11institutetext: Department of Physics (H&S), Govt. Women Engineering College,
Ajmer-305002, India 22institutetext: Department of Physics, Faculty of
Science, University of Zagreb, Bijeni$\breve{c}$ka c. 32, 10000 Zagreb,
Croatia 33institutetext: Department of Physics, School of Basic Sciences,
Manipal University Jaipur, Jaipur-303007, India 44institutetext: Department of
Physics, S. S. Jain Subodh P.G.(Autonomous) College, Jaipur-302004, India

# Cluster radioactivity from trans-tin to superheavy region using an improved
empirical formula

G. Saxena 1122 A. Jain 3344

(Received: date / Revised version: date)

###### Abstract

A simple relation $(aZ_{c}+b)(Z_{d}/Q)^{1/2}+(cZ_{c}+d)$ of estimation of the
half-life of cluster emission is further improved for cluster and
$\alpha$-decays, separately, by incorporating isospin of parent nucleus as
well as angular momentum taken away by the emitted particle. This improved
version is not only found robust in producing experimental half-lives
belonging to the trans-tin and trans-lead regions but also elucidates cluster
emission in superheavy nuclei over the usual $\alpha$-decay. Considering
daughter nuclei around the doubly magic 100Sn and 208Pb nuclei for trans-tin
and trans-lead (including superheavy) parents, respectively, a systematic and
extensive study of 56$\leq$Z$\leq$120 isotopes is performed for the light and
heavy cluster emissions. A fair competition among cluster emission,
$\alpha$-decay, spontaneous fission, and $\beta$-decay is observed in this
broad range resulting in a substantial probability of C to Sr clusters from
several nuclei, which demonstrates the adequacy of shell effects. The present
article proposes a single, improved, latest-fitted, and effective formula of
cluster radioactivity that can be used to estimate precise half-lives for a
wide range of the periodic chart from trans-tin to superheavy nuclei.

###### pacs:

21.10.-k, 21.10.Ft, 21.10.Dr, 21.10.Gv, 21.10.-n, 21.60.Jz

##  1 Introduction

An exotic decay mode, in which a heavy nucleus fragmentizes itself by emitting
a nucleus with the mass between the fission product and $\alpha$-particle, is
envisioned already in 1980 sandulescu1980  and named ’cluster decay’. After
its first experimental confirmation rose1984  by emission of 14C from actinide
parent nucleus 223Ra, this decay mode has been observed from many trans-lead
nuclei as the emission of light 14C to heavy 32Si clusters Bonetti2007 . From
the last two decades it has gained healthy attention Kumar2003 ; Kumar2009 ;
Gupta2003 ; Gupta1994 ; Price1989 ; Santhosh2012 ; Hourani1989 ; Royer2001 ;
Singh2022 ; kulkin2005  due to the direct signature of the crucial role of
shell effects as all known cluster emissions have daughter nuclei near to the
closed-shell nucleus 208Pb.

Except for the trans-lead region, there are two potential and new islands of
cluster emitters that are still less explored and require a comprehensive
investigation. The first one is the trans-tin region, regulated by doubly
magic nucleus 100Sn, where only one experimental observation of 12C cluster
decay from 114Ba exists Og1994 ; Guglielmetti1995 ; Guglielmetti1995npa ,
though it is not observed in the later measurement Guglielmetti1995 . The
other region is the superheavy region in which many heavy clusters are
speculated to compete with $\alpha$-decay poenaru2012 ; zhang2018 ;
poenaru2018 ; poenaru2018EPJA ; matheson2018 ; Warda2018 ; santhosh2008 ;
prathapan2021 ; Santhosh2022prc ; Royer2022 ; Nithya2022 , however, still
requires experimental confirmation santhosh2018 ; soylu2019 ; jain2023PS .
Therefore, an exhaustive study of cluster radioactivity including trans-tin,
trans-lead as well as superheavy regions can lead to important structure
information for the detection of heavy and superheavy nuclei in future
experimental investigations.

The estimation of the half-life of cluster decay was made in the same decade
of its experimental confirmation by several theoretical models which are
primarily based on the quantum tunneling effect through a potential barrier
Poenaru1982 ; Delion2001 ; Poenaru1996 ; Poenaru1996IPPB ; arun2009  or cold
asymmetric fission process Brack1972 ; Wahl1988 . Alternatively and
concomitantly, the very first estimation of the half-life of heavy ion
(cluster) emission was given by Poenaru et al. using an analytical formula
based on an analytical superheavy asymmetric fission model (ASAFM) Poenaru1985
; Poenaru1986 . Since then, cluster radioactivity has been studied by several
models out of which a few recent ones are the preformed-cluster-decay model
(PCM) Joshua2022 ; Yahya2022 , Coulomb and proximity potential model for
deformed nuclei Hosseini2023 , Wentzel-Kramers-Brilliouin (WKB) method
Dagtas2022 , and modified Gamow-like model Liu2021 . Similarly, by fitting
available experimental data, many (semi)empirical/analytical formulas related
to cluster decay were proposed from time to time such as Horoi formula
Horoi2004 , BKAG formula Balasubramaniam2004 , RenA formula Ren2004 , NRDX
formula NRDX2008 , UDL formula UDL2009 , and UNIV formula Poenaru2011 . Later
on, Tavares and Medeiros proposed a semiempirical relationship for the
accurate estimation of exotic decay from trans-lead nuclei Tavares2013 .
Recently, a few of above mentioned semiempirical relationships are modified in
Refs. jain2023 ; soylu2021 ; UF2022 ; Wang2021CPC ; cheng2022 ; Lin2023  which
endorse the crucial contributions of asymmetry of parent nucleus and
centrifugal barrier effect in the estimation of half-lives of cluster
radioactivity. It is, however, important to point out here that almost all the
formulas (older or modified) work fairly well in the trans-lead region, but
act somewhat unfavourable for the simultaneous estimation of half-lives of
cluster emitters in trans-tin and superheavy regions nature ; zhang2018 . As a
matter of fact, especially for the trans-tin region, the UDL formula
overestimates the half-life of 114Ba by four orders of magnitude whereas it
works well in the superheavy region due to its fission-like mechanisms
zhang2018 . On the other hand, Horoi and NRDX formulas are capable to
reproduce the half-life of 114Ba nature ; Guglielmetti1997PRC  but are found
not suitable for the heavy clusters in the superheavy region zhang2018 .
Therefore, there is a need for a unique formula that can be applied for the
accurate estimation of cluster decay half-life emulated with $\alpha$-decay
half-life for the whole periodic chart (from trans-tin to superheavy nuclei),
which is precisely the aim of the present investigation.

In this article, we present a single formula that can be applied for the
accurate estimation of half-lives of cluster emission for a systematic study
of nuclei in the range 56$\leq$Z$\leq$120 considering daughter nuclei around
the doubly magic nuclei 100Sn and 208Pb for the trans-tin and trans-lead
(including superheavy region) regions, respectively. The most probable
clusters are found as a result of its contest with $\alpha$-decay,
$\beta$-decay (referring for $\beta^{-}$, $\beta^{+}$, and EC throughout the
paper), and spontaneous fission (SF) for this wide region of the periodic
chart.

