# Modified empirical formulas and machine learning for $\alpha$-decay
systematics

G. Saxena1,†, P. K. Sharma2 and Prafulla Saxena3 1Department of Physics (H&S),
Govt. Women Engineering College, Ajmer-305002, India  2Govt. Polytechnic
College, Rajsamand-313324, India  3Department of Computer Science and
Engineering, Indian Institute of Technology, Kanpur-208016, India.
<EMAIL_ADDRESS>
###### Abstract

Latest experimental and evaluated $\alpha$-decay half-lives between
82$\leq$Z$\leq$118 have been used to modify two empirical formulas: (i) Horoi
scaling law [J. Phys. G 30, 945 (2004)], and Sobiczewski formula [Acta Phys.
Pol. B 36, 3095 (2005)] by adding asymmetry dependent terms ($I$ and $I^{2}$)
and refitting of the coefficients. The results of these modified formulas are
found with significant improvement while compared with other 21 formulas, and,
therefore, are used to predict $\alpha$-decay half-lives with more precision
in the unknown superheavy region. The formula of spontaneous fission (SF)
half-life proposed by Bao et al. [J. Phys. G 42, 085101 (2015)] is further
modified by using ground-state shell-plus-pairing correction taken from
FRDM-2012 and using latest experimental and evaluated spontaneous fission
half-lives between 82$\leq$Z$\leq$118\. Using these modified formulas, contest
between $\alpha$-decay and SF is probed for the nuclei within the range
112$\leq$Z$\leq$118 and consequently probable half-lives and decay modes are
estimated. Potential decay chains of 286-302Og and 287-303119
(168$\leq$N$\leq$184: island of stability) are analyzed which are found in
excellent agreement with available experimental data. In addition, four
different machine learning models: XGBoost, Random Forest (RF), Decision Trees
(DTs), and Multilayer Perceptron (MLP) neural network are used to train a
predictor for $\alpha$-decay and SF half-lives prediction. The prediction of
decay modes using XGBoost and MLP are found in excellent agreement with
available experimental decay modes along with our predictions obtained by
above mentioned modified formulas.

††: J. Phys. G: Nucl. Phys.

  * September 2020

Keywords: Superheavy nuclei; $\alpha$-decay; Empirical formulas; Machine
learning.

##  1 Introduction

The journey of $\alpha$-decay, starting from Rutherford followed by Geiger-
Nuttall law [1] in 1911 along with quantum mechanical formulation by Gamow [2]
and, Guerney and Condon [3] in 1928 to till this year 2020 [4, 5, 6, 7, 8]
dominates the investigations of superheavy nuclei in such a way that most of
the experimental and theoretical studies of superheavy nuclei are concentric
primarily on the $\alpha$-decay mode. Study of $\alpha$-decay chains plays a
crucial role for synthesis of new elements in the laboratory and also to
investigate the nuclear structure properties [9, 10, 11, 12, 13, 14, 15, 16,
17, 18]. The synthesis of superheavy elements in laboratories at GSI [9, 19],
RIKEN [20] and JINR [10, 21, 22] is mainly governed by cold-fusion reactions
between 208Pb, 209Bi and beams of A$>$50 [10] or hot-fusion reactions between
long-lived actinide nuclei from 238U to 249Cf and 48Ca beams [11]. As an
extension of the periodic chart, decay chains of 293117 and 294117 have been
reported by fusion reactions between 48Ca and 249Bk with 11 new nuclei [21]
and also of 294118 by 249Cf+48Ca reaction [23]. In addition, afterward, eleven
new heaviest isotopes of elements Z$=$105 to Z$=$117 are identified validating
the concept of the island of enhanced stability for superheavy nuclei (SHN)
[24]. For the synthesis of elements with Z$\geq$118 the attempts and progress
are still going on and in near future few more SHN are expected to be detected
[15, 25].

In recent times, experimental works are performed on the Dubna gas-filled
magnetic recoil separator (GFS) to study the reactions of the formation of
294Ts and 294Og nuclides along with future possibilities of synthesizing new
elements with Z$=$119 and Z$=$120 [8]. Similarly, the observation of a new
decay chain of 294Og along with prospects for reaching new isotopes 295,296Og
are reported [26]. Many properties of the decay of the heaviest nuclei with
Z$=$112–118 indicate a considerable stabilizing effect upon approaching the
island of stability N$\sim$184 [11]. However, up to now the heaviest
experimentally known nuclei 294Ts and 294Og have neutron number N$=$177 and
N$=$176, respectively. Therefore, future experiments are expected to
synthesize heavier elements which may have the neutron number between
N$\sim$170 and N$\sim$184\. With this in view, the study of decay chains of
Z$=$118 and 119 with neutron number 168$\leq$N$\leq$184 may be of extreme
importance and crucial for experimental point of view, which is precisely the
aim of this article.

The study of $\alpha$-decay chains consists of the determination of
$\alpha$-decay half-life ($log_{10}T_{1/2}$) which is first formulated as
linear dependence on reciprocal of the square root of $\alpha$-decay energy
($\sqrt{Q}$) [1]. In 1966, Viola and Seaborg [27] predicted a simple formula
that is based on the Gamow model by including the intercept parameters of
linear dependence on the charge number of the daughter nucleus and also
modified in Refs. [28, 29]. Afterwards, Brown [30], Royer [31] and Ren et al.
[32] have formulated the empirical formulas by studying the experimental
variation of the logarithmic half-lives. Later on, Ni et al. [33] have derived
the empirical formula (NRDX formula) for alpha decay half-lives from the WKB
barrier penetration probability with certain approximations. Likewise, Qi et
al. [34] proposed a linear universal decay law (UDL) from the microscopic
mechanism of the charged-particle emission. In addition to the formulas
mentioned above, some semi-empirical relationships including microscopic
effects, such as the universal (UNIV) curve, and a semi-empirical formula
based on fission theory (SemFIS), were proposed by Poenaru et al. which are
subsequently modified in Refs. [35] and [36] proposing new names NUNIV and
SemFIS2, respectively. Recently, many of these formulas have been modified by
adding asymmetry terms and referred as modified Royer formula (Akre formula)
[37], modified UDL formula (MUDL) [38], modified scaling law of Brown (MSLB)
[39], modified Viola-Seaborg formula (MVS) [39], and new Ren A formula [40].
Few of the above mentioned formulas are refitted using latest experimental
alpha decay half-lives viz. (i) modified Royer GLDM (Rm), modified Viola-
Seaborg (VSm1 and VSm2) and modified Sobiczewski-Parkhomenko (SPm1 and SPm2)
formulas [41], (ii) modified version of the Brown empirical formula (mB1 and
mB2) [42] (iii) new universal decay law (NUDL), new Royer (NRo) and new
Sobiczewski-Parkhomenko (NSP) formulas [43]. Moreover, Manjunatha et al. have
presented a formula [44] based on second-order polynomial fitting of
$Z^{0.4}_{d}/\sqrt{Q}$ which is subsequently modified in one of our recent
work [45].

Besides the above-mentioned formulas, in 2004, Horoi et al. [46] have proposed
the scaling law for cluster decay and in 2005, Parkhomenko and Sobiczewski
[29] have included excitation energy for odd-A and odd-odd nuclei in an
empirical formula for Z$>$82\. Both these formulas may be modified further and
the parameters may be adjusted with the fitting of updated experimental data
which is one of the motives of the present work.

Another decay mode that competes with $\alpha$-decay and acts as a decider to
terminate the $\alpha$-decay chains is spontaneous fission (SF). The first
semi-empirical formula for SF was proposed by Swiatecki [47] in 1955 and
afterward in 2005, Ren et al. [48] have generalized formula for spontaneous
fission half-lives of even-even nuclei in their ground state to both the case
of odd nuclei and the case of fission isomers. Later on, Xu et al. [49] have
proposed empirical formula of SF for even-even nuclei which is followed by the
semi-empirical formula given by Santosh et al. [50]. In 2012, Karpov et al.
[51] have proposed a new formula based on barrier height which is modified
with even-odd corrections by Silisteanu et al. [52]. In 2015, a modified
formula is proposed by Bao et al. [53] for determining the spontaneous fission
half-lives based on Swiatecki’s formula, including the microscopic shell
correction from FRDM-1995 [54] and isospin effect. This formula was able to
reproduce the experimental data quite well, however, it may be further
modified with shell plus pairing correction from FRDM-2012 [55] which is also
an object of present work.

In recent times, machine learning (ML) algorithms [56, 57] have participated
as an alternate and powerful tool to study and predict a complex data of
physics [58, 59, 60]. In particular, machine learning methods are employed in
nuclear physics for prediction of nuclear masses [61, 62], binding energies
[63], $\beta$-decay [64] and charge radii [65]. Since the predictions in the
superheavy realm by various theories and empirical/semi-empirical formulas are
still not so precise, therefore, machine learning methods could be very useful
to predict decay modes, decay chains, and half-lives of unknown territory of
superheavy nuclei. This possibility has invoked us to apply machine learning
methods for the prediction of $\alpha$-decay, SF and for the determination of
half-lives of known and unknown nuclei to visualize the merit of this
alternate approach in superheavy region.

The paper is organized as follows. In section II we provide the original
formulas and their modified versions with concerned coefficients for
$\alpha$-decay and spontaneous fission (SF) half-lives. These formulas are
fitted using $scipy.optimize.curve\\_fit$ library [66] along with the
Levenberg-Marquardt algorithm which is used for non-linear least-squares fit
[67]. In Section III we give a brief outline of the all machine learning
algorithms employed. In section IV we first compare results of modified
formulas with already established formulas and then compare our results of
half-lives for a few experimentally known decay chains. Then, we predict
probable decay modes for the nuclei in the range 112$\leq$Z$\leq$118 which is
followed by a prediction of potential decay chains of 286-302Og and
287-303119.

##  2 Formalism for $\alpha$-decay and spontaneous fission (SF) half-lives

###  2.1 Horoi Formula for $\alpha$-decay

In 2004, Horoi et al. [46] have generalized scaling law for the decay time of
alpha particles for cluster decay and proposed that the $log_{10}$T1/2 depends
linearly on the scaling variable S = (ZcZd)0.6/$\sqrt{Q}_{c}$ and on the
square root of the reduced mass ($\mu$) of cluster and daughter. From a
detailed analysis, he proposed a model-independent law for the entire set of
cluster decay data of the following form:

$\displaystyle log_{10}T_{1/2}$ $\displaystyle=$
$\displaystyle(a_{1}\sqrt{\mu}+b_{1})[(Z_{c}Z_{d})^{0.6}Q^{-1/2}-7]+(a_{2}\sqrt{\mu}+b_{2})$
(1)

here, Zc and Zd are the charges of the cluster and daughter nuclei, $\mu$ is
the reduced mass of the cluster–daughter system.

###  2.2 Modified Horoi Formula (MHF-2020) for $\alpha$-decay

In the present work, we modify the Horoi formula [46] for $\alpha$-decay by
adding asymmetry dependent terms ($I$ and $I^{2}$) which are linearly related
to the logarithm of $\alpha$-decay half-lives [38]. This kind of modification
has been successfully established by few other studies [39, 68, 40] and one of
our recent study [45] in which the addition of $I$ and $I^{2}$ is found in a
good match with experimental systematics. The modified Horoi formula
(MHF-2020) is given by:

$\displaystyle log_{10}T_{1/2}$ $\displaystyle=$
$\displaystyle(a\sqrt{\mu}+b)[(Z_{c}Z_{d})^{0.6}Q^{-1/2}-7]+(c\sqrt{\mu}+d)+eI+fI^{2}$
(2)

Here, new coefficients are obtained by fitting the half-lives from the latest
evaluated database [69] of 366 nuclei in the range 82$\leq$Z$\leq$118\. The
set of coefficients which is obtained by fitting 103 even-even (e-e), 101
even-odd (e-o), 76 odd-even (o-e) and 86 odd-odd (o-o) data separately is the
set 1 for which coefficients are given in the Table 1 along with the sample
size in each category. Whereas, set 2 is obtained by fitting of all the data
together and the coefficients are given in the Table 2. Set 1 comprises total
24 coefficients and can be used separately for all even(odd)-odd(even)
combinations of Z-N, whereas set 2 contains only 9 coefficients, out of which
5 coefficients are same for all sets of nuclei and only 1 coefficient is
different for even(odd)-odd(even) combinations of Z-N (refer to Tables 1 & 2).

