# Manifestation of deformation and nonlocality in $\alpha$ and cluster decay

D. F. Rojas-Gamboa<EMAIL_ADDRESS>Departamento de Física,
Universidad de Los Andes, Carrera 1 No.18A-10, Bogotá 111711, Colombia  J. E.
Perez Velasquez<EMAIL_ADDRESS>Departamento de Física,
Universidad de Los Andes, Carrera 1 No.18A-10, Bogotá 111711, Colombia  N. G.
Kelkar<EMAIL_ADDRESS>Departamento de Física, Universidad de Los
Andes, Carrera 1 No.18A-10, Bogotá 111711, Colombia  N. J. Upadhyay
<EMAIL_ADDRESS>Department of Sciences, Amrita School of Engineering,
Amrita Vishwa Vidyapeetham, Chennai 601103, India

###### Abstract

It is common to study the strong decay of a heavy nucleus as a tunneling
phenomenon where the $\alpha$ (4He) or a light nuclear cluster tunnels through
the Coulomb barrier formed by its interaction with the heavier daughter
nucleus. The position and width of the Coulomb barrier is determined by the
total interaction potential between the two daughter nuclei. We examine the
effects of including nonlocality and deformation in the interaction potential
by calculating the half-lives, $t_{1/2}$, and thereby phenomenological
preformation factors of several nuclei which have the possibility of decaying
by emitting either an $\alpha$ or a light nuclear cluster. The effect of
deformation manifests itself by a decrease in $t_{1/2}$ for all the decays
studied. The effect of nonlocality is studied within two different models of
the nuclear potential: the energy independent but angular momentum ($l$)
dependent Mumbai (M) potential and the energy dependent Perey-Buck (PB)
potential. The nonlocal nuclear interaction leads to a decrease in all half-
lives studied. Though the decrease is larger due to the Perey-Buck potential,
half-lives evaluated using the Mumbai potential show a strong sensitivity to
the $l$ value in the decay. This feature of the $\alpha$ and cluster decay
half-lives can provide a complementary tool in addition to scattering data
which are more commonly used to fix the parameters of a nonlocal potential in
literature.

###### pacs:

23.60.+e, 21.10.Tg, 21.30.-x

††preprint: APS/123-QED

##  I Introduction

$\alpha$ decay of radioactive nuclei, discovered in 1899 by Rutherford
Rutherford (1899), is one of the most important decay modes in providing
information about the structure of heavy nuclei such as the radius,
deformation, and shell effects Ni and Ren (2009); Qian  _et al._ (2011); Mohr
(2017). In addition to this, cluster radioactivity (CR) has also been very
instrumental in providing structure information of heavy nuclei Seif  _et al._
(2021). The spontaneous emission of light fragments heavier than the $\alpha$
particle, but lighter than a typical mass of a light fragment in the fission
process (A $\gtrsim$ 60) is referred to as cluster radioactivity or cluster
decay. Theoretically predicted cluster decay Sandulescu  _et al._ (1980) was
experimentally confirmed in 1984 by Rose and Jones Rose and Jones (1984).
Study of both the $\alpha$ and cluster decays for the isotopes of the same
element provides insight about the impact of shell effects and deformations on
the probabilities of these processes. In an investigation involving the
comparison of different models for cluster radioactivity in superheavy nuclei
Zhang and Wang (2018), the authors found that a universal decay law which can
predict light and heavy CR leads to the shortest half-lives when the daughter
nuclei are around the doubly magic 208Pb nucleus. Another interesting
conclusion was that the CR dominates over $\alpha$ decay for $Z\geq$ 118
nuclei. Because of the large difference in the masses of the cluster and the
daughter nucleus, just like the $\alpha$ decay, the cluster decay is also
treated as a tunneling problem for the calculation of half-lives of nuclei.

In a recent work Perez Velasquez  _et al._ (2019), we have investigated the
effects of nonlocality on $\alpha$-decay half-lives of nuclei. It is observed
that the effective potential obtained between the $\alpha$ and the daughter
nucleus within the nonlocal framework decreases the half-lives. This study was
limited to $\alpha$ decay involving spherical nuclei and deformation was not
considered. In the present work, we study the nonlocal effect for the $\alpha$
and cluster decay modes of a given parent nucleus. Furthermore, we also study
the impact of nonlocality in the presence of deformation which modifies the
barrier characteristics. In the $\alpha$ decay, the shape of the daughter
nucleus contributes to the deformation term, whereas in the cluster decay, the
shape of the lighter cluster in the decay channel contributes to the
deformation.