##  2 Formalism

Table 1: The coefficients of ITM formula proposed in the present work. The
formula is fitted separately for trans-tin and tans-lead regions (see the text
for details).

Coefficient | Trans-tin | Trans-lead
---|---|---
| ITM-CT | ITM-AT | ITM-CL | ITM-AL
a | 14.5427$\pm$0.8701 | 294.8314$\pm$52.3194 | 11.4570$\pm$0.0185 | 217.4129$\pm$42.6150
b | -1.5239$\pm$0.5639 | -578.2390$\mp$104.6389 | -8.9167$\mp$0.0446 | -420.0786$\mp$ 85.2188
c | -3.9917$\mp$0.0639 | -10.4338$\mp$0.1024 | -4.2778 $\mp$0.0131 | -12.5941$\mp$ 0.1005
d | -109.0040$\mp$10.4162 | -22.8685$\mp$0.1952 | -103.9499$\mp$0.2137 | -27.1875$\mp$ 0.2003
e | -10.8146$\mp$3.6985 | 11.8588$\mp$1E-6 | 93.2694$\mp$0.4313 | 4.2401$\mp$0.0845
f | 0.6419$\pm$0.1954 | 0.1495$\mp$1E-6 | 0.3527$\pm$0.0009 | 0.2630$\mp$0.0009

In 2013, Tavares and Medeiros found that experimental cluster decay half-lives
show straight lines when plotted as a function of $\xi$ = $(Z_{d}/Q)^{1/2}$,
where $Z_{d}$ is the atomic number of daughter and $Q$ is the energy of the
two-body disintegrating system Tavares2013 . A formula (TM), shown below, was
proposed for cluster decay half-lives dependent on only 4 parameters and was
able to reproduce the majority of the available experimental half-life data
reasonably well which belong to the trans-lead region.

$\displaystyle
log_{10}T_{1/2}^{TM}(s)=(aZ_{c}+b)\sqrt{\frac{Z_{d}}{Q}}+(cZ_{c}+d),$ (1)

In our investigation, this relationship while refitted, is also found to
reproduce $\alpha$-decay half-lives of the latest evaluated database
NUBASE2020 audi20201  with a good accuracy for the wide range of the periodic
chart. Similar to the crucial roles of (i) angular momentum ($l$) taken away
by the emitted particle, and (ii) isospin ($I=(N-Z)/A$) of parent nucleus in
$\alpha$-emission sharma2021npa , these two quantities are reported equally
important in determining the cluster decay half-lives by new UDL formula of
Soylu and Qi soylu2021  as well as by improved NRDX formula (named as improved
unified formula (IUF)) of Ismail et al. UF2022 . In addition, the crucialness
of the isospin effect is also probed by improving the semi-empirical formula
(ISEM) for the cluster radioactivity in Ref. cheng2022 . Recently, a few
semiempirical relationships are modified for the trans-lead region in Ref.
jain2023  which endorse the crucial contributions of asymmetry of parent
nucleus and centrifugal barrier effect in estimating half-lives of cluster
radioactivity. In order to elucidate the effect of these terms in Eqn. (1), we
have initially fitted our data set for (i) cluster decay and (ii)
$\alpha$-decay in the trans-tin region together with (iii) cluster decay and
(iv) $\alpha$-decay in the trans-lead region by incorporating only the
$I$-dependent term ($\sqrt{I(I+1)}$), which accounts for the asymmetry. This
addition of term has yielded a reduction of RMSE values from 1.53, 1.21, 0.97,
and 0.92 to 1.37, 0.74, 0.84, and 0.89, respectively. Subsequently, when we
have introduced only the $l$-dependent term ($\sqrt{l(l+1)}$), which reflects
the hindrance effect of the centrifugal barrier, then the RMSE values reduce
to 1.01, 1.09, 0.75, and 0.78, respectively. Finally, when both the $I$ and
$l$ dependent terms are incorporated in Eqn. (1) then RMSE values
significantly reduced to 0.82, 0.71, 0.62, and 0.76, respectively. Hence, the
Eqn. (1) can be improvised by adding isospin dependent term as well as a
centrifugal barrier term which lead to the following improved formula (named
as improved Tavares and Medeiros formula or ITM formula):

$\displaystyle log_{10}T_{1/2}^{ITM}(s)$ $\displaystyle=$
$\displaystyle(aZ_{c}+b)\sqrt{\frac{Z_{d}}{Q}}+(cZ_{c}+d)$ (2)
$\displaystyle+e\sqrt{I(I+1)}$ $\displaystyle+f\sqrt{l(l+1)},$

where, a, b, c, d, e, and f are fitting coefficients, tabulated in Table 1. Zc
and Zd are the proton number of emitted cluster and daughter nucleus,
respectively. $Q$ and $I$ are the disintegration energy and isospin,
respectively. $l$ is the minimum angular momentum of the cluster particle,
which is obtained by following selection rules:

$l=\left\\{\begin{array}[]{ll}\triangle_{j}&\mbox{for
even}\,\,\triangle_{j}\,\,\mbox{and}\,\,\pi_{i}=\pi_{f}\\\
\triangle_{j}+1&\mbox{for
even}\,\,\triangle_{j}\,\,\mbox{and}\,\,\pi_{i}\neq\pi_{f}\\\
\triangle_{j}&\mbox{for
odd}\,\,\triangle_{j}\,\,\mbox{and}\,\,\pi_{i}\neq\pi_{f}\\\
\triangle_{j}+1&\mbox{for
odd}\,\,\triangle_{j}\,\,\mbox{and}\,\,\pi_{i}=\pi_{f},\\\ \end{array}\right.$
(3)

here, $\triangle_{j}$ = $|j_{p}-j_{d}-j_{c}|$ with jp, $\pi_{i}$, are the spin
and parity values of the parent nucleus, respectively. jd is the spin of the
daughter nucleus. $\pi_{f}=(\pi_{d})(\pi_{c})$, in which, $\pi_{d}$ and
$\pi_{c}$ are the parities of the daughter nucleus and cluster, respectively.
For the purpose of fitting, the data of spin and parity are taken from
NUBASE2020 audi20201 .

Before discussing the results obtained by the ITM formula, there are two
important facts that are worth mentioning here. The first fact is related to
the inclusion of deformation which is an important ingredient for the
estimation of half-lives as also has been described in our recent article
saxenajpg2023 . We have followed a similar analysis by incorporating the
deformation of the parent nucleus but the reduction in RMSE values is found
very insignificant on the cost of the addition of one more parameter. In
several articles Soylu2012epja ; Hosseini2023 , it is demonstrated that in
addition to the deformation of the parent nucleus, the deformation of cluster
and daughter nuclei needs to be considered to visualize the complete picture
of this exotic decay. Such kind of investigation requires a comprehensive and
detailed analysis with the inclusion of quadrupole $(\beta_{2})$ and
hexadecapole $(\beta_{4})$ deformations, which will be reported in our
subsequent work on the deformation effect of $\alpha$ and cluster decays along
with the effect of shape-coexistence on half-lives. However, when compared to
the half-lives of cluster emission considering the deformation effect from
Refs. Soylu2012epja ; Hosseini2023 , the present form of the ITM formula
results reasonably well with the RMSE value of 0.62 instead of RMSE values
2.30 of Ref. Soylu2012epja  and 0.74 of Ref. Hosseini2023  on the provided
experimental data.