Table 1: The coefficients of modified Horoi formula (MHF-2020: Set 1).

Set 1 | a | b | c | d | e | f
---|---|---|---|---|---|---
Even-even(103) | 834.367 | -1648.187 | -875.905 | 1722.168 | 90.116 | -267.214
Even-odd(101) | 425.941 | -837.910 | -349.093 | 683.399 | 23.602 | -79.824
Odd-even(76) | -32.607 | 71.208 | 206.158 | -411.086 | -47.871 | 116.354
Odd-odd(86) | 145.412 | -282.793 | 45.602 | -90.823 | -53.034 | 128.653

Table 2: The coefficients of modified Horoi formula (MHF-2020: Set 2).

Set 2 | a | b | c | e | f
---|---|---|---|---|---
For all nuclei | 643.540 | -1269.906 | -480.259 | 33.377 | -116.008
|  | Even-even | Even-odd | Odd-even | Odd-odd
| d | 943.118 | 943.534 | 943.440 | 943.810

###  2.3 Sobiczewski formula for $\alpha$-decay

In 2005, Parkhomenko and Sobiczewski [29] proposed a phenomenological formula
for the description of $\alpha$-decay half-lives for the nuclei heavier than
208Pb which is referred as Sobiczewski formula and given by:

$log_{10}T_{1/2}=aZ(Q_{\alpha}-\overline{E}_{i})^{-1/2}+bZ+c$ (3)

###  2.4 Modified Sobiczewski formula (MSF-2020) for $\alpha$-decay

In the present work, we modify the Sobiczewski formula [29] for $\alpha$-decay
by adding asymmetry dependent terms ($I$ and $I^{2}$) which are linearly
related to the logarithm of $\alpha$-decay half-lives [38]. The modified
Sobiczewski formula (MSF-2020) is given by:

$log_{10}T_{1/2}=aZ(Q_{\alpha}-\overline{E}_{i})^{-1/2}+bZ+c+dI+eI^{2}$ (4)

As mentioned above, the new coefficients are obtained by fitting the half-
lives from the latest evaluated database [69] of 366 nuclei in the range
82$\leq$Z$\leq$118 and for this modified formula, coefficients are given in
the Table 3.

Table 3: The coefficients of modified Sobiczewski formula (MSF-2020).

Set | Ei | a | b | c | d | e
---|---|---|---|---|---|---
Even-even | 0.000 | 1.266 | -0.144 | -23.304 | -81.216 | 275.650
Even-odd | 0.171 | 1.539 | -0.188 | -32.353 | -19.488 | 56.520
Odd-even | 0.113 | 1.566 | -0.171 | -32.748 | -38.831 | 99.568
Odd-odd | 0.284 | 1.265 | -0.130 | -25.116 | -58.267 | 157.311

###  2.5 Bao formula for SF

In 2015, Bao et al. [53] have modified Swiatecki’s formula [47] of spontaneous
fission half-lives considering role of shell effects. Taking into account the
dependence of half-lives on the shell correction and isospin effect, the
formula was given as:

$\displaystyle
log_{10}T_{1/2}^{SF}(yr)=c_{1}+c_{2}\left(\frac{Z^{2}}{(1-kI^{2})A}\right)+c_{3}\left(\frac{Z^{2}}{(1-kI^{2})A}\right)^{2}+c_{4}E_{sh}+h_{i}$
(5)

Values of the parameters are: c1$=$1174.353441, c2$=$$-$47.666855,
c3$=$0.471307, c4$=$3.378848 and k$=$2.6. The values of hi for various sets of
nuclei are: he-e$=$0, he-o$=$2.609374, ho-e$=$2.619768,
ho-o$=$he-o+ho-e$=$5.229142. Esh is the shell correction energy taken from
FRDM-1995 [54].

###  2.6 Modified Bao formula (MBF-2020) for SF

For the first time, recently, it has been shown by Poenaru et al. [70, 71]
that in a spontaneous fission process the shell plus pairing corrections
calculated with Strutinsky’s procedure may give a strong argument for
preformation of a light fission fragment near the nuclear surface. With this
in view, in the present work, we have modified the Bao formula [53] by
introducing shell plus pairing correction energy taken from FRDM-2012 [55] and
fitting with latest evaluated spontaneous fission half-lives [69]. The
modified Bao formula (MBF-2020) is given by:

$\displaystyle log_{10}T_{1/2}^{SF}(s)$ $\displaystyle=$ $\displaystyle
c_{1}+c_{2}\left(\frac{Z^{2}}{(1-kI^{2})A}\right)+c_{3}\left(\frac{Z^{2}}{(1-kI^{2})A}\right)^{2}+c_{4}E_{s+p}$
(6)

Here k$=$2.6 and other coefficients are readjusted and fitted by considering
49 nuclei in the range 82$\leq$Z$\leq$118 and are c2$=$$-$37.051, c3$=$0.374,
c4$=$3.110. The values of c1 for various sets of nuclei are c1(e-e)$=$893.264,
c1(e-o)$=$895.415, c1(o-e)$=$896.845 and c1(o-o)$=$897.019.

##  3 Machine learning methods

###  3.1 Introduction to Machine learning

Machine learning (ML) is a field of study which explores algorithms where the
machine can learn itself from data. ML methods learn hidden patterns, rules,
and make intelligent decisions. Typical ML methods leverage optimization
strategies that iteratively learns a model by minimizing loss calculated on
target or actual ground truth during the training phase. Machine learning
methods are categorized into various sub-categories based on their learning
style.
Supervised learning: Supervised learning methods learn a function that maps
input to an output based on example input-output pairs. In other words, labels
or ground-truths are required in these methods, which leads to model
enhancement by minimizing loss incurred in previous iterations. For example,
Random forest (RF), XGBoost and decision tree (DT) are based on supervised
learning.
Unsupervised learning: Unsupervised learning methods follow algorithms which
groups input data based on their similarity. No label is required for
unsupervised learning as input features are processed to analyze the
similarity between data.
Online learning: Data is increasing continuously, where the older trained
model may no longer be good enough for newer data with different patterns as
the model does not learn their properties. Online learning approaches
incrementally update the model, which can improve model performance for newer
samples with different patterns.
Ensemble learning: The ensemble method leverages multiple learning algorithms
to obtain better predictive performance. Many weak learners like decision
trees play their part, and a collective decision is taken to get an
appropriate prediction. Random forest and XGBoost methods are ensemble
learning methods we have used in this work. Bagging and boosting strategies
are used by ensemble learning methods to train multiple decision models.
Techniques used in ensemble learning, i.e. bagging and others, tend to reduce
problems related to over-fitting. Further, in small-sized datasets, these
methods are more beneficial than neural networks in terms of training time and
accuracy.
Model Training: Machine learning methods leverages optimization methods which
iteratively learn hidden data pattern by updating on previous learning model.
The goal of optimization methods, as well as the final model, is to minimize
training loss. The model is trained for some set of epochs until loss does not
decrease further. To validate the learned model, validation and test set are
kept separate from the training set. K-fold cross-validation is one of the
highly practiced validation approaches. According to K-fold cross-validation,
data is partitioned into K folds. Each fold is treated as a validation set,
and the remaining folds are considered for training. This methodology is well
suited with small-sized data as the model may or may not perform well on the
specific or random split. We have used a 5-fold cross-validation approach in
this work. ML problems or task are broadly divided into two categories known
as Classification and Regression. In Classification problems, the learned
model classifies input among a set of predefined classes, i.e., predicting a
vehicle class by looking at its features. Whereas, in Regression, task input
is bounded by values. It predicts the possible value of the given input, i.e.,
predicting a person’s age by looking at physical features.

###  3.2 Machine learning algorithms

In this work, we have used Supervised learning methods and framed our problem
as a Regression one. We use three tree-based supervised ML methods and one
neural network method. Considering the requirement of predicting life time,
which is a real value, the problem becomes a Regression problem.

####  3.2.1 Decision Trees

A decision tree (DT) [72] uses a tree-like data structure to learn the hidden
data pattern based on some set of rules. Decisions are taken at tree nodes
based on attribute importance. The importance of attributes is decided by some
metrics, i.e. entropy or gini coefficients. The final decision is taken at the
leaf of the tree. The decision tree approach is similar to conditional rule-
based decision-making, but it is highly prone to overfitting with large
datasets. As a single tree may overfit on some set of samples..

####  3.2.2 Random Forest

Random Forest (RF) [73] algorithm is an ensemble learning technique in which
the number of uncorrelated decision trees are constructed to avoid overfitting
to the training data. RF uses bagging techniques in which a random subset of
features (attributes) and data are sampled to fit various decision trees. It
creates a forest of decision trees. A collective decision is then taken from
the generated forest for a final prediction.

####  3.2.3 Multi-layer perceptrons

Figure 1: Multi-layer perceptron.

Multilayer perceptron (MLP)[74] belongs to a class of feed-forward artificial
neural network (ANN). It uses a supervised learning technique called
backpropagation for training. MLP contains three layers named input layers,
hidden layer(s), and output layer, as shown in Fig. 1. Even MLPs with a single
hidden layer are powerful models that can approximate a continuous function
under some constraints. Hidden layers can be increased to learn a complex
function best suited to training data. An activation function is used at
hidden layers to introduce non-linearity to distinguish data that is not
linearly separable. In backpropagation based on error optimization, weights of
hidden layers are adjusted iteratively. MLPs are powerful to approximate any
kind of function based on training data but highly prone to overfitting due to
complex model structure and less data availability.

####  3.2.4 XGBoost

XGBoost [75] stands for eXtreme Gradient Boosting, an ensemble machine
learning algorithm that uses a gradient boosting framework. XGBoost uses
gradient boosted decision trees designed for performance and speed with
parallel tree boosting. It supports classification, regression, and various
rank objective functions that provide a more efficient solution than its
counterparts like Decision trees and Random forest. Gradient boosting is an
ensemble learning technique in which many weak learners or models (i.e.,
Decision Trees) are combined to produce a final decision. All learned models
and their weighted sum are taken into consideration for the final results. For
large and unstructured datasets, artificial neural networks usually outperform
all other algorithms. However, with small-to-medium structured or low-
dimensional data, neural networks suffer in performance where XGBoost
(decision tree-based algorithms) perform well and considered best-in-class.

###  3.3 Employed approach

Figure 2: Flow of model training and prediction of final result.

We have used K-fold cross-validation in all machine learning models and
reported the average RMSE and mean deviation in Table 6. Considering 5-fold
cross validation, models are trained and tested five times so that each group
gets a chance to be the test set. For one group, the training set consists 80%
of the dataset, and the remaining 20% is used for the test. We separate
prediction data from training data and merge it with some newer samples for
which we do not have experimental results. We use XGBoost and MLP models for
alpha data prediction. For SF data prediction we use XGBoost and Random
forest. We take 5 prediction from 5 models trained during 5-fold cross
validation. The average of all five predictions is reported as the final
answer. The reason for using five model average is that a single model may
give excellent performance on a specific input data if the model properly
learns its similar features, but if the model does not learn similar
embedding, then it may drastically degrade the prediction result whether it is
possible that other model trained on different split may give a more accurate
prediction. To overcome the issue of best prediction due to biased training
and worst prediction due to improper training, we use an average of all five
splits. A detailed flow of our approach is shown in Fig. 2.

##  4 Results and discussions

In the beginning, to test the predictive power of few known and mostly used
$\alpha$-decay empirical/semi-empirical formulas, we have calculated
$\alpha$-decay half-lives using experimental and evaluated Qα [69] for 366
nuclei within the range 82$\leq$Z$\leq$118\. The formulas used are Sobiczewski
[29], NRDX [33], new universal curve (NUNIV) [35], semi-empirical formula
based on fission theory (SemFIS2) [36], Akre formula given by Akrawy et al.
[37], modified UDL formula [38], MSLB formula [39], modified Viola-Seaborg
formula [39], Manjunatha present formula [44], Horoi [46], new Ren A formula
[40], modified Royer GLDM (Rm) [41], modified Viola-Seaborg (VSm1 and VSm2)
[41], modified Sobiczewski-Parkhomenko (SPm1 and SPm2) formulas [41], modified
version of the Brown empirical formula (mB1 and mB2) [42], new universal decay
law (NUDL), new Royer (NRo) and new Sobiczewski-Parkhomenko (NSP) formula
[43]. In addition, the modified formulas in the present work i.e. modified
Horoi formula (MHF-2020) (set-1 and set-2) and modified Sobiczewski formula
(MSF-2020) are also tested for the same set of nuclei.

