distribution of dominant decay mode has clear boundary, the
minimum partial half-life is smooth.

There are 341 (104) nuclides are with known (unknown) corresponding reaction
energies among all 445 considered nuclides. Those nuclides with unmeasured
mass take that calculated by WS4 [40] and UNEDF0 [41] for estimating the
partial half-lives. The results are depicted in FIG. 2(c-f). The calculation
accords well with the experiment, as the dominant decay mode is correctly
described for 431 and 427 (96.9% and 96.0%) nuclei when the RMSE of
log${}_{10}T_{1/2}$ of the dominant decay mode is 0.62 and 0.67, respectively.
Those nuclides, of which the dominant decay mode is inconsistently described,
are generally with two competitive decay modes. For example, the $\alpha$ and
SF branch ratios of 255Rf, 262Db, and 286Fl are around 50%. Meanwhile, the
liquid drop model trained by RF [38] is also applied to provide the energies,
which presents consistent results and is not shown here.

The accuracy of energy is important to the half-life calculation. If
calculated energies of WS4 and UNEDF0 replace all the experimental ones, the
number with consistent decay mode comparing to the experiment reduces to 72.6%
and 64%, and the RMSE of log${}_{10}T_{1/2}$ also increases to 2.07 and 2.64.
The difference of the results between using energies of the two models comes
from the accuracy since the RMSE of mass of WS4 is about 0.3 MeV [40] while
that of UNEDF0 is about 1.45 MeV [41]. This also leads to difference during
extrapolation. The consistency rate of dominant decay mode between using
energies calculated by these two models decreases from 82.2% to 66.2%. More
accurate and precise measurements of decay energy will aid the theoretical
prediction. Besides, WS4 and UNEDF0 may lose prediction power after training
through machine learning. If the WS4 and UNEDF0 binding energies are trained
with features $Z$, $N$, $\delta$ and $P$, which well describe the residuals in
Ref. [15], though the description of energy improves, the consistency in the
dominant decay mode decreases several percent, which is considerable compared
with the rate 23.4% of theoretical energies among all (104/445).

The SF is important for investigating the half-lives of superheavy nuclei. As
shown by FIG. 2(c, e), the dominant decay mode of the unknown nuclides is
determined through the competition between SF, $\alpha$ decay, and $\beta^{-}$
decay. The major competition is between SF and $\beta^{-}$ decay for neutron-
rich nuclides, while it is between SF and $\alpha$ decay for neutron-deficient
ones. The existing experimental data show a long-lived $\alpha$ decay region
from ${}^{226}_{88}\rm{Ra}_{138}$ to ${}^{251}_{98}\rm{Cf}_{153}$, lying
between the $\beta^{+}$ and $\beta^{-}$ decay regions, and being ended by the
SF. The present models correctly describe such phenomenon. Along the long-
lived region, and after $N$ exceeds 154, a blue band is shown in FIG. 2(d, f),
which indicates half-lives of about $10^{2}\sim 10^{7}$ s. At the southwest
corner of $Z=114$ and $N=184$, nuclides in a circle are with longer half-life
than those surrounded. This is because the fission barrier is high in this
region and lead to longer $T_{1/2,\mathrm{SF}}$. FIG. 3 compares the evolution
of FB and measured $T_{\mathrm{1/2,SF}}$ along the mass number. The FB
decreases with $A$ before $A=230$, then behaves like a sinusoidal wave
oscillating between 2 and 10 MeV. It seems that there exists a FB threshold,
below which could the nuclide fissure spontaneously. Nuclides with rather long
$T_{1/2,\mathrm{SF}}$ are generally with small SF branch ratio. Besides, the
FB of nuclides with SF branch ratio less than 1% is mostly higher than those
with SF branch ratio greater than 1%, which implies that higher the FB, weaker
the SF. However, if one concentrates only on nuclides with SF branch ratio
less than 1% or greater than 1%, the correspondence between FB and
$T_{1/2,\mathrm{SF}}$ becomes much more complex.

Figure 3:  The evolution of $T_{1/2,\mathrm{SF}}$ and FB along the mass
number. The datasets are divided according to whether the corresponding
$T_{1/2,\mathrm{SF}}$ is measured and whether the branch ratio (BR) of SF is
less than 1%.

Figure 4: Comparison of partial half-lives of isotopes with $Z=117-122$.
IMELDM is extracted from Ref. [42], GLDM+RHF, KPS and Xu are extracted from
[43], Sarriguren is extracted from Ref. [44], Sridhar is extracted from Ref.
[45].