Another fact is related to the internal structure of the heavy cluster. For
simplicity, in general, cluster formation is assumed similar to the formation
of $\alpha$-particle inside the nucleus but this approximation is valid for
the light clusters only. For the heavy clusters, one needs to consider their
internal structure together with the neutron-to-proton ratio which eventually
can affect cluster preformation in the parent nuclei and consequently the
half-life Wei2017prc . With this in view, a separate study considering
internal structure is required for the clusters with the mass number
$A_{c}$$>$28.

##  3 Results and discussion

Considering the role of the shell effect on cluster decay, we divide our
region of interest (56$\leq$Z$\leq$120) into two parts: (i) the trans-tin
region in which daughter nuclei are reckoned around doubly magic nucleus
100Sn, and (ii) the trans-lead and superheavy regions where daughter nuclei
are counted around doubly magic nucleus 208Pb. For the trans-tin region only
one data for the cluster (114Ba) is available nature ; Guglielmetti1997PRC ,
therefore, to fit the formula for this region we include 199 GLDM data nature
for cluster decay. For the $\alpha$-decay fitting, we use 107 experimental
data taken from NUBASE2020 audi20201 . On the other side, for the trans-lead
region, we fit the formula for 37 parent-cluster combinations (total of 61
data points due to multiple $Q$-value corresponding to various detection
systems) Bonetti2007 ; Gupta1994 ; Price1989 ; Santhosh2012 ; Hourani1989  and
308 $\alpha$-decay data from NUBASE2020 audi20201 , separately. In this way,
the same ITM formula can be utilized to estimate both clusters and
$\alpha$-decay half-lives for both regions (trans-tin and trans-lead) which
are speculated to show different decay properties of nuclei dengplb2021 ;
saxenanew2023 . The corresponding coefficients are mentioned in Table 1 for
trans-tin region (ITM-CT and ITM-AT) and for trans-lead region (ITM-CL and
ITM-AL), separately. For the nomenclature of the formula ’C’ and ’A’ refer to
cluster and alpha decay whereas last letters ’T’ and ’L’ refer to trans-tin
and trans-lead region, respectively (please see Table 1). Since, in heavy and
superheavy regions, uncertainties play a crucial role therefore we have
calculated all the fitting parameters with their respective uncertainties by
taking into account experimental uncertainties audi20201 . These uncertainties
in parameters are mentioned in Table 1 for all the present formulas i.e. ITM-
CT, ITM-AT, ITM-CL, and ITM-AL which lead to uncertainties in the theoretical
half-live with $\pm 0.71$, $\pm 0.40$, $\pm 0.56$, and $\pm 0.41$,
respectively.

Table 2: The RMSE and ${\chi}^{2}$ of various formulas viz. ITM (present
work), IUF UF2022 , ISEF cheng2022 , MBKAG jain2023 , UDL UDL2009 , TM
Tavares2013 , and New UDL soylu2021  for 62 cluster decay data.

Formula | No. of | RMSE | ${\chi}^{2}$
---|---|---|---
| coefficient |  |
ITM (present work) | 6+6 | 0.66 | 0.55
IUF UF2022  | 7 | 0.87 | 0.85
ISEF cheng2022  | 4 | 0.99 | 1.04
MBKAG jain2023  | 5 | 1.14 | 1.41
UDL UDL2009  | 3 | 1.52 | 2.44
TM Tavares2013  | 4 | 1.81 | 3.51
New UDL soylu2021  | 6 | 1.94 | 4.16

Figure 1: (Colour online) Comparison of logarithmic half-lives of cluster
decay, calculated by using ITM (present work), New UDL soylu2021 , and UDL
UDL2009  formulas with available experimental data. X-axis refers to the
clusters for which experimental data are available. Inset shows one data for
trans-tin region i.e. decay of 12C cluster from 114Ba. Figure 2: (Colour
online) Cluster decay half-lives for various isotopes of Ba (Np=56-62), Ce
(Np=61-71), Ra (Np=113-128), Th (Np=120-132), Og (Np=162-192), and Z=120
(Np=166-192) calculated by the ITM formula.

The applicability of this kind of separate fitting for trans-tin and trans-
lead regions can be judged by comparing the results of ITM formula with the
few latest and well-established formulas of cluster decay. In Table 2, we list
the calculated values of root mean square error (RMSE) as well as $\chi^{2}$
(formulas are mentioned in Ref. jain2023 )for ITM and other similar formulas
viz. improved unified formula (IUF) UF2022 , improved semi-empirical formula
(ISEF) cheng2022 , MBKAG jain2023 , UDL UDL2009 , TM formulaTavares2013 , and
New UDL soylu2021  formulas. The lowest value of RMSE of ITM formula among all
the considered formulas clearly demonstrates its accuracy and improvement over
the original TM formula (as RMSE improves from 1.81 to 0.61). In addition to
this, the least value of $\chi^{2}$ exhibits pertinency of ITM formula on the
ground of the number of parameters. Further, to analyze the estimation of
half-lives using the ITM formula with the available experimental data, we have
plotted half-lives of several clusters calculated by using the ITM formula,
New UDL soylu2021 , and UDL formula UDL2009  along with their experimental
half-lives Bonetti2007 ; Gupta1994 ; Price1989 ; Santhosh2012 ; Hourani1989
in Fig. 1. The figure contains all the experimental C, N, O, Ne, Mg, and Si
clusters (in different colour bands) that belong to the trans-lead region. In
the inset, we have depicted the half-life of 12C cluster from the only trans-
tin parent nucleus 114Ba. A reasonable match of theoretical half-lives from
ITM formula with the experimental half-lives endorses the utilization of ITM
formula for the unknown region to find the possibility of cluster decay for a
wide range of periodic chart from trans-tin to trans-lead as well as the
superheavy regions. For C clusters in Fig. 1, the theoretical half-lives of
all formulas are found the same for all the points due to the same $Q$-value
(31.85 MeV) for the single parent-cluster combination
${}^{223}Ra\longrightarrow^{209}Pb+^{14}C$.