Table 4: Root mean square error (RMSE), mean deviation ($\overline{\delta}$),
index F$=$10${}^{\overline{\delta}}$ (in the unit of second) and No. of
coefficients of various formulas for $\alpha$-decay half-lives calculated for
nuclei in the range with 82$\leq$Z$\leq$118.

Formula for $\alpha$-decay half-life | RMSE |  $\overline{\delta}$ | F | No. of Coefficients
---|---|---|---|---
M Horoi (2020) Set-1 | 0.8355 | 0.5752 | 3.7597 | 24
M Horoi (2020) Set-3 | 0.9383 | 0.6345 | 4.3098 | 9
M Sobic. (2020) | 0.9833 | 0.6416 | 4.3813 | 24
Sobiczewski (2005) | 1.1788 | 0.5821 | 3.8203 | 7
Akre (2017) | 1.1813 | 0.6038 | 4.0162 | 20
SPm1 (2009) | 1.2242 | 0.6964 | 4.9699 | 7
VSm1 (2009) | 1.2835 | 0.7403 | 5.4988 | 8
NUDL (2018) | 1.2860 | 0.6721 | 4.7004 | 4
MSLB (2019) | 1.2978 | 0.7111 | 5.1411 | 16
Rm (2009) | 1.3119 | 0.7220 | 5.2722 | 12
NSobicP (2018) | 1.3416 | 0.7779 | 5.9967 | 4
New RenA (2019) | 1.3462 | 0.6912 | 4.9116 | 20
NRDX (2008) | 1.3469 | 0.8347 | 6.8345 | 3
MVS (2019) | 1.3646 | 0.6931 | 4.9331 | 24
Universal curve (2011) | 1.4081 | 0.7932 | 6.2112 | 5
MUDL (2019) | 1.4357 | 0.8168 | 6.5581 | 20
N Ro (2018) | 1.4426 | 0.9017 | 7.9752 | 4
mB1 (2016) | 1.5392 | 0.9973 | 9.9387 | 7
SemFIS2 (2010) | 1.6678 | 1.0076 | 10.1755 | 11
mB2 (2016) | 1.8361 | 1.0918 | 12.3547 | 12
VSm2 (2009) | 1.9839 | 1.0858 | 12.1847 | 16
Horoi (2004) | 2.5237 | 2.3267 | 212.1864 | 5
Manjunatha (2019) | 2.6752 | 1.5039 | 31.9113 | 4
SPm2 (2009) | 2.8200 | 1.4614 | 28.9302 | 16

The root mean square error (RMSE), mean deviation ($\overline{\delta}$) and
the index F$=$10${}^{\overline{\delta}}$ expressed by the following equations
are deduced for all the above mentioned formulas and mentioned in Table 4. The
total number of fitted coefficients are also listed in front of each formulas.
A justified comparison among formulas with coefficients $>$10 leads the
modified formulas with a good merit. Also, the RMSE and $\overline{\delta}$
are plotted in Fig. 3 to depict a comparison of the above mentioned formulas.
It is worth to mention here that few formulas with comparatively lesser number
of coefficients (Sobiczewski, NUDL etc.) still possess a good accuracy for the
calculation of half lives.

$\displaystyle RMSE$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{N_{nucl}}\sum^{N_{nucl}}_{i=1}\left(log\frac{T^{i}_{th}}{T^{i}_{exp}}\right)^{2}}$
(7) $\displaystyle\overline{\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{nucl}}\sum^{N_{nucl}}_{i=1}\left|log\frac{T^{i}_{th}}{T^{i}_{exp}}\right|$
(8) $\displaystyle F$ $\displaystyle=$ $\displaystyle 10^{\overline{\delta}}$
(9)

Figure 3: Root mean square error (RMSE) and mean deviation
($\overline{\delta}$) of various formulas for $\alpha$-decay half-lives for
the nuclei in the range of 82$\leq$Z$\leq$118.

It is gratifying to note from Table 4 and Fig. 3 that the modified formulas
introduced in present work, result with minimum error and deviation
comparative to other 21 considered formulas. In fact, comparison shown in the
Table 4 for all the statistical parameters certifies the applicability of the
MHF-2020 (set-1 and set-2) and MSF-2020, and therefore will be employed for
prediction of decay modes, to calculate $\alpha$-decay half-lives and to
construct decay chains in the unknown territory of superheavy land.

Table 5: Root mean square error (RMSE), mean deviation ($\overline{\delta}$),
index F$=$10${}^{\overline{\delta}}$ (in the unit of hour) and No. of
coefficients of various formulas for SF half-lives calculated for nuclei in
the range with 82$\leq$Z$\leq$118.

Formula for SF half-life | RMSE |  $\overline{\delta}$ | F | No. of Coefficients
---|---|---|---|---
Modified Bao (MBF-2020) | 2.1103 | 1.7389 | 0.0152 | 7
Bao (2015) | 2.8379 | 2.3107 | 0.0568 | 8
Xu (2008) | 3.1387 | 2.5305 | 0.0942 | 7
Santosh (2010) | 3.3064 | 2.7297 | 0.1491 | 5
Karpov (2012) | 5.6784 | 5.1697 | 41.0536 | 9
Ren (2005) | 5.8511 | 4.0182 | 2.8967 | 7
Silisteanu (2015) | 6.0458 | 5.6374 | 120.5425 | 10

Figure 4: Root mean square error (RMSE) and mean deviation
($\overline{\delta}$) of various formulas for SF half-lives for the nuclei in
the range of 82$\leq$Z$\leq$118.

In a similar manner, to find more precise formula for spontaneous fission: the
mode which compete with $\alpha$-decay, we apply the formula given by Ren et
al. [48], Xu et al. [49], Santosh et al. [50], Karpov et al. [51], Silisteanu
et al. [52], Bao et al. [53] along with the modified Bao formula of present
work. Spontaneous fission half-lives are calculated by using above all
formulas for 49 nuclei for which SF-half-lives are known and consequently a
few statistical parameters are obtained which are listed in Table 5 and also
shown in Fig. 4. The comparison displays improvement in the prediction of SF
half-lives after the modification in the formula of Bao et al. [53] which
earmark the use of modified Bao formula (MBF-2020) to analyze the competition
between $\alpha$-decay and SF.

As mentioned above, four methods of machine learning i.e. Decision Trees
(DTs), Random Forest (RF), Multilayer Perceptron (MLP) neural network, and
XGBoost are tested for $\alpha$-decay and SF half-lives considering 366 and 49
nuclei, respectively. We apply 5-fold cross-validation, and accordingly, the
dataset is split into 5 groups. Using this 5-fold cross-validation, root mean
square error (RMSE) and mean deviation ($\overline{\delta}$) of considered
methods of machine learning are mentioned in Table 6.

Table 6: Root mean square error (RMSE) and mean deviation
($\overline{\delta}$) of various methods of machine learning for the
$\alpha$-decay half-lives and SF half-lives data in the range with
82$\leq$Z$\leq$118.

Machine learning | RMSE |  | $\overline{\delta}$
---|---|---|---
method |  $\alpha$-decay | SF |  |  $\alpha$-decay | SF
MLP neural network | 0.9774 | 2.5753 |  | 0.6459 | 2.5177
XGBoost | 1.0245 | 3.1719 |  | 0.6468 | 2.6695
Random Forest | 1.1134 | 3.2748 |  | 0.8035 | 2.9100
Decision Trees (DTs) | 1.4449 | 4.0971 |  | 1.0780 | 3.6270

The Table 6 establishes MLP neural network and XGBoost methods as the more
appropriate method of machine learning among the considered ones because for
these two methods the RMSE and $\overline{\delta}$ for $\alpha$-decay and SF
half-lives are found very similar to what are found for modified empirical
formulas (please see Tables 4 and 5). Therefore, in the following, we will
employ MLP and XGBoost for the prediction of $\alpha$-decay half-lives. For SF
half-lives, since the training data is very limited therefore predictions of
MLP neural network for such a small dataset are not found reasonable. As a
result, we opt the Random Forest (RF) method to predict SF half-lives along
with XGBoost algorithm.

With the above-mentioned selection of empirical formulas of $\alpha$-decay and
SF half-lives along with machine learning methods, in Table 7, we predict the
decay modes of nuclei in the range 112$\leq$Z$\leq$118 to demonstrate the
predictive power of modified formulas and machine learning methods. It is
important to mention here that in all the formulas of $\alpha$-decay
considered here (3 Proposed$+$21 others), the contribution of centrifugal
potential is not included. The calculations show that favored $\alpha$-decay
in which the orbital angular momentum taken by the $\alpha$-particle is zero
($l$$=$0), can be efficaciously described by the considered formulas. However,
for unfavored $\alpha$-decay ($l$$\neq$0) one needs to take into account the
$l$ dependence which is left for our upcoming work.

In Table 7, a few nuclei from 112$\leq$Z$\leq$118 are taken for which change
in angular momentum $l$$=$0 (favoured) and the values of Qα are evaluated from
latest database [69]. Few available $log_{10}$T1/2 and decay modes are also
tabulated for comparison. To predict $\alpha$-decay half-lives we use modified
Horoi formula (MHF-2020) (set-1 and set-2) and modified Sobiczewski formula
(MSF-2020) which are constructed in this present work. Also, to predict SF
half-lives we employ modified Bao formula (MBF-2020) reported in this work.
These formulas are utilized to analyze the competition of $\alpha$-decay with
SF and consequently to predict probable decay mode. The same kind of
predictions are also described in Table 7 by using machine learning methods as
an alternate approach.

Table 7: Prediction of decay modes and half-lives from modified formulas and
machine learning for the range 112$\leq$Z$\leq$118.

Z | A | Experimental Data [69] | Modified formulas of present work | Machine learning methods
---|---|---|---|---
|  | Qα | l | log10T1/2(s) | Decay | log10T${}_{1/2}^{\alpha}$(s) | log10T${}_{1/2}^{SF}$(s) | Predicted | log10T${}_{1/2}^{\alpha}$(s) |  | log10T${}_{1/2}^{SF}$(s) | Predicted
|  | (MeV) |  |  | modes | MHF (Set-1) | MHF (Set-2) | MSF | MBF | Decay-modes | MLP | Xgboost |  | RF | Xgboost | Decay-modes
112 | 276 | 11.9 | 0 |  |  | -6.27 | -5.32 | -3.84 | 3.14 | $\alpha$ | -3.54 | -4.87 |  | 0.29 | -0.61 | $\alpha$
112 | 278 | 11.3 | 0 |  |  | -4.78 | -3.92 | -2.64 | -1.79 | $\alpha$ | -2.52 | -3.19 |  | 0.21 | -0.65 | $\alpha$
112 | 280 | 10.7 | 0 |  |  | -3.18 | -2.40 | -1.36 | -5.10 | SF | -1.19 | -1.75 |  | 0.00 | -0.75 | $\alpha$
112 | 282 | 10.2 | 0 | -3.30 | SF | -1.48 | -0.78 | -0.01 | -4.26 | SF | -0.51 | -0.31 |  | -0.02 | -0.68 | SF
112 | 284 | 9.6 | 0 | -1.00 | SF | 0.44 | 1.03 | 1.48 | -2.49 | SF | 0.73 | 1.15 |  | 0.05 | -0.49 | SF
114 | 284 | 10.8 | 0 | -2.60 | SF | -2.67 | -1.88 | -1.10 | 0.13 | $\alpha$ | -1.32 | -1.28 |  | -0.33 | -1.24 | $\alpha$
114 | 285 | 10.6 | 0 | -0.82 | $\alpha$ | -1.07 | -0.78 | -1.03 | 3.48 | $\alpha$ | -0.95 | -0.66 |  | -0.22 | -0.92 |  $\alpha$/SF
114 | 286 | 10.4 | 0 | -0.80 | SF/$\alpha$ | -1.36 | -0.64 | -0.03 | 1.16 | $\alpha$ | -0.96 | -0.18 |  | -0.16 | -0.99 | SF/$\alpha$
114 | 287 | 10.2 | 0 | -0.29 | $\alpha$ | 0.12 | 0.41 | 0.07 | 4.64 | $\alpha$ | -0.04 | 0.36 |  | -0.16 | -0.99 | SF
114 | 288 | 10.1 | 0 | -0.28 | $\alpha$ | -0.43 | 0.25 | 0.80 | 2.42 | $\alpha$ | 0.02 | 0.56 |  | -0.16 | -0.99 | SF
114 | 289 | 10.0 | 0 | -0.01 | $\alpha$ | 0.70 | 0.98 | 0.63 | 6.00 | $\alpha$ | 0.05 | 0.80 |  | -0.16 | -0.99 | SF
116 | 290 | 11.0 | 0 | -1.82 | $\alpha$ | -2.67 | -1.81 | -0.94 | 4.41 | $\alpha$ | -1.44 | -1.20 |  | -0.16 | -0.99 | $\alpha$
116 | 292 | 10.8 | 0 | -1.74 | $\alpha$ | -2.04 | -1.20 | -0.31 | 4.81 | $\alpha$ | -1.46 | -0.65 |  | -0.16 | -0.99 | $\alpha$
118 | 294 | 11.6 | 0 | -3.05 | $\alpha$ | -3.80 | -2.82 | -1.72 | 6.15 | $\alpha$ | -1.59 | -0.40 |  | -0.16 | -0.99 | $\alpha$