Nuclides with partial half-life predicted to be longer than $10^{4}$ s are
marked by star in FIG. 2(d, f), which suggests ${}^{250,252,254}\rm{Cm}$,
${}^{260,261}\rm{Es}$, ${}^{261\sim 264}\rm{Md}$, and ${}^{265}\rm{Lr}$ for
future measurement. It is a coincidence that no experimental value of the
half-life of 250Cm is suggested in NUBASE2020 and thus extrapolated in the
present work. In NNDC, SF is shown to be its dominant decay mode, and the
half-life is recommended to be 8300 years, which is rather long. Though the
calculation of the present work underestimates the NNDC value, the long half-
life property and the dominant decay mode are reproduced. Besides, the upper
limit of the half-life of 252Cm is 2 days, proposed in 1966 by Ref. [46] and
not updated, while the present work estimates a value of 1.43 days. No
experimental half-lives are published for 260,261Es, 261∼264Md, and 265Lr. But
their nearby isotopes are with long half-lives, e.g., 257Es (7.7 days), 260Md
(31.8 days), 259Md (1.6 hours), 258Md (51.5 days), 257Md (5.52 hours) and
266Lr (11 hours). Moreover, these Es, Md, and Lr isotopes locate at the
extension of the narrow long-lived region from 226Ra to 251Cf, which makes it
convincing that these Es, Md, and Lr isotopes are candidates with long partial
half-lives. More measurement is also suggested since, for example, the more
than 50 years of un-updated datum of 252Cm.

Comparing all possible decay channels is limited by the accurate description
of each channel and the observed data. We should note that the mechanism of SF
is still not fully understood. The effect of the quadrupole deformation
parameter ($\varepsilon_{2}$) [47] on the half-life estimation is also
investigated. If one replaces FB by $\varepsilon_{2}$ during training the RF,
more nuclides in $N>184$ are calculated to be dominated by $\alpha$ and
$\beta^{-}$ decay. The $\alpha$ decay leads to shorter half-life. Besides, the
relatively long-lived circle at the southwest of $Z=114,N=184$ is no longer
locally. The blue region is much more extended and more long-lived candidates
are marked by star. This is because the deformation decreases, which
compensates the Coulomb repulsion, that is increased by $Z$. Furthermore, the
FB combines contribution of multipole deformations and thus presents stronger
quantum effect in FIG. 2(d, f) than $\varepsilon_{2}$.

The extrapolation stops at the single neutron (proton) and two-neutron (two-
proton) drip lines. Dataset of UNEDF0 stops at $Z=120$. From the existent
region to the neutron-deficient side, $\alpha$ decay and SF are predicted to
compete with each other. On the neutron-rich side, calculations predict
$\beta^{-}$ decay as the dominant mode, while SF competes in specific
nuclides. Up to date, results of most theoretical calculations of partial
half-lives [44, 42, 43, 48, 49, 50, 28, 51, 27, 45, 52] support that the
$\alpha$ decay is the dominant decay mode for new elements at $N\leqslant
184$. But those calculated $T_{1/2,\rm{SF}}$ diverge when it is far from
shells. In FIG. 4, we compare the partial half-lives of isotopes with
$Z=117-122$ predicted in the present work and the corresponding results of
Refs. [44, 42, 43]. Even though the partial half-lives of $\beta^{+}$ decay
and EC in the present work are not globally longer than that in Ref. [44],
they are still about five orders of magnitude greater than that of $\alpha$
decay in this region, which does not change the dominant decay mode.

Figure 5:  The odd-even staggering of $T_{\mathrm{1/2,SF}}$ of U, Pu, Cm, Cf,
Fm, Rf, Sg, Ds, Cn and Db isotopes and comparing with $T_{1/2,\alpha}$.