The article is organized as follows. In Sec. II, we give a brief overview of
the density-dependent double-folding (DF) model used to evaluate the potential
between the cluster and the daughter nucleus followed by the formalism for the
evaluation of decay half-lives. The two models used to study the nonlocal
effects in the $\alpha$ and cluster decay of some heavy nuclei are also
discussed. In order to extend the half-life calculations in the presence of
deformation, we provide the details about the deformed DF nuclear potential
and calculation of the half-lives from the deformed decay width. In Sec. III,
we present the results and discuss them. Finally, in Sec. IV we summarize our
findings.

##  II Formalism for $\alpha$ and cluster decay

In previous studies on $\alpha$ and cluster decay Ni and Ren (2009); Qian  _et
al._ (2011); Mohr (2017), it has been found that the half-life is an important
observable related to the nuclear structure. In most of the theoretical
models, the $\alpha$ and cluster decay half-life of a nucleus is evaluated as

$t_{1/2}=\frac{\hbar\ln 2}{\Gamma}\,,$ (1)

where the decay width $\Gamma$ can be calculated within the framework of the
semiclassical Jeffreys-Wentzel–Kramers–Brillouin (JWKB) approximation Gurvitz
and Kalbermann (1987); Kelkar and Castañeda (2007); Kelkar and Nowakowski
(2016), as

$\Gamma=P_{i}\,\frac{\hbar^{2}}{2\mu}\left[\int_{r_{1}}^{r_{2}}\frac{{\rm
d}r}{k(r)}\right]^{-1}\exp\left[-2\int_{r_{2}}^{r_{3}}k(r)\,{\rm
d}r\right]\,,$ (2)

where $k(r)=\sqrt{\frac{2\mu}{\hbar^{2}}|V(r)-E|}$. $P_{i}$ (with $i$ =
$\alpha$ or $c$) is the preformation factor in $\alpha$ or cluster ($c$)
decay, $\mu$ is the reduced mass of the cluster-daughter system and $r_{i}$
($i=1,2,3$) are the three classical turning points. These latter are solutions
of $V(r_{i})=E$, where the energy $E$ of the $\alpha$ particle or cluster is
usually taken to be the same as the $Q$-value for the decay. The exponential
factor $\exp{[-2\int_{r_{2}}^{r_{3}}k(r)\,{\rm d}r]}$ is known as the
penetration probability and the factor $[\int_{r_{1}}^{r_{2}}{\rm
d}r/k(r)]^{-1}$ in front of this arises from the normalization of the bound-
state wave function in the region between the turning points $r_{1}$ and
$r_{2}$ Gurvitz and Kalbermann (1987); Kelkar and Castañeda (2007); Kelkar and
Nowakowski (2016). An essential ingredient for evaluating both factors is the
interaction potential $V(r)$ between the emitted particle and the daughter
nucleus that will be explained below. The appearance of a preformation factor
$P_{i}$ ($i$ = $\alpha$ or $c$) in Eq. (2) takes into account the non-zero
probability for the existence of the preformed cluster or $\alpha$ particle
inside the parent as suggested by several theoretical models Xu  _et al._
(2016, 2017). In the present study, we compare the effects of nonlocalities
and deformation in different models. Hence, we shall set $P_{i}=1$ when we
compare the half-lives calculated within the different models.

###  II.1 Double folding model

$\alpha$ and cluster decay are treated as a tunneling problem where the
emitted particle, preformed inside the parent nucleus, tunnels through the
potential barrier. The interaction potential between the daughter nucleus and
the cluster consists of the nuclear $V_{N}$, Coulomb $V_{C}$, and centrifugal
potentials and is given by

$V(r)=\lambda\,V_{N}(r)+V_{C}(r)+\frac{\hbar^{2}\left(l+\frac{1}{2}\right)^{2}}{2\mu
r^{2}}\,,$ (3)

where $r$ is the separation between the center of mass of the cluster and the
center of mass of the daughter nucleus, $l$ is the angular momentum carried by
the emitted cluster, and $\lambda$ is the strength of the nuclear interaction
(see Fig. 1). The latter has been left as a parameter in order to ensure that
the Bohr-Sommerfeld quantization condition Kelkar and Castañeda (2007) is
satisfied. Thus $\lambda$ fixes the strength of the nuclear potential. The
centrifugal term containing, $l(l+1)$, is replaced by the correction,
$(l+\frac{1}{2})^{2}$, introduced by Langer when one uses the semiclassical
approximation Langer (1937). This is essential to ensure the correct behavior
of the JWKB wave function near the origin. The potential used in the
calculation then has three turning points, $r_{1}$, $r_{2}$, and $r_{3}$.

Figure 1: Coordinates used in the DF calculations. Vectors $\mathbf{r}_{c}$
and $\mathbf{r}_{d}$ correspond to the position of a nucleon in the cluster
and daughter nucleus, respectively.