Figure 5: (Colour online) $\alpha$ half-lives for the experimentally known
decay chains [21, 23, 76, 77]: 294Og, 294Ts, 293Ts, 291Lv, 288Mc and 287Mc.

Table 7 itself expresses the predictive power of modified formulas and also
the use of machine learning methods as the decay modes are anticipated
effectively. In fact, not only the decay modes are found in an excellent match
with available experimental/estimated decay modes but also the half-lives are
found in agreement with experimental half-lives. In addition, we have compared
our estimation of half-lives for the experimentally known decay chains [21,
23, 76, 77]: 294Og, 294Ts, 293Ts, 291Lv, 288Mc and 287Mc in Fig. 5. The
$\alpha$-decay half-lives are calculated by employing MHF-2020 (set-1 and
set-2) and MSF-2020 which are constructed in this present work along with
machine learning methods: XGBoost and MLP. Satisfyingly, these half-lives are
found in a reasonable match with the experimental decay half-lives mentioned
in Fig. 5 with error bars.

This agreement countenances the use of formulas and methods reported in the
present work to estimate decay modes, decay chains, and half-lives for the
unknown territory of superheavy nuclei to accompaniment the crucial
experiments eyeing on heavier elements with Z$\geq$118 [15, 25, 8, 26]. With
this in view, we employ the above-mentioned formulas and machine learning
methods to construct the possible decay chains for isotopes of Og and Z$=$119
which are of current interest. The decay chains are constructed for neutron
number lying between 168$\leq$N$\leq$184 as observed from experimental decay
chains [21, 23, 76, 77] and speculated for island of stability at N$\sim$184
[15, 25, 8, 26]. To apply empirical formulas for calculations of
$\alpha$-decay half-lives the experimental and evaulated Qα [69] are used. For
the rest of the nuclei, the values of Qα can be taken from theoretical
approaches. In this regard, we have performed calculations using relativistic
mean-field theory [78, 79, 80, 81] for the nuclei in the range of
111$\leq$Z$\leq$118\. The values of Qα are also taken directly from FRDM [82],
RCHB [83] and KTUY [84] mass tables for comparison. For determination of Qα,
one can also use the very efficient five-parameter formula derived from LDM
considerations by Dong et al. [85, 86], which is given by:

$\displaystyle
Q_{\alpha}(MeV)=aZA^{-4/3}(3A-Z)+b\left(\frac{N-Z}{A}\right)^{2}$
$\displaystyle+c\left[\frac{|N-152|}{N}-\frac{|N-154|}{N-2}\right]$
$\displaystyle+d\left[\frac{|Z-110|}{N}-\frac{|Z-112|}{Z-2}\right]+e$ (10)

In addition to the above treatments for Qα, we have applied machine learning
methods as an alternate path. To train a machine, we have used all
experimental/evaulated Qα values ($\approx$709) between 82$\leq$Z$\leq$110
with the 5-fold cross-validation as described above. Once again XGBoost is
found with minimum RMSE among the considered machine learning methods and,
therefore, we have predicted Qα values for the nuclei in the range
111$\leq$Z$\leq$118 using XGBoost algorithm. The errors in the values of Qα
for the nuclei in the range 111$\leq$Z$\leq$118 calculated by RMF [78, 79, 80,
81], FRDM [82], RCHB [83], KTUY [84], Dong et al. formula [85, 86] and XGBoost
are plotted in Fig. 6. RMSE for all these approaches are also mentioned in the
corresponding panels.

Figure 6: Errors in the values of Qα for the nuclei in range
111$\leq$Z$\leq$118.

It is clear from Fig. 6 and mentioned RMSE values that XGBoost is found closer
with experimental/evaulated Qα values comparative to the theoretical
approaches and the formula. Hence, for the prediction of half-lives for Og
isotopes and Z$=$119 isotopes, we will opt Qα values from XGBoost algorithm
where experimental Qα are not available. In this way, with the more accurate
Qα, we expect a more reliable prediction of $\alpha$-decay half-lives using
more precise formulas (MHF-2020 & MSF-2020, please see Table 4) reported in
the present work. In Tables 8 & 9, we show decay chains of Og isotopes in the
range 286$\leq$A$\leq$302\. As mentioned above, to speculate chances of
favoured and unfavoured $\alpha$-decay, we have also mentioned change in the
angular momentum ($l$) calculated from selection rule [87, 88] based on parity
and spin of parent and daughter nuclei taken from latest evaluated nuclear
properties table NUBASE2016 [89] or P. Möller [90]. More detailed analysis on
favoured and unfavoured $\alpha$-decay will be covered in our upcoming work.

Asterisk (*) values of Qα are taken from the XGBoost algorithm whereas all
other values are experimental/evaluated [69]. Using these Qα values,
$\alpha$-decay half-lives ($log_{10}$T${}_{1/2}^{\alpha}$(s)) are deduced by
modified Horoi formula (MHF-2020) (set-1 and set-2) and modified Sobiczewski
formula (MSF-2020), and also with MLP and XGBoost algorithm of machine
learning. Spontaneous fission half-lives ($log_{10}$T${}_{1/2}^{SF}$(s)) are
also mentioned which are calculated by using modified Bao formula (MBF-2020).
Consequently, decay modes are predicted for all the chains which are compared
with available experimental/estimated decay modes [69]. Similar calculations
and prediction are tabulated for Z$=$119 isotopes in the range
287$\leq$A$\leq$303 in Tables 10 & 11.

From Tables 8 to 11, it is indulging to note that our prediction of decay
modes and half-lives are in an excellent match with available experimental
data. 286-291Og and 287-296,302,303119 are found with very long $\alpha$-decay
chains (6$\alpha$/7$\alpha$) and are found as potential candidates for future
experiments. As far as the current experimental facilities are concerned,
50Ti+249Bk and 50Ti+249-251Cf reactions could lead to the discovery of
295,296119 and 291Og [8] which are found with 7$\alpha$ decay chains as per
our predictions. For 48Ca beam, the available heaviest target material is
251Cf (N = 153) which could lead to the production of 295Og, 296Og, and 297Og
[22] and from our prediction, these isotopes of Og are indeed found with
3$\alpha$/4$\alpha$ chain and their half-lives are found within the range of
current experimental facilities. Therefore, our predictions from the modified
formulas are expected to provide more extensive and precise information for
the upcoming experiments to reach the boundaries of the island of stability.

Table 8: Prediction of decay chains from modified formulas and machine
learning for Og isotopes. Qα and log10T1/2 are taken from Ref. [69], whereas
(*) values of Qα are taken from XGBoost algorithm as described in text. The
angular momentum ($l$) is calculated from selection rule [87, 88] based on
parity and spin of parent and daughter nuclei, which are taken from latest
evaluated nuclear properties table NUBASE2016 [89] or P. Möller [90].