To explore the full periodic chart, we perform a systematic and comprehensive
analysis for all the parent isotopes in the range 56$\leq$Z$\leq$120 using ITM
formula by varying their corresponding daughters in such a way that cluster
decay leads daughter nuclei around 100Sn for the trans-tin region and around
208Pb for rest of the chart (including trans-lead and superheavy regions). As
per Eqn. (2), half-lives of cluster decay mainly depend on $Q$-value which is
calculated by using the following relation:

$\displaystyle Q(MeV)$ $\displaystyle=$ $\displaystyle
B.E.(d)+B.E.(c)-B.E.(p)$ (4) $\displaystyle+k[Z_{p}^{\beta}-Z_{d}^{\beta}],$

here, the term $k[Z_{p}^{\epsilon}-Z_{d}^{\epsilon}]$ indicates the screening
effect caused by the surrounding electrons around the nucleus Denisov2009prc
where k=8.7 eV and $\epsilon$=2.517 for Z $\geq$ 60 and k=13.6 eV and
$\epsilon$ =2.408 for Z $<$ 60 have been deducted from the data shown by Huang
et al. huang1976 . For accurate prediction of theoretical $Q$-values, we have
used binding energies (for daughter(d), cluster(c), and parent(p) nuclei) from
Weizsacker-Skyrme mass model (WS4) ws42014  which was found more precise
jain2023  compared to other theories viz. Finite Range Droplet Model (FRDM)
moller2019 , nonrelativistic Skyrme Hartree-Fock-Bogoliubov (HFB) hfb2004 ,
and relativistic mean-field theory (RMF) saxenaPLB2019 ; saxena2017 ;
singh2020 ; saxenaplb2017 .

Table 3: The comparison of branching ratio (BR) with available experimental
results audi20201 ; Bonetti2007 ; Gupta1994 ; soylu2021  for various types of
decay viz. cluster decay, $\alpha$-decay, and SF.

Parent | Emitted | Daughter | Expt. BR |  | Theoretical BR
---|---|---|---|---|---
nucleus | cluster | nucleus | Cluster | $\alpha$ | SF |  | Cluster | $\alpha$ | SF
221Fr |  14C |  207Tl | 8.8e-11 | $100.00$ | 0.0048 |  | 8e-15 | $100.00$ | $0.00$
221Ra |  14C |  207Pb | 1.2e-10 | $100.00$ | - |  | 9e-09 | $100.00$ | $0.00$
222Ra |  14C |  208Pb | 3.0e-8 | $100.00$ | - |  | 9e-09 | $100.00$ | $0.00$
223Ra |  14C |  209Pb | 8.9e-8 | $100.00$ | - |  | 4e-08 | $100.00$ | $0.00$
223Ac |  14C |  209Bi | - | $99.00$ | - |  | 4e-07 | $100.00$ | $0.00$
223Ac |  15N |  208Pb | - | $99.00$ | - |  | 8e-11 | $100.00$ | $0.00$
224Ra |  14C |  210Pb | 4.0e-9 | $100.00$ | - |  | 6e-07 | $100.00$ | $0.00$
225Ac |  14C |  211Bi | 5.3e-10 | $100.00$ | - |  | 5e-09 | $100.00$ | $0.00$
226Ra |  14C |  212Pb | 2.6e-9 | $100.00$ | - |  | 2e-05 | $100.00$ | $0.00$
226Th |  18O |  208Pb |  $<$3.2e-12 | $100.00$ | - |  | 3e-11 | $100.00$ | $0.00$
228Th |  20O |  208Pb | 1.13e-11 | $100.00$ | - |  | 3e-08 | $100.00$ | $0.00$
230Th |  24Ne |  206Hg | 5.8e-11 | $100.00$ | 4e-12 |  | 4e-10 | $100.00$ | $0.00$
230U |  22Ne |  208Pb | 4.8e-12 | $100.00$ | $?$ |  | 6.18e-13 | $100.00$ | $0.00$
231Pa |  24Ne |  207Tl | 13.4e-10 | $100.00$ |  $<$3e-10 |  | 6.18e-13 | $100.00$ | $0.00$
231Pa |  24Ne |  207Tl | 13.4e-10 | $100.00$ |  $<$3e-10 |  | 4.91e-11 | $100.00$ | $0.00$
232Th |  24Ne |  208Hg |  $<$2.78e-10 | $100.00$ | 1.1e-9 |  | 3.02e-09 | $100.00$ | $0.00$
232Th |  26Ne |  206Hg |  $<$2.78e-10 | $100.00$ | 1.1e-9 |  | 3.02e-09 | $100.00$ | $0.00$
232U |  24Ne |  208Pb | 8.9e-10 | $100.00$ | 2.7e-12 |  | 6.18e-10 | $100.00$ | $0.00$
233U |  24Ne |  209Pb | 7.2e-11 | $100.00$ |  $<$6e-11 |  | 5.21e-10 | $100.00$ | $0.00$
233U |  25Ne |  208Pb | 7.2e-13 | $100.00$ |  $<$6e-11 |  | 5.21e-10 | $100.00$ | $0.00$
233U |  28Mg |  205Hg |  $<$1.3e-13 | $100.00$ |  $<$6e-11 |  | 2.85e-12 | $100.00$ | $0.00$
234U |  24Ne |  210Pb | 9e-12 | $100.00$ | 1.64e-9 |  | 5.85e-11 | $100.00$ | $0.00$
234U |  26Ne |  208Pb | 9e-12 | $100.00$ | 1.64e-9 |  | 2.69e-10 | $99.97$ | $0.03$
234U |  28Mg |  206Hg | 1.4e-11 | $100.00$ | 1.64e-9 |  | 3.07e-11 | $99.97$ | $0.03$
235U |  24Ne |  211Pb | 8e-10 | $100.00$ | 7e-9 |  | 2.88e-12 | $100.00$ | $0.00$
235U |  25Ne |  210Pb | 8e-10 | $100.00$ | 7e-9 |  | 1.78e-12 | $100.00$ | $0.00$
235U |  28Mg |  207Hg | 8e-10 | $100.00$ | 7e-9 |  | 1.01e-12 | $100.00$ | $0.00$
235U |  29Mg |  206Hg | 8e-10 | $100.00$ | 7e-9 |  | 4.61e-13 | $100.00$ | $0.00$
235U |  24Ne |  211Pb | 8e-10 | $100.00$ | 7e-9 |  | 2.87e-12 | $100.00$ | $0.00$
236U |  28Mg |  208Hg | 2e-13 | $100.00$ | 9.4e-8 |  | 2.23e-12 | $99.58$ | $0.42$
236U |  30Mg |  206Hg | 2e-13 | $100.00$ | 9.4e-8 |  | 1.31e-11 | $99.58$ | $0.42$
236Pu |  28Mg |  208Pb | 2e-12 | $100.00$ | 1.9e-7 |  | 6.69e-12 | $100.00$ | $0.00$
237Np |  30Mg |  207Tl |  $<$4e-12 | $100.00$ |  $<$2e-10 |  | 1.28e-12 | $100.00$ | $0.00$
238Pu |  28Mg |  210Pb |  $\approx$6e-15 | $100.00$ | 1.9e-7 |  | 1.66e-14 | $99.87$ | $0.13$
238Pu |  30Mg |  208Pb |  $\approx$6e-15 | $100.00$ | 1.9e-7 |  | 1.50e-13 | $99.87$ | $0.13$
238Pu |  32Si |  206Hg |  $\approx$1.4e-14 | $100.00$ | 1.9e-7 |  | 3.28e-14 | $99.87$ | $0.13$
240Pu |  34Si |  206Hg |  $<$1.3e-11 | $100.00$ | 5.796e-6 |  | 2.25e-13 | $93.28$ | $6.72$
241Am |  34Si |  207Tl |  $<$2.6e-13 | $100.00$ | 3.6e-10 |  | 2.08e-14 | $100.00$ | $0.00$
242Cm |  34Si |  208Pb | 1.1e-14 | $100.00$ | 6.2e-6 |  | 5.54e-14 | $99.75$ | $0.25$