$T_{1/2,\alpha}$ predicted in the present work are longer than the results of
Refs. [44, 42, 43], which does not change the dominant decay mode of odd-$Z$
isotopes but enhances the competition of SF in even-$Z$ isotopes. Furthermore,
the prediction of the present work shows strong odd-even staggering of
$T_{1/2,\rm{SF}}$ of even-$Z$ isotopes, i.e., $T_{1/2,\rm{SF}}$ of even-even
nuclei is several times of magnitude shorter than its two isotopic neighbors,
which differs from the weak or not predicted odd-even staggering effect of
other SF models in FIG. 4. In fact, all measured $T_{1/2,\rm{SF}}$ of even-$Z$
isotopes show such odd-even staggering. FIG. 5 draws $T_{1/2,\mathrm{SF}}$ and
$T_{1/2,\alpha}$ of isotopes with $Z\geqslant 92$. When $Z$ is small, for
example in U, Pu, Cm and Cf isotopes, SF is not competitive to $\alpha$ decay
because the Coulomb repulsion is not enough strong. But when $Z$ is large,
this odd-even staggering makes SF competitive with $\alpha$ decay in these
even-even nuclides. The $\alpha$ decay is thus suggested to be a key signal
detected for $Z=119$ and $121$ isotopes, while the SF should also be taken
into account for even-$N$ isotopes of $Z=120$ and $122$. Note that the odd-
even staggering also exists in the odd-$Z$ isotopes. It can only be verified
by 260∼263Db because the data are rare. This is why the odd-even staggering of
odd-$Z$ isotopes is not predicted in the present work. The DNS model predicted
$\sigma_{\rm{ER}}$ of hundreds of fb for the 3$n$ or 2$n$ channels producing
${}^{293}119_{174}$ on 243Am target [53], which can be examined on the new
facility CAFE2 and SHANS2 in Lanzhou [54]. Considering the odd-even effect of
partial half-lives, candidates of nuclide for new superheavy elements still
need analysis through the production cross section and the partial half-life.

Figure 6:  The comparison of $T_{1/2,\rm{SF}}$ calculated by formulas of
Ren2005 [24], Xu2008 [25], Santhosh2010 [26], Soylu2019 [27] and the present
work ($\mathrm{SF_{FB}}$). The extrapolation in the upper panel is constrained
by $104\leqslant Z\leqslant 121$ and $56\leqslant N-Z\leqslant 62$, the bottom
panel is constrained by $Z=112-116$.

There are several formulas of the SF with more parameters [24, 25, 26, 27],
but the extrapolation diverges, sometimes to tens of orders of magnitude, as
drawn in Figure 6, because of the higher order terms. There is no hint of such
divergence of $T_{\rm{1/2,SF}}$ from experiments. If the SF is not considered
in the extrapolation because of its possible large uncertainty, a long-lived
region is predicted along the boundary of competition between the $\alpha$ and
$\beta^{-}$ decays. Particularly, results from UNEDF0 show a new long-lived
region just around $Z=114$ and $N=184$, which is also mainly contributed by
the $\alpha$ decay. Thus, the SF is key for investigating the superheavy
stable island but still not well understood.

##  IV Conclusion

In summary, the decay modes of superheavy nuclei are investigated through the
Random Forest algorithm. The partial half-lives of $\alpha$ decay, $\beta^{-}$
decay, $\beta^{+}$ decay, EC, and SF are studied and compared with each other.
The dominance of $\alpha$ decay in the neutron-deficient region is more
convinced. $\beta^{-}$ decay is predicted to be dominant in the neutron-rich
region. SF contributes to a long-lived circle at the southwest corner of
$Z=114$ and $N=184$. More accurate and precise measurement on nuclear mass and
decay energy can improve the prediction of decay mode. After correcting the
divergence of up-to-date SF formulas, the odd-even effect of SF is found in
even-$Z$ nuclides, which induces possible competition between SF and $\alpha$
decay in even-even nuclides. The $\alpha$ decay is thus suggested to be key
probe of isotopes with $Z=119$ and $121$, while the competition of SF should
be taken into account in even-even isotopes with $Z=120$ and $122$.

${}^{250,252,254}\rm{Cm}$, ${}^{260,261}\rm{Es}$, ${}^{261\sim 264}\rm{Md}$,
and ${}^{265}\rm{Lr}$ with half-life predicted to be longer than $10^{4}$ s
are suggested for future measurement. The SF influenced by the fission barrier
and the Coulomb repulsion leads to a long-lived region during extrapolation.
The present results indicate that study on SF, especially beyond 286Fl, which
is the presently heaviest nuclide with significant SF branch ratio, would be
of key importance for further studies to be performed on new facilities, such
as CAFE2 and SHANS2 in Lanzhou.

###### Acknowledgements.

The authors acknowledge useful discussion with Professors Zaiguo Gan,
Zhongzhou Ren, and Zhiyuan Zhang. This work has been supported by the
Guangdong Major Project of Basic and Applied Basic Research under Grant No.
2021B0301030006, and the computational resources from SYSU and National
Supercomputer Center in Guangzhou.

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