Nucleus | Expt. | log10T${}_{1/2}^{\alpha}$(s) | log10T${}_{1/2}^{SF}$(s) | Decay Modes
---|---|---|---|---
| Qα | l | log10T1/2(s) | MHF (Set-1) | MHF (Set-2) | MSF | XGBoost | MLP | MBF | Predicted | Expt.
286Og | 11.17* | 0 |  | -2.19 | -1.39 | -1.37 | -1.07 | -2.83 | 1.59 | $\alpha$ |
282Lv | 11.18* | 0 |  | -2.92 | -2.12 | -1.83 | -1.71 | -2.97 | -0.45 | $\alpha$ |
278Fl | 11.35* | 0 |  | -4.06 | -3.23 | -2.57 | -2.74 | -3.21 | -2.71 | $\alpha$ |
274Cn | 10.84* | 0 |  | -3.21 | -2.50 | -1.99 | -1.97 | -2.72 | 3.01 | $\alpha$ |
270Ds | 11.12 | 0 | -4.00 | -4.66 | -3.92 | -2.97 | -3.34 | -3.33 | 0.21 | $\alpha$ | $\alpha$
266Hs | 10.35 | 0 | -2.64 | -3.09 | -2.50 | -1.87 | -1.91 | -2.29 | -1.39 | $\alpha$ | $\alpha$
262Sg | 9.60 | 0 | -2.16 | -1.48 | -1.04 | -0.71 | -0.55 | -0.29 | -2.09 | SF | SF
287Og | 11.15* | 0 |  | -1.31 | -0.94 | -1.45 | -0.96 | -2.77 | 5.05 | $\alpha$ |
283Lv | 11.09* | 2 |  | -1.80 | -1.45 | -1.84 | -1.39 | -2.84 | 3.33 | $\alpha$ |
279Fl | 11.29* | 5 |  | -2.99 | -2.67 | -2.87 | -2.53 | -2.98 | 0.31 | $\alpha$ |
275Cn | 11.12* | 5 |  | -3.18 | -2.88 | -3.01 | -2.69 | -2.78 | 6.14 | $\alpha$ |
271Ds | 10.87 | 2 | -2.79 | -3.17 | -2.88 | -2.98 | -2.66 | -2.92 | 5.42 | $\alpha$ | $\alpha$
267Hs | 10.04 | 4 | -1.28 | -1.50 | -1.23 | -1.44 | -1.09 | -1.11 | 3.11 | $\alpha$ | $\alpha$
263Sg | 9.40 | 2 | 0.00 | -0.25 | -0.03 | -0.27 | -0.02 | 0.29 | 0.27 | $\alpha$ | $\alpha$
288Og | 11.12* | 0 |  | -2.06 | -1.28 | -1.16 | -0.81 | -2.62 | 3.17 | $\alpha$ |
284Lv | 10.99* | 0 |  | -2.38 | -1.61 | -1.34 | -1.09 | -2.51 | 1.44 | $\alpha$ |
280Fl | 11.14* | 0 |  | -3.53 | -2.73 | -2.06 | -2.12 | -2.81 | -1.40 | $\alpha$ |
276Cn | 11.90 | 0 |  | -6.27 | -5.32 | -3.84 | -4.87 | -3.54 | 3.14 | $\alpha$ |
272Ds | 10.80 | 0 |  | -3.86 | -3.14 | -2.22 | -2.49 | -2.40 | 3.93 | $\alpha$ |
268Hs | 9.62 | 0 | -0.40 | -0.89 | -0.42 | -0.16 | 0.01 | -0.38 | 1.57 | $\alpha$ | $\alpha$
264Sg | 9.21 | 0 | -1.43 | -0.26 | 0.13 | 0.35 | 0.44 | 0.93 | -1.10 | SF | SF
289Og | 11.09* | 0 |  | -1.17 | -0.81 | -1.30 | -0.71 | -2.59 | 6.65 | $\alpha$ |
285Lv | 10.99* | 2 |  | -1.56 | -1.23 | -1.60 | -1.07 | -2.40 | 4.69 | $\alpha$ |
281Fl | 10.96* | 2 |  | -2.14 | -1.83 | -2.07 | -1.63 | -2.41 | 1.41 | $\alpha$ |
277Cn | 11.62 | 5 |  | -4.53 | -4.26 | -4.16 | -4.03 | -2.86 | 3.36 | $\alpha$ |
273Ds | 11.37 | 5 | -3.77 | -4.53 | -4.26 | -4.13 | -4.00 | -3.11 | 6.83 | $\alpha$ | $\alpha$
269Hs | 9.34 | 2 | 0.99 | 0.66 | 0.89 | 0.55 | 0.72 | 0.66 | 5.11 | $\alpha$ | $\alpha$
265Sg | 9.05 | 4 | 1.16 | 0.87 | 1.05 | 0.80 | 0.86 | 1.08 | 2.47 | $\alpha$ | $\alpha$
290Og | 11.05* | 0 |  | -1.92 | -1.13 | -0.92 | -0.59 | -2.32 | 4.50 | $\alpha$ |
286Lv | 10.92* | 0 |  | -2.23 | -1.46 | -1.07 | -0.89 | -2.21 | 2.97 | $\alpha$ |
282Fl | 10.69* | 0 |  | -2.23 | -1.51 | -1.03 | -0.94 | -1.69 | -0.95 | $\alpha$ |
278Cn | 11.31 | 0 |  | -4.78 | -3.92 | -2.64 | -3.19 | -2.52 | -1.79 | $\alpha$ |
274Ds | 11.70 | 0 |  | -6.47 | -5.54 | -3.73 | -5.05 | -3.01 | 4.96 | $\alpha$ |
270Hs | 9.07 | 0 | 1.34 | 0.98 | 1.36 | 1.33 | 1.40 | 0.84 | 4.52 | $\alpha$ | $\alpha$
266Sg | 8.80 | 0 | 1.32 | 1.13 | 1.47 | 1.55 | 1.48 | 1.31 | 0.62 | SF | SF
291Og | 11.07* | 2 |  | -1.15 | -0.80 | -1.24 | -0.66 | -2.24 | 7.93 | $\alpha$ |
287Lv | 10.82* | 0 |  | -1.08 | -0.76 | -1.14 | -0.63 | -2.07 | 5.86 | $\alpha$ |
283Fl | 10.59* | 4 |  | -1.11 | -0.81 | -1.11 | -0.68 | -1.62 | 2.21 | $\alpha$ |
279Cn | 11.04 | 4 |  | -3.07 | -2.81 | -2.81 | -2.52 | -1.69 | -0.76 | $\alpha$ |
275Ds | 11.40 | 0 |  | -4.67 | -4.43 | -4.18 | -4.18 | -2.45 | 6.15 | $\alpha$ |
271Hs | 9.51 | 0 |  | 0.06 | 0.28 | 0.07 | 0.22 | 0.41 | 7.21 | $\alpha$ |
267Sg | 8.63 | 0 |  | 2.32 | 2.48 | 2.16 | 1.93 | 1.53 | 4.90 | $\alpha$ |
292Og | 11.02* | 0 |  | -1.90 | -1.10 | -0.74 | -0.55 | -1.99 | 6.14 | $\alpha$ |
288Lv | 10.77* | 0 |  | -1.82 | -1.06 | -0.62 | -0.53 | -1.79 | 3.40 | $\alpha$ |
284Fl | 10.80 | 0 | -2.60 | -2.67 | -1.88 | -1.10 | -1.28 | -1.32 | 0.13 | $\alpha$ | SF
280Cn | 10.73 | 0 |  | -3.18 | -2.40 | -1.36 | -1.75 | -1.19 | -5.10 | SF |
276Ds | 11.11 | 0 |  | -5.01 | -4.14 | -2.49 | -3.39 | -2.17 | 2.08 | $\alpha$ |
272Hs | 9.78 | 0 |  | -1.63 | -1.03 | -0.17 | -0.53 | 0.08 | 3.58 | $\alpha$ |
268Sg | 8.30 | 0 |  | 3.01 | 3.28 | 3.08 | 2.97 | 2.14 | 2.98 | $\alpha$ |
293Og | 11.01* | 2 |  | -0.99 | -0.66 | -1.06 | -0.54 | -2.08 | 9.89 | $\alpha$ |
289Lv | 11.10 | 2 |  | -1.94 | -1.65 | -1.84 | -1.44 | -1.58 | 6.64 | $\alpha$ |
285Fl | 10.56 | 0 | -0.82 | -1.07 | -0.78 | -1.03 | -0.66 | -0.95 | 3.48 | $\alpha$ | $\alpha$
281Cn | 10.45 | 4 | -0.89 | -1.45 | -1.19 | -1.31 | -1.03 | -0.98 | -2.64 | SF | $\alpha$
277Ds | 10.83 | 4 | -2.39 | -3.22 | -3.01 | -2.83 | -2.68 | -1.99 | 1.90 | $\alpha$ | $\alpha$
273Hs | 9.70 | 5 | -0.12 | -0.59 | -0.40 | -0.45 | -0.34 | 0.11 | 4.68 | $\alpha$ | $\alpha$
269Sg | 8.65 | 5 | 2.27 | 2.21 | 2.36 | 2.12 | 1.88 | 2.06 | 5.35 | $\alpha$ | $\alpha$
294Og | 11.60 | 0 | -3.05 | -3.80 | -2.82 | -1.72 | -0.40 | -1.59 | 6.15 | $\alpha$ | $\alpha$
290Lv | 11.00 | 0 | -1.82 | -2.67 | -1.81 | -0.94 | -1.20 | -1.44 | 4.41 | $\alpha$ | $\alpha$
286Fl | 10.37 | 0 | -0.80 | -1.36 | -0.64 | -0.03 | -0.18 | -0.96 | 1.16 | $\alpha$ |  $\alpha$/SF
282Cn | 10.17 | 0 | -3.30 | -1.48 | -0.78 | -0.01 | -0.31 | -0.51 | -4.26 | SF | SF
278Ds | 10.47 | 0 |  | -3.24 | -2.44 | -1.05 | -1.78 | -1.20 | -4.39 | SF |
274Hs | 9.57 | 0 |  | -1.04 | -0.43 | 0.51 | -0.01 | -0.01 | 0.45 | $\alpha$ |
270Sg | 9.00 | 0 |  | 0.16 | 0.67 | 1.44 | 0.83 | 1.65 | 2.74 | $\alpha$ |

Table 9: Continue to Table 8.

Nucleus | Expt. | log10T${}_{1/2}^{\alpha}$(s) | log10T${}_{1/2}^{SF}$(s) | Decay Modes
---|---|---|---|---
| Qα | l | log10T1/2(s) | MHF (Set-1) | MHF (Set-2) | MSF | XGBoost | MLP | MBF | Predicted | Expt.
295Og | 10.88* | 0 |  | -0.65 | -0.32 | -0.72 | -0.24 | -1.62 | 8.45 | $\alpha$ |
291Lv | 10.89 | 2 | -2.20 | -1.38 | -1.10 | -1.30 | -0.94 | -1.69 | 7.59 | $\alpha$ | $\alpha$
287Fl | 10.16 | 0 | -0.29 | 0.12 | 0.41 | 0.07 | 0.36 | -0.04 | 4.64 | $\alpha$ | $\alpha$
283Cn | 9.94 | 7 | 0.60 | 0.08 | 0.34 | 0.10 | 0.29 | 0.32 | -0.56 | SF |  $\alpha$/SF
279Ds | 10.08 | 7 | -0.74 | -1.10 | -0.89 | -0.88 | -0.76 | -0.22 | -3.10 | SF | SF
275Hs | 9.44 | 0 | -0.82 | 0.20 | 0.38 | 0.32 | 0.30 | 0.13 | 1.54 | $\alpha$ | $\alpha$
271Sg | 8.89 | 5 | 2.16 | 1.31 | 1.44 | 1.37 | 1.08 | 2.00 | 3.74 | $\alpha$ | $\alpha$
296Og | 10.71* | 0 |  | -1.02 | -0.23 | 0.22 | 0.19 | -1.36 | 5.65 | $\alpha$ |
292Lv | 10.77 | 0 | -1.74 | -2.04 | -1.20 | -0.31 | -0.65 | -1.46 | 4.81 | $\alpha$ | $\alpha$
288Fl | 10.07 | 0 | -0.28 | -0.43 | 0.25 | 0.80 | 0.56 | 0.02 | 2.42 | $\alpha$ | $\alpha$
284Cn | 9.60 | 0 | -1.00 | 0.44 | 1.03 | 1.48 | 1.15 | 0.73 | -2.49 | SF | SF
280Ds | 9.81 | 0 |  | -1.18 | -0.49 | 0.58 | -0.04 | 0.25 | -5.25 | SF |
276Hs | 9.28 | 0 |  | -0.13 | 0.47 | 1.40 | 0.63 | 0.41 | -4.96 | SF |
272Sg | 8.70 | 0 |  | 1.18 | 1.68 | 2.43 | 1.69 | 1.13 | -1.75 | SF |
297Og | 10.52* | 0 |  | 0.41 | 0.76 | 0.25 | 0.66 | -1.12 | 8.31 | $\alpha$ |
293Lv | 10.68 | 2 | -1.28 | -0.81 | -0.53 | -0.73 | -0.43 | -1.19 | 8.10 | $\alpha$ | $\alpha$
289Fl | 9.97 | 0 | -0.01 | 0.70 | 0.98 | 0.63 | 0.80 | 0.05 | 6.00 | $\alpha$ | $\alpha$
285Cn | 9.32 | 2 | 1.48 | 2.14 | 2.41 | 1.96 | 1.91 | 1.06 | 1.55 | $\alpha$ | $\alpha$
281Ds | 9.51 | 0 | 1.11 | 0.69 | 0.90 | 0.76 | 0.67 | 0.70 | -1.83 | SF | SF/$\alpha$
277Hs | 9.05 | 0 | -2.52 | 1.50 | 1.66 | 1.54 | 1.24 | -0.71 | -2.69 | SF | $\alpha$
273Sg | 8.20* | 0 |  | 3.89 | 4.00 | 3.73 | 3.52 | 2.32 | -0.51 | SF |
298Og | 10.40* | 0 |  | -0.04 | 0.70 | 1.06 | 0.93 | -1.11 | 5.18 | $\alpha$ |
294Lv | 9.90* | 0 |  | 0.98 | 1.61 | 1.81 | 1.61 | 0.07 | 5.75 | $\alpha$ |
290Fl | 9.39* | 0 |  | 2.07 | 2.60 | 2.63 | 2.34 | 0.68 | 4.16 | $\alpha$ |
286Cn | 9.07* | 0 |  | 2.41 | 2.90 | 3.01 | 2.55 | 1.11 | -0.60 | SF |
282Ds | 9.04* | 0 |  | 1.57 | 2.11 | 2.64 | 1.98 | -0.09 | -4.17 | SF |
278Hs | 8.47* | 0 |  | 2.96 | 3.40 | 3.72 | 3.28 | 0.57 | -7.12 | SF |
274Sg | 8.13* | 0 |  | 3.42 | 3.83 | 4.23 | 3.82 | 2.32 | -6.37 | SF |
299Og | 10.02* | 2 |  | 2.04 | 2.42 | 1.71 | 1.89 | 0.05 | 7.57 | $\alpha$ |
295Lv | 9.54* | 0 |  | 2.91 | 3.27 | 2.53 | 2.52 | 0.59 | 8.86 | $\alpha$ |
291Fl | 9.23* | 0 |  | 3.24 | 3.57 | 2.90 | 2.75 | 0.86 | 7.52 | $\alpha$ |
287Cn | 9.03* | 2 |  | 3.15 | 3.42 | 2.90 | 2.68 | 0.48 | 3.39 | $\alpha$ |
283Ds | 8.84* | 2 |  | 3.04 | 3.25 | 2.88 | 2.65 | 0.54 | -0.62 | SF |
279Hs | 8.48* | 0 |  | 3.60 | 3.76 | 3.45 | 3.30 | 0.67 | -3.67 | SF |
275Sg | 8.07* | 4 |  | 4.40 | 4.50 | 4.25 | 4.11 | 2.33 | -3.33 | SF |
300Og | 9.75* | 0 |  | 2.36 | 2.94 | 2.77 | 2.58 | 0.34 | 3.98 | $\alpha$ |
296Lv | 9.33* | 0 |  | 3.14 | 3.64 | 3.40 | 3.11 | 0.80 | 5.74 | $\alpha$ |
292Fl | 9.09* | 0 |  | 3.18 | 3.67 | 3.59 | 3.23 | 0.49 | 5.71 | $\alpha$ |
288Cn | 8.83* | 0 |  | 3.30 | 3.77 | 3.85 | 3.45 | 0.51 | -0.16 | SF |
284Ds | 8.86* | 0 |  | 2.20 | 2.76 | 3.35 | 2.64 | 0.54 | -2.83 | SF |
280Hs | 8.37* | 0 |  | 3.29 | 3.78 | 4.26 | 3.77 | 1.09 | -6.01 | SF |
276Sg | 7.98* | 0 |  | 3.98 | 4.42 | 4.93 | 4.50 | 2.37 | -7.43 | SF |
301Og | 9.47* | 2 |  | 3.95 | 4.38 | 3.40 | 3.36 | 0.62 | 6.06 | $\alpha$ |
297Lv | 9.19* | 2 |  | 4.19 | 4.58 | 3.69 | 3.62 | 0.62 | 8.49 | $\alpha$ |
293Fl | 8.92* | 0 |  | 4.41 | 4.75 | 3.96 | 3.89 | 0.16 | 8.68 | $\alpha$ |
289Cn | 8.85* | 0 |  | 3.82 | 4.09 | 3.54 | 3.44 | 0.63 | 2.21 | $\alpha$ |
285Ds | 8.80* | 2 |  | 3.16 | 3.36 | 3.05 | 2.91 | 0.49 | 1.34 | $\alpha$ |
281Hs | 8.33* | 2 |  | 4.17 | 4.32 | 4.02 | 3.96 | 1.17 | -1.79 | SF |
277Sg | 7.92* | 0 |  | 4.98 | 5.07 | 4.83 | 4.80 | 2.40 | -5.15 | SF |
302Og | 9.34* | 0 |  | 3.98 | 4.46 | 4.00 | 3.82 | 0.70 | 1.96 | $\alpha$ |
298Lv | 9.00* | 0 |  | 4.48 | 4.91 | 4.48 | 4.33 | -0.06 | 5.01 | $\alpha$ |
294Fl | 8.91* | 0 |  | 3.87 | 4.36 | 4.30 | 3.98 | 0.37 | 5.69 | $\alpha$ |
290Cn | 8.75* | 0 |  | 3.55 | 4.07 | 4.31 | 3.87 | 0.64 | 0.09 | SF |
286Ds | 8.77* | 0 |  | 2.41 | 3.03 | 3.81 | 3.07 | 0.49 | -1.44 | SF |
282Hs | 8.25* | 0 |  | 3.67 | 4.19 | 4.83 | 4.27 | 1.19 | -4.08 | SF |
278Sg | 7.74* | 0 |  | 4.98 | 5.43 | 5.93 | 5.71 | 2.88 | -7.64 | SF |