As representative examples for the trans-tin region, in Fig. 2 (a) and (b), we
have shown the estimated half-lives of Ba and Ce isotopes by choosing
experimentally known isotopes of C and O as clusters, where each line
represents half-lives of all considered clusters from the corresponding one
parent isotope. Similarly, for the trans-lead region, we have shown plots (c)
and (d) for Ra and Th isotopes for the emission of isotopes of C and O,
respectively. Towards the superheavy region, heavy clusters Kr and Sr isotopes
are estimated in plots (e) and (f) for Z=118 (Og) and Z=120 isotopes,
respectively. A parabolic trend is clearly visible from all the plots and with
all the extensive analysis in the range 56$\leq$Z$\leq$120 (not shown here),
from which one can select a minimum (indicating least half-life) of each
parabola corresponding to the potential cluster nucleus for a particular
parent nucleus. From this selection, we have extracted the most probable
cluster for each considered isotope in the chosen range.

Table 4: Competition among various decay modes viz. cluster decay,
$\alpha$-decay, and $\beta$-decay for trans-tin nuclei in the range
56$\leq$Z$\leq$81\. Experimental BR and half-life are taken from NUBASE2020
audi20201  and Guglielmetti1997PRC , respectively. Cluster decay,
$\alpha$-decay half-lives are calculated by using ITM-CT and ITM-AT formulas,
respectively (present work), and for the $\beta$-decay half-lives formula
given by Fiset and Nix Fiset1972  has been used.

Parent | Expt. Data | Theoretical Estimates for Cluster Emission
---|---|---
Nucleus | logT1/2 | BR | Emitted | Daughter |  $Q$-value | $l$ | logT1/2 (s) | BR (in $\%$)
| (s) | (in $\%$) | Cluster | Nucleus | (MeV) |  | Cluster | $\alpha$ | $\beta$ | Cluster | $\alpha$ | $\beta$
114Ba |  $>$4.10 |  $\beta^{+}\approx$100;$\beta^{+}$p=20; |  12C |  102Sn | 19.00 | 0 | $4.68$ | $2.48$ | $-0.34$ | $0.0010$ | $0.1431$ | $99.8560$
|  |  $\alpha$=0.9;12C$<$0.0034 |  |  |  |  |  |  |  |  |  |
111La | - | - |  13N |  98Sn | 25.64 | 1 | $1.27$ | $-5.00$ | $2.19$ | $0.0001$ | $99.9999$ | $0.0000$
112La | - | - |  13N |  99Sn | 26.15 | 3 | $0.26$ | $-3.09$ | $1.84$ | $0.0438$ | $99.9550$ | $0.0012$
113La | - | - |  13N |  100Sn | 26.61 | 1 | $-0.51$ | $-1.33$ | $2.41$ | $13.1268$ | $86.8573$ | $0.0160$
112Ce | - | - |  17O |  95Sn | 32.55 | 2 | $0.85$ | $-5.12$ | $2.77$ | $0.0001$ | $99.9999$ | $0.0000$
113Ce | - | - |  17O |  96Sn | 33.58 | 2 | $-1.05$ | $-3.41$ | $2.13$ | $0.4308$ | $99.5689$ | $0.0003$
114Ce | - | - |  17O |  97Sn | 34.73 | 2 | $-3.04$ | $-2.79$ | $2.95$ | $63.9793$ | $36.0206$ | $0.0001$
115Ce | - | - |  17O |  98Sn | 35.88 | 2 | $-4.84$ | $-2.76$ | $2.36$ | $99.1812$ | $0.8188$ | $0.0000$
116Ce | - | - |  16O |  100Sn | 32.30 | 0 | $1.92$ | $-2.65$ | $3.29$ | $0.0027$ | $99.9972$ | $0.0001$
117Ce | - | - |  16O |  101Sn | 31.45 | 3 | $5.06$ | $-0.52$ | $2.58$ | $0.0003$ | $99.9208$ | $0.0790$
118Ce | - | - |  16O |  102Sn | 30.07 | 0 | $5.71$ | $3.22$ | $3.48$ | $0.2104$ | $64.2798$ | $35.5098$
114Pr | - | - |  15F |  99Sn | 37.38 | 4 | $2.62$ | $-2.90$ | $1.63$ | $0.0003$ | $99.9967$ | $0.0030$
115Pr | - | - |  15F |  100Sn | 37.04 | 1 | $4.50$ | $-2.44$ | $2.11$ | $0.0001$ | $99.9953$ | $0.0046$
116Pr | - | - |  17F |  99Sn | 36.65 | 6 | $4.74$ | $-1.51$ | $1.75$ | $0.0001$ | $99.9460$ | $0.0540$
117Pr | - | - |  17F |  100Sn | 36.92 | 1 | $5.53$ | $-1.10$ | $2.33$ | $0.0003$ | $99.9383$ | $0.0614$
116Nd | - | - |  18Ne |  98Sn | 43.32 | 0 | $3.63$ | $-0.54$ | $2.70$ | $0.0067$ | $99.9357$ | $0.0576$
117Nd | - | - |  18Ne |  99Sn | 43.11 | 2 | $5.87$ | $1.22$ | $2.08$ | $0.0020$ | $87.8970$ | $12.1011$
118Nd | - | - |  18Ne |  100Sn | 42.68 | 0 | $5.42$ | $2.30$ | $2.86$ | $0.0584$ | $78.4906$ | $21.4510$
121Sm | - | - |  22Mg |  99Sn | 54.21 | 0 | $7.51$ | $4.23$ | $1.94$ | $0.0003$ | $0.5094$ | $99.4903$

This systematic study indeed provides the most probable cluster for each
considered isotope, however, its lowest half-life does not guarantee the
chances of cluster decay due to its contest with other probable decay modes.
For example, as per experimental systematics, $\beta$-decay and $\alpha$-decay
are the principal decay modes for the trans-tin region whereas $\alpha$-decay
and spontaneous fission (SF) play the primary roles in the trans-lead and
superheavy region. As a result, the chances of cluster decay are always
dependent on its competition with other primary decay modes. With this in
view, we have calculated half-lives of other decay modes using (i) ITM-AT and
ITM-AL formulas (mentioned in Table 1) for the estimation of half-lives of
$\alpha$-decay in trans-tin and trans-lead region, respectively, (ii) Fisset
and Nix formula Fiset1972  for $\beta$-decay in trans-tin region, and (iii)
Modified Bao Formula (MBF) Saxena2021jpg  for spontaneous fission in trans-
lead and superheavy regions. These formulas for $\beta$-decay and SF are
recently been applied and found very successful in determining the respective
half-lives in the heavy and superheavy regions singh2020 ; Saxena2021jpg ;
jain2023PS ; rsharma2022 ; saxena2021QF . The competition can be quantized in
the form of branching ratios (BR) which are defined as:

$BR=\frac{T^{Th.}_{1/2}}{T^{Cluster/\alpha/\beta/SF}_{1/2}},$ (5)

where, $T^{Th.}_{1/2}$ is the total half-life calculated by considering half-
lives of all decay modes (cluster, $\alpha$, and $\beta$/SF), and the relation
is given by:

$\frac{1}{T^{Th.}_{1/2}}=\frac{1}{T^{Cluster}_{1/2}}+\frac{1}{T^{\alpha}_{1/2}}+\frac{1}{T^{\beta/SF}_{1/2}},$
(6)

where the superscripts refer to the half-lives of concerned decay modes. The
superscript $\beta$/SF refers to the consideration of $\beta$-decay (for
trans-tin region) or SF (for trans-lead and superheavy regions). A good test
of our predictions of all the used formulas can be performed using Eqn. (5)
from which we can compare values of BR of various decay modes from our
theoretical half-lives with the BR available in NUBASE2020 audi20201 . Taking
this into consideration, we have listed the theoretical and experimental BR
(in percentage) for several known cluster decay audi20201 ; Bonetti2007 ;
Gupta1994 ; soylu2021  in Table 3. The dominance of $\alpha$-decay can be
easily verified by Table 3 because the value of concern BR is near to 100$\%$.
In contrast, the values of BR for cluster decay are very less, however, the
experimental values of BR for cluster decay are indeed reproduced very well
from the half-life obtained by using the ITM formula. This excellent match
endorses our approach of determining BR by using ITM and other formulas for
our considered unknown domain of the periodic chart.

In Table 4, a worthful competition among various decay modes viz. cluster
decay, $\alpha$-decay, and $\beta$-decay is listed for the trans-tin region in
the range 56$\leq$Z$\leq$81\. As mentioned earlier this whole systematic study
provides many clusters i.e. one probable cluster for each considered isotope
(as obtained by the minimum of each parabola of Fig. 2) but only those are
mentioned in the Table 4 which are found with reasonable BR as compared to
other decay modes. More than 15 cluster decays are found probable in the
trans-tin region, and in the future, looking into the experimental progress
one may expect more observation of cluster decay in the trans-tin region where
so far only one candidate i.e. 114Ba is detected. It is noticeable from Table
4 that two mentioned clusters have values of BR more than 60$\%$ or more than
$\alpha$ or $\beta$ decay modes and at the same time the values of BR for a
few nuclei are several orders of magnitude larger than the BR of
experimentally known clusters (please see Table 3). Therefore, this comparison
certainly anticipates cluster decay as a prominent mode of decay in several
nuclei of the trans-tin region.

Table 5: Same as Table 4, but for trans-lead and superheavy regions in the
range 88$\leq$Z$\leq$120\. Cluster decay and $\alpha$-decay half-lives are
calculated by using ITM-CL and ITM-AL formulas, respectively (present work).
For the SF half-lives MBF formula Saxena2021jpg  has been used.