Table 10: Prediction of decay chains from modified formulas and machine
learning for Z$=$119 isotopes. Qα and log10T1/2 are taken from Ref. [69],
whereas (*) values of Qα are taken from XGBoost algorithm as described in
text. The angular momentum ($l$) is calculated from selection rule [87, 88]
based on parity and spin of parent and daughter nuclei, which are taken from
latest evaluated nuclear properties table NUBASE2016 [89] or P. Möller [90].

Nucleus | Expt. | log10T${}_{1/2}^{\alpha}$(s) | log10T${}_{1/2}^{SF}$(s) | Decay Modes
---|---|---|---|---
| Qα | l | log10T1/2(s) | MHF (Set-1) | MHF (Set-2) | MSF | XGBoost | MLP | MBF | Predicted | Expt.
287119 | 11.21* | 2 |  | -0.73 | -0.82 | -0.83 | -0.92 | -2.70 | 6.02 | $\alpha$ |
283Ts | 11.29* | 2 |  | -1.49 | -1.73 | -1.64 | -1.75 | -3.12 | 4.00 | $\alpha$ |
279Mc | 11.39* | 4 |  | -2.30 | -2.68 | -2.50 | -2.62 | -3.37 | 1.64 | $\alpha$ |
275Nh | 11.05* | - |  | -2.13 | -2.41 | -2.28 | -2.28 | -2.78 | 4.87 | $\alpha$ |
271Rg | 10.95* | 1 |  | -2.49 | -2.79 | -2.64 | -2.60 | -2.78 | 3.36 | $\alpha$ |
267Mt | 10.90 | 3 |  | -2.99 | -3.31 | -3.15 | -3.08 | -3.02 | 3.32 | $\alpha$ |
263Bh | 10.10 | 2 |  | -1.71 | -1.80 | -1.76 | -1.56 | -1.14 | 1.42 | $\alpha$ |
288119 | 11.92* | - |  | -1.71 | -2.42 | -1.80 | -0.82 | -2.71 | 7.49 | $\alpha$ |
284Ts | 11.88* | - |  | -2.15 | -2.96 | -2.21 | -1.23 | -3.21 | 5.75 | $\alpha$ |
280Mc | 11.63* | - |  | -2.19 | -2.95 | -2.23 | -1.54 | -3.44 | 2.98 | $\alpha$ |
276Nh | 11.40* | - |  | -2.27 | -2.99 | -2.29 | -1.65 | -3.51 | 7.67 | $\alpha$ |
272Rg | 11.20 | - | -2.42 | -2.40 | -3.11 | -2.40 | -1.76 | -3.58 | 6.42 | $\alpha$ | $\alpha$
268Mt | 10.67 | - | -1.68 | -1.86 | -2.36 | -1.86 | -1.63 | -1.92 | 5.22 | $\alpha$ | $\alpha$
264Bh | 9.96 | - | -0.36 | -0.87 | -1.05 | -0.87 | -0.56 | -0.92 | 3.55 | $\alpha$ | $\alpha$
289119 | 11.19* | 2 |  | -0.72 | -0.80 | -0.81 | -0.76 | -2.74 | 7.65 | $\alpha$ |
285Ts | 11.17* | 2 |  | -1.24 | -1.42 | -1.36 | -1.30 | -2.73 | 5.98 | $\alpha$ |
281Mc | 11.14* | 2 |  | -1.77 | -2.05 | -1.91 | -1.85 | -2.86 | 3.08 | $\alpha$ |
277Nh | 11.36* | 0 |  | -2.83 | -3.30 | -3.04 | -3.04 | -3.19 | 5.32 | $\alpha$ |
273Rg | 10.90 | 5 |  | -2.40 | -2.72 | -2.54 | -2.42 | -2.65 | 6.90 | $\alpha$ |
269Mt | 10.50 | 3 |  | -2.07 | -2.30 | -2.16 | -2.01 | -2.06 | 5.62 | $\alpha$ |
265Bh | 9.68 | 2 | -0.05 | -0.63 | -0.62 | -0.61 | -0.46 | -0.14 | 3.34 | $\alpha$ | $\alpha$
290119 | 11.16* | - |  | -0.19 | -0.35 | -0.29 | -0.62 | -2.64 | 9.45 | $\alpha$ |
286Ts | 11.12* | - |  | -0.65 | -0.95 | -0.73 | -1.14 | -2.66 | 7.52 | $\alpha$ |
282Mc | 11.00* | - |  | -0.93 | -1.29 | -0.98 | -1.42 | -2.73 | 4.71 | $\alpha$ |
278Nh | 11.85 | - | -3.62 | -3.15 | -4.20 | -3.14 | -4.37 | -3.64 | 6.88 | $\alpha$ | $\alpha$
274Rg | 11.48 | - | -2.19 | -2.97 | -3.90 | -2.94 | -3.97 | -2.85 | 9.62 | $\alpha$ | $\alpha$
270Mt | 10.18 | - | -2.30 | -0.80 | -1.05 | -0.81 | -1.17 | -1.54 | 7.08 | $\alpha$ | $\alpha$
266Bh | 9.43 | - | 0.23 | 0.40 | 0.50 | 0.38 | 0.19 | 0.49 | 4.94 | $\alpha$ | $\alpha$
291119 | 11.10* | 2 |  | -0.53 | -0.57 | -0.60 | -0.43 | -2.59 | 9.72 | $\alpha$ |
287Ts | 11.11* | 2 |  | -1.12 | -1.30 | -1.23 | -1.07 | -2.65 | 7.35 | $\alpha$ |
283Mc | 10.89* | 2 |  | -1.19 | -1.37 | -1.28 | -1.11 | -2.47 | 4.93 | $\alpha$ |
279Nh | 11.50 | 0 |  | -3.13 | -3.73 | -3.37 | -3.36 | -2.94 | -0.81 | $\alpha$ |
275Rg | 11.80 | 5 |  | -4.33 | -5.11 | -4.62 | -4.93 | -3.30 | 8.23 | $\alpha$ |
271Mt | 9.91 | 3 |  | -0.60 | -0.64 | -0.59 | -0.47 | -0.47 | 8.81 | $\alpha$ |
267Bh | 9.23 | 2 | 1.23 | 0.62 | 0.76 | 0.71 | 0.69 | 0.98 | 6.00 | $\alpha$ | $\alpha$
292119 | 11.11* | - |  | -0.11 | -0.24 | -0.19 | -0.41 | -2.62 | 11.28 | $\alpha$ |
288Ts | 11.05* | - |  | -0.53 | -0.80 | -0.59 | -0.92 | -2.33 | 8.87 | $\alpha$ |
284Mc | 10.74* | - |  | -0.40 | -0.61 | -0.45 | -0.75 | -1.90 | 6.16 | $\alpha$ |
280Nh | 11.20 | - |  | -1.90 | -2.60 | -1.89 | -1.45 | -2.01 | 0.32 | $\alpha$ |
276Rg | 11.50 | - |  | -2.12 | -2.87 | -2.10 | -2.85 | -1.90 | 6.42 | $\alpha$ |
272Mt | 10.05 | - |  | -0.50 | -0.72 | -0.50 | -0.87 | -0.34 | 9.74 | $\alpha$ |
268Bh | 9.00 | - |  | 1.52 | 1.88 | 1.50 | 1.27 | 1.07 | 7.56 | $\alpha$ |
293119 | 11.09* | 2 |  | -0.51 | -0.57 | -0.57 | -0.36 | -2.49 | 11.75 | $\alpha$ |
289Ts | 10.99* | 2 |  | -0.85 | -1.01 | -0.94 | -0.76 | -2.06 | 8.50 | $\alpha$ |
285Mc | 10.63* | 4 |  | -0.58 | -0.66 | -0.62 | -0.46 | -1.58 | 6.02 | $\alpha$ |
281Nh | 11.00 | 2 |  | -2.03 | -2.45 | -2.18 | -2.10 | -1.70 | 0.44 | $\alpha$ |
277Rg | 11.20 | 1 |  | -3.06 | -3.68 | -3.27 | -3.26 | -1.86 | 4.56 | $\alpha$ |
273Mt | 10.81 | 3 |  | -2.78 | -3.30 | -2.94 | -2.90 | -1.83 | 8.08 | $\alpha$ |
269Bh | 8.60 | 2 |  | 2.53 | 2.93 | 2.74 | 2.31 | 1.58 | 7.61 | $\alpha$ |
294119 | 11.04* | - |  | 0.03 | -0.08 | -0.04 | -0.26 | -2.22 | 13.24 | $\alpha$ |
290Ts | 10.98* | - |  | -0.37 | -0.63 | -0.42 | -0.76 | -1.96 | 10.03 | $\alpha$ |
286Mc | 10.58* | - |  | -0.03 | -0.16 | -0.07 | -0.35 | -1.14 | 7.36 | $\alpha$ |
282Nh | 10.75 | - | -1.15 | -0.96 | -1.41 | -0.96 | -1.48 | -1.20 | 2.13 | $\alpha$ | $\alpha$
278Rg | 10.47 | - | -2.38 | -0.88 | -1.31 | -0.87 | -1.39 | -1.25 | 1.72 | $\alpha$ | $\alpha$
274Mt | 10.60 | - | -0.36 | -1.72 | -2.40 | -1.66 | -2.38 | -1.27 | 8.86 | $\alpha$ | $\alpha$
270Bh | 9.06 | - | 1.78 | 1.39 | 1.63 | 1.38 | 1.09 | 1.20 | 9.45 | $\alpha$ | $\alpha$
295119 | 11.01* | 2 |  | -0.33 | -0.39 | -0.38 | -0.21 | -2.12 | 12.05 | $\alpha$ |
291Ts | 11.50 | 2 |  | -1.99 | -2.53 | -2.20 | -2.14 | -2.16 | 10.10 | $\alpha$ |
287Mc | 10.76 | 2 | -1.49 | -0.89 | -1.11 | -0.96 | -0.87 | -1.23 | 6.97 | $\alpha$ | $\alpha$
283Nh | 10.51 | 2 | -1.00 | -0.88 | -1.10 | -0.93 | -0.86 | -0.92 | 2.13 | $\alpha$ | $\alpha$
279Rg | 10.52 | 1 | -0.77 | -1.49 | -1.86 | -1.58 | -1.55 | -0.95 | 0.36 | $\alpha$ | $\alpha$
275Mt | 10.48 | 5 | -2.01 | -1.99 | -2.47 | -2.10 | -2.10 | -1.40 | 5.14 | $\alpha$ | $\alpha$
271Bh | 9.42 | 0 |  | 0.12 | 0.02 | 0.16 | 0.09 | 0.88 | 7.60 | $\alpha$ |

Table 11: Continue to Table 10.