Parent | Expt. Data | Theoretical Estimates for Cluster Emission
---|---|---
Nucleus | logT1/2 | BR | Emitted | Daughter |  $Q$-value | $l$ | logT1/2 (s) | BR (in $\%$)
| (s) | (in $\%$) | Cluster | Nucleus | (MeV) |  | Cluster | $\alpha$ | SF | Cluster | $\alpha$ | SF
285Mc | - | - |  81As |  204Pb | 272.66 | 2 | $2.95$ | $-0.91$ | $6.02$ | $0.01$ | $99.99$ | $0.00$
289Mc | - | - |  81As |  208Pb | 272.95 | 2 | $4.33$ | $0.16$ | $7.91$ | $0.01$ | $99.99$ | $0.00$
275Lv | - | - |  80Se |  195Pb | 284.92 | 6 | $-3.26$ | $-5.81$ | $-5.24$ | $0.22$ | $78.64$ | $21.14$
276Lv | - | - |  80Se |  196Pb | 284.87 | 0 | $-5.09$ | $-7.47$ | $-6.05$ | $0.40$ | $95.92$ | $3.69$
277Lv | - | - |  80Se |  197Pb | 284.48 | 2 | $-3.65$ | $-6.19$ | $-1.04$ | $0.29$ | $99.71$ | $0.00$
278Lv | - | - |  80Se |  198Pb | 284.28 | 0 | $-4.02$ | $-6.77$ | $-5.64$ | $0.16$ | $92.93$ | $6.91$
279Lv | - | - |  80Se |  199Pb | 284.06 | 6 | $-1.23$ | $-4.93$ | $-1.86$ | $0.02$ | $99.89$ | $0.09$
280Lv | - | - |  80Se |  200Pb | 283.90 | 0 | $-3.04$ | $-6.71$ | $-2.25$ | $0.02$ | $99.98$ | $0.00$
281Lv | - | - |  80Se |  201Pb | 283.53 | 1 | $-2.00$ | $-3.64$ | $1.02$ | $2.25$ | $97.75$ | $0.00$
282Lv | - | - |  80Se |  202Pb | 283.39 | 0 | $-2.04$ | $-5.68$ | $-0.45$ | $0.02$ | $99.98$ | $0.00$
283Lv | - | - |  82Se |  201Pb | 283.49 | 1 | $-1.18$ | $-4.30$ | $3.33$ | $0.07$ | $99.93$ | $0.00$
284Lv | - | - |  82Se |  202Pb | 283.64 | 0 | $-1.33$ | $-4.63$ | $1.44$ | $0.05$ | $99.95$ | $0.00$
285Lv | - | - |  82Se |  203Pb | 283.48 | 1 | $-0.39$ | $-3.19$ | $4.69$ | $0.16$ | $99.84$ | $0.00$
286Lv | - | - |  82Se |  204Pb | 283.41 | 0 | $-0.47$ | $-3.55$ | $2.97$ | $0.08$ | $99.92$ | $0.00$
287Lv | - | - |  82Se |  205Pb | 283.44 | 1 | $0.40$ | $-3.47$ | $5.86$ | $0.01$ | $99.99$ | $0.00$
288Lv | - | - |  82Se |  206Pb | 283.60 | 0 | $0.22$ | $-3.47$ | $3.40$ | $0.02$ | $99.98$ | $0.00$
289Lv | -1.80 |  $\alpha$$?$ |  82Se |  207Pb | 283.43 | 3 | $1.88$ | $-2.35$ | $6.64$ | $0.01$ | $99.99$ | $0.00$
290Lv | -2.05$\pm$0.15 |  $\alpha$$\approx$100;SF$?$ |  84Se |  206Pb | 283.57 | 0 | $0.97$ | $-3.00$ | $4.41$ | $0.01$ | $99.99$ | $0.00$
291Lv | -1.59$\pm$0.22 |  $\alpha$$\approx$100; SF$?$ |  84Se |  207Pb | 283.77 | 1 | $1.77$ | $-2.20$ | $7.59$ | $0.01$ | $99.99$ | $0.00$
279Ts | - | - |  83Br |  196Pb | 294.22 | 0 | $-6.32$ | $-6.37$ | $-0.88$ | $47.04$ | $52.96$ | $0.00$
281Ts | - | - |  83Br |  198Pb | 293.88 | 0 | $-5.35$ | $-6.13$ | $2.40$ | $14.32$ | $85.68$ | $0.00$
283Ts | - | - |  83Br |  200Pb | 293.58 | 0 | $-4.42$ | $-5.54$ | $4.00$ | $7.13$ | $92.87$ | $0.00$
285Ts | - | - |  83Br |  202Pb | 293.17 | 0 | $-3.48$ | $-4.73$ | $5.98$ | $5.25$ | $94.75$ | $0.00$
287Ts | - | - |  85Br |  202Pb | 292.75 | 0 | $-2.54$ | $-3.97$ | $7.35$ | $3.58$ | $96.42$ | $0.00$
289Ts | - | - |  85Br |  204Pb | 292.92 | 0 | $-1.84$ | $-3.81$ | $8.50$ | $1.04$ | $98.96$ | $0.00$
291Ts | -2.70 |  $\alpha$$?$;SF$?$ |  85Br |  206Pb | 293.18 | 0 | $-1.18$ | $-3.26$ | $10.10$ | $0.82$ | $99.18$ | $0.00$
293Ts | -1.60$\pm$0.11 |  $\alpha$=100 |  85Br |  208Pb | 292.85 | 0 | $-0.33$ | $-3.04$ | $9.45$ | $0.20$ | $99.80$ | $0.00$
295Ts | - | - |  87Br |  208Pb | 291.24 | 2 | $1.83$ | $-2.34$ | $10.13$ | $0.01$ | $99.99$ | $0.00$
280Og | - | - |  84Kr |  196Pb | 305.72 | 0 | $-9.19$ | $-7.67$ | $-1.14$ | $97.09$ | $2.91$ | $0.00$
281Og | - | - |  86Kr |  195Pb | 304.59 | 1 | $-7.87$ | $-5.80$ | $-1.02$ | $99.16$ | $0.84$ | $0.00$
282Og | - | - |  84Kr |  198Pb | 304.93 | 0 | $-8.06$ | $-7.29$ | $-1.80$ | $85.57$ | $14.43$ | $0.00$
283Og | - | - |  84Kr |  199Pb | 304.54 | 1 | $-7.01$ | $-4.37$ | $1.54$ | $99.77$ | $0.23$ | $0.00$
284Og | - | - |  84Kr |  200Pb | 304.28 | 0 | $-7.01$ | $-6.82$ | $-0.25$ | $60.71$ | $39.29$ | $0.00$
285Og | - | - |  86Kr |  199Pb | 304.17 | 1 | $-6.06$ | $-6.54$ | $3.42$ | $24.99$ | $75.01$ | $0.00$
286Og | - | - |  86Kr |  200Pb | 304.14 | 0 | $-6.15$ | $-6.26$ | $1.59$ | $43.80$ | $56.20$ | $0.00$
287Og | - | - |  86Kr |  201Pb | 303.98 | 1 | $-5.20$ | $-6.04$ | $5.05$ | $12.50$ | $87.50$ | $0.00$
288Og | - | - |  84Kr |  204Pb | 302.62 | 0 | $-4.83$ | $-5.70$ | $3.17$ | $11.90$ | $88.10$ | $0.00$
289Og | - | - |  86Kr |  203Pb | 303.76 | 1 | $-4.34$ | $-5.64$ | $6.65$ | $4.75$ | $95.25$ | $0.00$
290Og | - | - |  86Kr |  204Pb | 303.69 | 0 | $-4.43$ | $-5.64$ | $4.50$ | $5.83$ | $94.17$ | $0.00$
291Og | - | - |  86Kr |  205Pb | 303.54 | 3 | $-2.78$ | $-4.43$ | $7.93$ | $2.16$ | $97.84$ | $0.00$
292Og | - | - |  86Kr |  206Pb | 303.53 | 0 | $-3.62$ | $-4.95$ | $6.14$ | $4.44$ | $95.56$ | $0.00$
293Og | -3.00 |  $\alpha$$?$ |  87Kr |  206Pb | 302.51 | 0 | $-2.89$ | $-4.07$ | $9.89$ | $6.21$ | $93.79$ | $0.00$
294Og | -3.15$\pm$0.20 |  $\alpha$$\approx$100;SF$?$ |  86Kr |  208Pb | 302.91 | 0 | $-2.66$ | $-4.84$ | $6.15$ | $0.67$ | $99.33$ | $0.00$
295Og | -0.175$\pm$0.47 |  $\alpha$$\approx$100 |  87Kr |  208Pb | 301.96 | 0 | $-1.97$ | $-4.26$ | $8.45$ | $0.51$ | $99.49$ | $0.00$
296Og | - | - |  88Kr |  208Pb | 301.70 | 0 | $-1.52$ | $-3.95$ | $5.65$ | $0.37$ | $99.63$ | $0.00$
299Og | - | - |  91Kr |  208Pb | 298.62 | 0 | $0.62$ | $-3.61$ | $7.57$ | $0.01$ | $99.99$ | $0.00$
287120 | - | - |  88Sr |  199Pb | 325.93 | 3 | $-10.67$ | $-5.90$ | $3.70$ | $100.00$ | $0.00$ | $0.00$
288120 | - | - |  88Sr |  200Pb | 325.54 | 0 | $-11.35$ | $-7.27$ | $2.03$ | $99.99$ | $0.01$ | $0.00$
289120 | - | - |  89Sr |  200Pb | 324.18 | 0 | $-10.50$ | $-7.23$ | $5.48$ | $99.95$ | $0.05$ | $0.00$
289120 | - | - |  90Sr |  199Pb | 323.31 | 1 | $-9.71$ | $-7.23$ | $5.48$ | $99.67$ | $0.33$ | $0.00$
290120 | - | - |  88Sr |  202Pb | 324.70 | 0 | $-10.27$ | $-7.19$ | $3.33$ | $99.92$ | $0.08$ | $0.00$
291120 | - | - |  88Sr |  203Pb | 324.10 | 1 | $-9.17$ | $-6.00$ | $7.39$ | $99.93$ | $0.07$ | $0.00$
292120 | - | - |  88Sr |  204Pb | 323.63 | 0 | $-9.13$ | $-6.78$ | $5.39$ | $99.55$ | $0.45$ | $0.00$
293120 | - | - |  90Sr |  203Pb | 322.58 | 1 | $-7.89$ | $-5.79$ | $9.56$ | $99.22$ | $0.78$ | $0.00$
294120 | - | - |  88Sr |  206Pb | 322.79 | 0 | $-8.08$ | $-6.38$ | $7.69$ | $98.07$ | $1.93$ | $0.00$
295120 | - | - |  89Sr |  206Pb | 322.26 | 0 | $-7.53$ | $-5.54$ | $10.85$ | $98.97$ | $1.03$ | $0.00$
296120 | - | - |  90Sr |  206Pb | 322.29 | 0 | $-7.16$ | $-6.51$ | $7.61$ | $81.86$ | $18.14$ | $0.00$
297120 | - | - |  89Sr |  208Pb | 321.13 | 0 | $-6.40$ | $-6.16$ | $10.10$ | $63.71$ | $36.29$ | $0.00$
298120 | - | - |  90Sr |  208Pb | 321.34 | 0 | $-6.11$ | $-5.91$ | $5.25$ | $61.19$ | $38.81$ | $0.00$
299120 | - | - |  91Sr |  208Pb | 320.72 | 2 | $-4.68$ | $-6.32$ | $7.91$ | $2.22$ | $97.78$ | $0.00$
300120 | - | - |  92Sr |  208Pb | 320.87 | 0 | $-5.23$ | $-6.40$ | $4.72$ | $6.34$ | $93.66$ | $0.00$
301120 | - | - |  93Sr |  208Pb | 320.00 | 0 | $-4.59$ | $-5.10$ | $7.08$ | $23.54$ | $76.46$ | $0.00$
302120 | - | - |  94Sr |  208Pb | 319.86 | 0 | $-4.19$ | $-5.65$ | $3.41$ | $3.38$ | $96.62$ | $0.00$
303120 | - | - |  95Sr |  208Pb | 318.35 | 1 | $-2.84$ | $-4.63$ | $5.39$ | $1.58$ | $98.42$ | $0.00$
304120 | - | - |  96Sr |  208Pb | 317.35 | 0 | $-2.65$ | $-5.39$ | $1.20$ | $0.18$ | $99.82$ | $0.00$
305120 | - | - |  94Sr |  211Pb | 315.96 | 0 | $-1.84$ | $-5.39$ | $0.60$ | $0.03$ | $99.97$ | $0.00$