Nucleus | Expt. | log10T${}_{1/2}^{\alpha}$(s) | log10T${}_{1/2}^{SF}$(s) | Decay Modes
---|---|---|---|---
| Qα | l | log10T1/2(s) | MHF (Set-1) | MHF (Set-2) | MSF | XGBoost | MLP | MBF | Predicted | Expt.
296119 | 11.03* | - |  | 0.05 | -0.11 | -0.01 | -0.30 | -1.67 | 11.68 | $\alpha$ |
292Ts | 10.14 | - |  | 1.56 | 1.97 | 1.48 | 1.33 | -0.19 | 11.21 | $\alpha$ |
288Mc | 10.75 | - | -1.06 | -0.41 | -0.74 | -0.42 | -0.87 | -1.18 | 8.31 | $\alpha$ | $\alpha$
284Nh | 10.28 | - | -0.32 | 0.11 | -0.05 | 0.10 | -0.27 | -0.53 | 3.82 | $\alpha$ | $\alpha$
280Rg | 11.20 | - | -2.38 | -2.41 | -3.44 | -2.33 | -0.37 | -0.62 | -0.61 | $\alpha$ | $\alpha$
276Mt | 10.10 | - | -0.14 | -0.59 | -1.02 | -0.54 | -1.12 | -0.39 | 4.29 | $\alpha$ | $\alpha$
272Bh | 9.30 | - | 1.00 | 0.79 | 0.75 | 0.81 | 0.36 | 1.08 | 6.69 | $\alpha$ | $\alpha$
297119 | 10.93* | 2 |  | -0.14 | -0.19 | -0.17 | -0.05 | -1.61 | 9.97 | $\alpha$ |
293Ts | 11.29 | 2 | -1.85 | -1.53 | -2.01 | -1.69 | -1.65 | -1.87 | 9.45 | $\alpha$ | $\alpha$
289Mc | 10.51 | 4 | -0.66 | -0.28 | -0.41 | -0.29 | -0.25 | -0.91 | 7.91 | $\alpha$ | $\alpha$
285Nh | 10.01 | 3 | 0.74 | 0.39 | 0.41 | 0.45 | 0.43 | 0.28 | 3.52 | $\alpha$ | $\alpha$
281Rg | 9.90 | 7 | 1.23 | 0.08 | -0.02 | 0.12 | 0.07 | 0.22 | -0.17 | SF | SF
277Mt | 9.91 | 0 | 0.70 | -0.55 | -0.84 | -0.56 | -0.63 | -0.13 | -0.33 | $\alpha$ | SF
273Bh | 9.10 | 2 |  | 1.05 | 1.03 | 1.15 | 0.85 | 1.09 | 4.31 | $\alpha$ |
298119 | 10.88* | - |  | 0.39 | 0.31 | 0.34 | 0.06 | -1.64 | 10.64 | $\alpha$ |
294Ts | 10.81 | - | -1.10 | 0.01 | 1.02 | -0.01 | 0.68 | -1.08 | 10.82 | $\alpha$ | $\alpha$
290Mc | 10.40 | - | -1.80 | 0.40 | 0.28 | 0.38 | 0.02 | -1.15 | 9.19 | $\alpha$ | $\alpha$
286Nh | 9.79 | - | 1.30 | 1.31 | 1.49 | 1.29 | 0.96 | 0.59 | 5.37 | $\alpha$ | $\alpha$
282Rg | 9.64 | - | -0.30 | 0.81 | 0.47 | 0.77 | 0.71 | 0.44 | 0.72 | $\alpha$ | $\alpha$
278Mt | 9.63 | - | 0.90 | 0.57 | 0.40 | 0.60 | 0.12 | 0.34 | -0.81 | SF | $\alpha$
274Bh | 8.95 | - | -0.05 | 1.75 | 1.90 | 1.77 | 1.22 | 0.98 | 2.24 | $\alpha$ | $\alpha$
299119 | 10.73* | 2 |  | 0.33 | 0.37 | 0.35 | 0.43 | -1.30 | 9.44 | $\alpha$ |
295Ts | 10.48* | 2 |  | 0.39 | 0.40 | 0.42 | 0.44 | -1.15 | 10.13 | $\alpha$ |
291Mc | 10.30 | 2 |  | 0.26 | 0.20 | 0.29 | 0.27 | -0.83 | 8.85 | $\alpha$ |
287Nh | 9.54 | 3 |  | 1.68 | 1.96 | 1.85 | 1.63 | 0.75 | 5.07 | $\alpha$ |
283Rg | 9.36 | 2 |  | 1.58 | 1.75 | 1.73 | 1.47 | 0.91 | 0.61 | SF |
279Mt | 9.40 | 0 |  | 0.85 | 0.76 | 0.94 | 0.59 | 0.46 | -2.07 | SF |
275Bh | 8.80* | 3 |  | 1.99 | 2.04 | 2.14 | 1.72 | 0.67 | -0.93 | SF |
300119 | 10.47* | - |  | 1.35 | 1.57 | 1.29 | 1.07 | -1.15 | 9.90 | $\alpha$ |
296Ts | 10.25* | - |  | 1.33 | 1.50 | 1.29 | 1.01 | -0.73 | 11.10 | $\alpha$ |
292Mc | 9.64* | - |  | 2.29 | 2.79 | 2.25 | 1.97 | 0.33 | 10.30 | $\alpha$ |
288Nh | 9.21* | - |  | 2.85 | 3.50 | 2.81 | 2.50 | 0.94 | 6.39 | $\alpha$ |
284Rg | 8.94* | - |  | 3.03 | 3.65 | 3.00 | 2.64 | 1.06 | 2.25 | $\alpha$ |
280Mt | 8.83* | - |  | 2.72 | 3.15 | 2.71 | 2.30 | 0.27 | -0.56 | SF |
276Bh | 8.42* | - |  | 3.33 | 3.85 | 3.32 | 3.12 | 1.67 | -0.79 | SF |
301119 | 10.25* | 0 |  | 1.56 | 1.92 | 1.72 | 1.60 | -0.73 | 8.12 | $\alpha$ |
297Ts | 9.92* | 2 |  | 1.85 | 2.24 | 2.04 | 1.83 | 0.06 | 10.02 | $\alpha$ |
293Mc | 9.46* | 2 |  | 2.55 | 3.07 | 2.80 | 2.45 | 0.62 | 10.18 | $\alpha$ |
289Nh | 9.11* | 7 |  | 2.95 | 3.47 | 3.23 | 2.76 | 0.91 | 5.63 | $\alpha$ |
285Rg | 8.82* | 2 |  | 3.21 | 3.69 | 3.50 | 3.08 | 0.36 | 2.79 | $\alpha$ |
281Mt | 8.83* | 0 |  | 2.57 | 2.76 | 2.77 | 2.35 | 0.55 | -0.12 | SF |
277Bh | 8.27* | 3 |  | 3.73 | 4.04 | 3.99 | 3.72 | 1.81 | -2.78 | SF |
302119 | 9.92* | - |  | 2.73 | 3.43 | 2.65 | 2.43 | 0.06 | 8.20 | $\alpha$ |
298Ts | 9.70* | - |  | 2.74 | 3.39 | 2.68 | 2.41 | 0.35 | 10.56 | $\alpha$ |
294Mc | 9.27* | - |  | 3.31 | 4.10 | 3.26 | 2.93 | 0.83 | 11.39 | $\alpha$ |
290Nh | 8.93* | - |  | 3.69 | 4.54 | 3.64 | 3.45 | 0.07 | 7.21 | $\alpha$ |
286Rg | 8.87* | - |  | 3.26 | 3.85 | 3.24 | 2.94 | 0.56 | 4.18 | $\alpha$ |
282Mt | 8.72* | - |  | 3.09 | 3.52 | 3.09 | 2.79 | 0.58 | 1.30 | $\alpha$ |
278Bh | 8.21* | - |  | 4.02 | 4.64 | 4.01 | 4.00 | 1.85 | -2.40 | SF |
303119 | 9.68* | 2 |  | 3.11 | 3.92 | 3.44 | 3.06 | 0.35 | 5.99 | $\alpha$ |
299Ts | 9.44* | 2 |  | 3.23 | 3.98 | 3.56 | 3.12 | 0.65 | 9.02 | $\alpha$ |
295Mc | 9.15* | 2 |  | 3.49 | 4.21 | 3.83 | 3.42 | -0.24 | 10.31 | $\alpha$ |
291Nh | 8.85* | 7 |  | 3.80 | 4.48 | 4.15 | 3.81 | 0.58 | 6.75 | $\alpha$ |
287Rg | 8.79* | 3 |  | 3.37 | 3.79 | 3.65 | 3.32 | 0.59 | 3.07 | $\alpha$ |
283Mt | 8.70* | 2 |  | 3.03 | 3.22 | 3.25 | 2.94 | 0.59 | 0.99 | $\alpha$ |
279Bh | 8.21* | 0 |  | 3.96 | 4.22 | 4.23 | 4.01 | 1.87 | -2.84 | SF |

##  5 Conclusions

Empirical formulas for $\alpha$-decay half-life (Horoi scaling law and
Sobiczewski formula) and spontaneous fission half-life (Bao et al.) are
modified and the coefficients are fitted using latest evaluated half-lives in
the range 82$\leq$Z$\leq$118\. These formulas are found with a good accuracy
while compared with the original formulas and various other formulas, and
therefore, utilized to demonstrate the competition between $\alpha$-decay and
SF. As an alternate method, machine learning algorithms are also used to probe
the contest of $\alpha$-decay and SF for the range of 111$\leq$Z$\leq$118\.
These modified formulas i.e. modified Horoi formula (MHF-2020) (set-1 and
set-2) and modified Sobiczewski formula (MSF-2020), along with machine
learning methods i.e. MLP and XGBoost algorithm, provide an excellent match
with available experimental data and half-lives. These formulas and algorithms
are used to construct $\alpha$-decay chain of 286-302Og and 287-303119\. Among
them, 286-291Og and 287-296,302,303119 are found with a very long
$\alpha$-decay chain (6$\alpha$/7$\alpha$) and reported here as potential
candidates for future experiments in the superheavy domain.

## ACKNOWLEDGMENTS

Authors are thankful to Prof. H. F. Zhang and X. J. Bao for the valuable
communication. Authors are indebted to Amit Garg, GWECA regarding the various
discussion on machine learning. G. Saxena acknowledges the support provided by
SERB (DST), Govt. of India under CRG/2019/001851.

## References

## References

  * [1] H. Geiger et al., Phil. Mag. 22, 613 (1911).
  * [2] G. Gamow, Z. Phys. 51, 204 (1928).
  * [3] R. W. Gurney and E. U. Condon, Nature 122, 439 (1928).
  * [4] Boshuai Cai, Guangshang Chen, Jiongyu Xu, Cenxi Yuan, Chong Qi, and Yuan Yao, Phys. Rev. C 101, 054304 (2020).
  * [5] Jun-Gang Deng, Hong-Fei Zhang, and G. Royer, Phys. Rev. C 101, 034307 (2020).
  * [6] G. G. Adamian, N. V. Antonenko, H. Lenske, and L. A. Malov, Phys. Rev. C 101, 034301 (2020).
  * [7] O. N. Ghodsi and M. Hassanzad, Phys. Rev. C 101, 034606 (2020).
  * [8] A. A. Voinov, V. K. Utyonkov, Yu. Ts. Oganessian, F. Sh. Abdullin, A. N. Polyakov, Yu. S. Tsyganov et al., Bulletin of the Russian Academy of Sciences: Physics 84, 351 (2020).
  * [9] S. Hofmann and G Münzenberg, Rev. Mod. Phys. 72, 733 (2000).
  * [10] J. M. Hamilton, D. Hofmann, and Yu. Ts. Oganessian, Annu. Rev. Nucl. Part. Sci. 63, 383 (2013).
  * [11] Yu. Ts. Oganessian and K. P. Rykaczewski, Phys. Today 68, 32 (2015).
  * [12] P.-H.Heenen, J. Skalski, A. Staszczak, and D. Vretenar, Nucl. Phys. A 944, 415 (2015).
  * [13] Yu. Ts. Oganessian and V. K. Utyonkov, Rep. Prog. Phys. 78, 036301 (2015).
  * [14] Yu. Ts. Oganessian and K. Rykaczewski, Phys. Today 68, 32 (2015).
  * [15] S. Hofmann, S. Heinz, R. Mann, J. Maurer, G. Münzenberg, S. Antalic et al., Eur. Phys. J. A 52, 180 (2016).
  * [16] C. E. Dullmann and M. Block, Sci. Am. 318, 46 (2018).
  * [17] W. Nazarewicz, Nat. Phys. 14, 537 (2018).
  * [18] S.A. Giuliani et al., Rev. Mod. Phys. 91, 011001 (2019).
  * [19] S. Hofmann, Radiochimica Acta International journal for chemical aspects of nuclear science and technology 99, 405 (2011).
  * [20] Kosuke Morita, Kouji Morimoto, Daiya Kaji, Takahiro Akiyama, Sin-ichi Goto, Hiromitsu Haba et al., Journal of the Physical Society of Japan 76, 045001 (2007).
  * [21] Yu. Ts. Oganessian, F. Sh. Abdullin, P. D. Bailey, D. E. Benker, M. E. Bennett, S. N. Dmitriev et al., Phys. Rev. Lett. 104, 142502 (2010).
  * [22] Yu. Ts. Oganessian and V.K. Utyonkov, Nucl. Phys. A 944, 62 (2015).
  * [23] Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyakov, R. N. Sagaidak et al., Phys. Rev. C 74, 044602 (2006).
  * [24] Yu. Ts. Oganessian, F. Sh. Abdullin, P. D. Bailey, D. E. Benker, M. E. Bennett, S. N. Dmitriev et al. Phys. Rev. C 83, 054315 (2011).
  * [25] Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyakov, R. N. Sagaidak et al., Phys. Rev. C 79, 024603 (2009).
  * [26] N. T. Brewer, V. K. Utyonkov, K. P. Rykaczewski, Yu. Ts. Oganessian, F. Sh. Abdullin, R. A. Boll et al., Phys. Rev. C 98, 024317 (2018).
  * [27] V. Viola and G. Seaborg, Journal of Inorganic and Nuclear Chemistry 28, 741 (1966).
  * [28] A. Sobiczewski, Z. Patyk, and S. Cwiok, Phys. Lett. B 224, 1 (1989).
  * [29] A. Parkhomenko and A. Sobiczewski, Acta Phys. Pol. B 36, 3095 (2005).
  * [30] B. A. Brown, Phys. Rev. C 46, 811 (1992).
  * [31] G. Royer, J. Phys. G: Nucl. Part. Phys. 26, 1149 (2000).
  * [32] Z. Ren, C. Xu, and Z. Wang, Phys. Rev. C 70, 034304 (2004).
  * [33] D. Ni, Z. Ren, T. Dong, and C. Xu, Phys. Rev. C 78, 044310 (2008).
  * [34] C. Qi, F. R. Xu, R. J. Liotta, and R. Wyss, Phys. Rev. Lett. 103, 072501 (2009).
  * [35] D. N. Poenaru, R. A. Gherghescu, W. Greiner, Phys. Rev. C 85, 034615 (2012).
  * [36] J. Dong, H. Zhang, W. Wang, W. Zuo, J. Li, Nucl. Phys. A 832, 198 (2010).
  * [37] D. T. Akrawy and D. N. Poenaru, J. Phys. G 44, 105105 (2017).
  * [38] Dashty T. Akrawy, H. Hassanabadi, S. S. Hosseini, and K. P. Santhosh, Int. J. Mod. Phys. E 28, 1950075 (2019).
  * [39] Dashty T. Akrawy and Ali H. Ahmed, Phys. Rev. C 100, 044618 (2019).
  * [40] Dashty T. Akrawy, H.Hassanabadi, Yibin Qian, and K.P .Santhosh, Nucl. Phys. A 983, 310 (2019).
  * [41] N. Dasgupta-Schubert et al., Eur. Phys. J. A 42, 121 (2009).
  * [42] A. I. Budaca, R. Budaca, I. Silisteanu, Nucl. Phys. A 951, 60 (2016).
  * [43] T. L. Zhao et al., J. Phys. G: Nucl. Part. Phys. 45, 025106 (2018).
  * [44] H. C. Manjunatha, L. Seenappa, and K. N. Sridhar, Eur. Phys. J. Plus 134, 477 (2019).
  * [45] U. K. Singh, R. Sharma, P. K. Sharma, M. Kaushik, S. K. Jain, and G. Saxena, Accepted in Nucl. Phys. A (2020).
  * [46] M. Horoi, B. A. Brown, A. Sandulescu, J. Phys. G 30, 945 (2004).
  * [47] W. Swiatecki, Phys. Rev. 100, 937 (1955).
  * [48] Z. Ren and C. Xu, Nucl. Phys. A 759, 64 (2005).
  * [49] C. Xu, Z. Ren, and Y. Guo, Phys. Rev. C 78, 044329 (2008).
  * [50] K. P. Santhosh, R. K. Biju, and Sabina Sahadevan, Nucl. Phys. A 832, 220 (2010).
  * [51] A. V. Karpov, V. I. Zagrebaev, Y. Martinez Palenzuela, L. Felipe Ruiz, and W. Greiner, Int. J. Mod. Phys. E 21, 1250013 (2012).
  * [52] I. Silisteanu and C. I. Anghel, Rom. J. Phys. 60, 444 (2015).
  * [53] X. J. Bao, S. Q. Guo, H. F. Zhang, Y. Z. Xing, J. M. Dong, and J. Q. Li, J. Phys. G: Nucl. Part. Phys. 42, 085101 (2015).
  * [54] P. Möller, J. Nix, W. Myers, and W. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995).
  * [55] P. Möller, A. J. Sierk, T. Ichikawa, and H. Sagawa, At. Data Nucl. Data Tables 109–110, 1 (2016).
  * [56] Y. Lecun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
  * [57] J. Schmidhuber, Neural Networks 61, 85 (2015).
  * [58] G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborova, Rev. Mod. Phys. 91, 045002 (2019).
  * [59] P. Mehta, M. Bukov, C.-H.Wang, A. G. Day, C. Richardson, C. K. Fisher, and D. J. Schwab, Phys. Rept. 810, 1 (2019).
  * [60] D. Guest, K. Cranmer, and D. Whiteson, Ann. Rev. Nucl. Part. Sci. 68, 161 (2018).
  * [61] S. Athanassopoulos, E. Mavrommatis, K. Gernoth, and J. W. Clark, Nucl. Phys. A 743, 222 (2004).
  * [62] Z. M. Niu and H. Z. Liang, Phys. Lett. B 778, 48 (2018).
  * [63] Murarka Utsav Anil, Tuhin Malik, and Kinjal Banerjee, arXiv:2004.14196 [physics.comp-ph].
  * [64] N. J. Costiris, E. Mavrommatis, K. A. Gernoth, and J. W. Clark, Phys. Rev. C 80, 044332 (2009).
  * [65] R. Utama, J. Piekarewicz, and H. B. Prosper, Phys. Rev. C 93, 014311 (2016).
  * [66] https://github.com/scipy/scipy/blob/v1.5.2/scipy/
optimize/minpack.py#L532-L834.

  * [67] Matt A Wood, ’Python and Matplotlib Essentials for Scientists and Engineers’, Morgan & Claypool Publishers 11-1 (2015).
  * [68] Dashty T.Akrawy, H.Hassanabadi, S.S.Hosseini, K.P.Santhosh, Nuclear Physics A 975 19 (2018).
  * [69] http://www.nndc.bnl.gov/
  * [70] D. N. Poenaru and R. A. Gherghescu, ’Fission and Properties of Neutron-Rich Nuclei’, World Scintific Publishers 142-147 (2017).
  * [71] D. N. Poenaru and R. A. Gherghescu, Europhysics Letters 118, 22001 (2017).
  * [72] B. M. E. Moret, Computing Surveys, 14, 593 (1982).
  * [73] L. Breiman, Mach. Learn. 45, 5 (2001).
  * [74] Martin Riedmiller, Computer Standards & Interfaces 16, 265 (1994).
  * [75] T. Chen and C. Guestrin, Proceedings of the 22Nd ACM SIGKDD international conference on knowledge discovery and data mining. ACM, 785 (2016).
  * [76] Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyakov, I. V. Shirokovsky et al. Phys. Rev. C 69, 021601(R) (2004).
  * [77] Yu. Ts. Oganessian, F. Sh. Abdullin, C. Alexander, J. Binder, R. A. Boll, S. N. Dmitriev et al. Phys. Rev. Lett. 109, 162501 (2012).
  * [78] G. Saxena, M. Kumawat, M. Kaushik, S.K. Jain, and Mamta Aggarwal, Phys. Lett. B 788, 1 (2019).
  * [79] G. Saxena, M. Kumawat, B. K. Agarwal, and Mamta Aggarwal, Phys. Lett. B 789, 323 (2019).
  * [80] G. Saxena, M. Kumawat, M. Kaushik, U.K. Singh, S.K Jain, S. Somorendro Singh, and Mamta Aggarwal, Int. J. Mod. Phys. E 26, 1750072 (2017).
  * [81] U. K. Singh, P. K. Sharma, M. Kaushik, S.K Jain, Dashty T Akrawy, and G. Saxena, Nucl. Phys. A 1004, 122035 (2020).
  * [82] P. Möller, M. R. Mumpower, T. Kawano, and W.D. Myers, At. Data Nucl. Data Tables 125, 1 (2019).
  * [83] X. W. Xia, Y.Lim, P. W. Zhao, H. Z. Lian, X. Y. Qu, Y. Chen et al., At. Data Nucl. Data Tables 121–122, 1 (2018).
  * [84] H. Koura, T. Tachibana, M. Uno, and M. Yamada, Prog. Theor. Phys. 113 (2005) 305.
  * [85] J. Dong, W. Zuo, J. Gu, Y. Wang, and B. Peng, Phys. Rev. C 81, 064309 (2010).
  * [86] J. Dong, W. Zuo, and W. Scheid, Phys. Rev. Lett. 107, 012501 (2011).
  * [87] V. Y. Denisov and A. A. Khudenko, Phys. Rev. C 79, 054614 (2009).
  * [88] Jun-Gang Deng, Hong-Fei Zhang, and G. Royer, Phys. Rev. C 101, 034307 (2020)
  * [89] G. Audi, F. Kondev, M. Wang, W. Huang, and S. Naimi, Chin. Phys. C 41, 030001 (2017).
  * [90] https://t2.lanl.gov/nis/data/astro/molnix96/spidat.html.