In a similar way, the study has been performed for trans-lead nuclei together
with superheavy nuclei, and the probable clusters are mentioned in Table 5.
Various heavy clusters are found with comparable BR in this region of the
periodic chart and show a great probability of decay compared to
$\alpha$-decay and SF, especially for Z$=$118 and 120 isotopes similar to what
was found in Refs. zhang2018 ; soylu2019 ; santhosh2018 ; poenaru2018 ;
poenaru2018EPJA ; matheson2018 ; Warda2018 ; jain2023PS . In both the tables
(Tables 4 and 5), the theoretical half-lives of various decay modes are
mentioned in columns 8, 9, and 10 which by using Eqn. (6) provide total half-
lives with an excellent agreement with the available experimental half-lives
(mentioned in column 2). Hence, the prediction of various cluster decay in the
present article is reliable and of great importance due to its sizeable
probability over other prominent decay modes of heavy and superheavy regions.

##  4 Conclusions

An extensive analysis of cluster decay within the range 56$\leq$Z$\leq$120 is
carried out using an improved version of a semi-empirical formula in which
isospin and angular momentum effects are included. This improved Tavares and
Medeiros formula (ITM) is found with greater accuracy when compared with the
available data of cluster decay in the trans-lead region. The applicability of
ITM formula is demonstrated by considering daughter nuclei around the double
magic nucleus 100Sn for trans-tin nuclei and 208Pb for trans-lead and
superheavy nuclei. The formula is indeed found suitable for the trans-tin and
trans-lead regions including the superheavy domain of the periodic chart.
Competition of cluster decay mode with other probable decay modes is
visualized in the form of branching ratios (BR) which are also found in an
excellent match with the available experimental data. Various clusters with
substantial BR are reported separately for trans-tin, trans-lead, and
superheavy regions. This kind of comprehensive analysis is expected to
stimulate the wide region of the periodic chart (56$\leq$Z$\leq$120) in view
of cluster decay and to enforce theoretical and experimental studies eyeing
heavy-ion or cluster decays.

##  5 Acknowledgement

GS acknowledges the support provided by SERB (DST), Govt. of India under
SIR/2022/000566, and would like to thank Prof. Nils Paar for his kind
hospitality at the University of Zagreb, Croatia. AJ is indebted to Prof. S.
K. Jain, Manipal University, Jaipur, India for his guidance.

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##  Appendix A Materials with their Parameters

In order to simulate not only the degrader but also foils and collimator in
the beam transport line and the nozzle at the isocenter of the gantries, new
materials have been implemented in OPAL as listed in Tab. 6.

Material | Z | A |  $\rho$ [$g/cm^{3}$] |  X0 [$g/cm^{2}$] | A2 | A3 | A4 | A5
---|---|---|---|---|---|---|---|---
Aluminum | 13 | 26.98 | 2.7 | 24.01 | 4.739 | 2766 | 164.5 | 2.023E-02
Copper | 29 | 63.54 | 8.96 | 12.86 | 4.194 | 4649 | 81.13 | 2.242E-02
Graphite | 6 | 12 | 2.210 | 42.7 | 2.601 | 1701 | 1279 | 1.638E-02
GraphiteR6710 | 6 | 12 | 1.88 | 42.7 | 2.601 | 1701 | 1279 | 1.638E-02
Titan | 22 | 47.8 | 4.54 | 16.16 | 5.489 | 5260 | 651.1 | 8.930E-03
Alumina | 50 | 101.96 | 3.97 | 27.94 | 7.227 | 11210 | 386.4 | 4.474E-03
Air | 7 | 14 | 0.0012 | 37.99 | 3.350 | 1683 | 1900 | 2.513E-02
Kapton | 6 | 12 | 1.4 | 39.95 | 2.601 | 1701 | 1279 | 1.638E-02
Gold | 79 | 197 | 19.3 | 6.46 | 5.458 | 7852 | 975.8 | 2.077E-02
Water | 10 | 18 | 1 | 36.08 | 2.199 | 2393 | 2699 | 1.568E-02
Mylar | 6.702 | 12.88 | 1.4 | 39.95 | 3.350 | 1683 | 1900 | 2.5133E-02
Beryllium | 4 | 9.012 | 1.848 | 65.19 | 2.590 | 966.0 | 153.8 | 3.475E-02
Mo | 4.68 | 95.94 | 10.22 | 9.8 | 7.248 | 9545 | 480.2 | 5.376E-03
Table 6: List of materials with their parameters implemented in OPAL

The $A_{i}$ coefficients (i = 2…5) for the energy loss calculation at low
energy have been found on ICRU [39] (Table 3.1).