For the nuclear potential it is desirable to relate the nucleus-nucleus
interaction to the NN interaction. This can be obtained by folding an
effective NN interaction over the density distribution of the two nuclei. This
model is known as the double folding (DF) model Satchler and Love (1979). It
has been used widely and has been found to be quite successful in evaluating
the $\alpha$ decay half-lives Kelkar and Castañeda (2007); Kelkar and
Nowakowski (2016); Perez Velasquez  _et al._ (2019). The folded nuclear
potential is written as

$V_{N}(\mathbf{r})=\int\mathrm{d}\mathbf{r}_{c}\mathrm{d}\mathbf{r}_{d}\,\rho_{1}\left(\mathbf{r}_{c}\right)v_{N}\left(|\mathbf{s}|=\left|\mathbf{r}+\mathbf{r}_{c}-\mathbf{r}_{d}\right|\right)\rho_{2}\left(\mathbf{r}_{d}\right)\,.$
(4)

where $\rho_{i}$ ($i=$ 1, 2) are the densities of the cluster and the daughter
nucleus in a decay, and $v_{N}(|\mathbf{s}|)$ is the nucleon-nucleon (NN)
interaction. The matter density distribution of the daughter or cluster
nucleus can be calculated as

$\rho(r)=\frac{\rho_{0}}{1+\exp\left(\frac{r-R}{a}\right)}\,,$ (5)

where $\rho_{0}$ is obtained by normalizing $\rho(r)$ to the mass number,
$\int\rho(\mathbf{r})\,{\rm d}\mathbf{r}=A$, and the constants are given as
$R=1.07A^{1/3}$ fm and $a=0.54$ fm. If the emitted cluster is an $\alpha$
particle, its matter density distribution is given using a standard Gaussian
form Satchler and Love (1979), namely,

$\rho_{\alpha}(r)=0.4229\exp\left(-0.7024\,r^{2}\right)\,.$ (6)

A popular choice of the effective NN interaction is based on the M3Y-Reid-type
soft core potential,

$v_{N}\left(|\mathbf{s}|\right)=7999\frac{\exp\left(-4\left|\mathbf{s}\right|\right)}{4\left|\mathbf{s}\right|}-2134\frac{\exp\left(-2.5\left|\mathbf{s}\right|\right)}{2.5\left|\mathbf{s}\right|}\,,$
(7)

where $|\mathbf{s}|=\left|\mathbf{r}+\mathbf{r}_{c}-\mathbf{r}_{d}\right|$ is
the distance between a nucleon in the daughter nucleus and a nucleon in the
cluster (Fig. 1). This consists of a short-ranged repulsion and a long-ranged
attraction responsible for the direct component of the interaction. An
additional zero-range contribution $J_{00}\delta(\mathbf{s})$ with
$J_{00}=-276(1-0.005\,E/A_{c})$ called knock-on exchange term takes into
account the antisymmetrization of identical nucleons in the cluster and the
daughter nucleus. The latter represents a kind of nonlocality in the DF
potential and is usually not included in the calculation if one uses nonlocal
nuclear potentials, in order to avoid double counting. Hence, the results in
the present work will also be presented without the inclusion of the knock-on
exchange term.

As mentioned above, the nuclear potential is modified, multiplying by a
constant $\lambda$ such that the Bohr-Sommerfeld quantization condition is
satisfied:

$\int_{r_{1}}^{r_{2}}k(r)\,{\rm d}r=\left(n+\frac{1}{2}\right)\pi\,,$ (8)

where $k(r)=\sqrt{\frac{2\mu}{\hbar^{2}}|V(r)-Q|}$ and $n$ is the number of
nodes of the quasibound wave function of the cluster nucleus relative motion.
This is expressed as $n=(G-l)/2$, where $G$ is a global quantum number and $l$
is the orbital angular momentum quantum number. The calculated optimal value
of $G$ for cluster decay is chosen according to the Wildermuth-Tang condition
Wildermuth (2013); Ni and Ren (2010),

$G=2n+l=\sum_{i=1}^{A_{c}}\left(g_{i}^{(A_{c}+A_{d})}-g_{i}^{(A_{c})}\right)\,,$
(9)

where $g_{i}^{(Ad+Ac)}$ are the oscillator quantum numbers of the nucleons
forming the cluster required to ensure that the cluster is completely outside
the shell occupied by the daughter nucleus, and $g_{i}^{(Ac)}$ are the
internal quantum numbers of the nucleons in the emitted cluster. The values of
$g_{i}$ are taken as $g_{i}=4$ for nucleons in the $50\leq Z$, $N\leq 82$
shell, $g_{i}=5$ for nucleons in the $82<Z$, $N\leq 126$ shell, $g_{i}=6$ for
nucleons in the $N\leq 184$, and $g_{i}=7$ for nucleons outside the $N=184$
neutron shell closure. These values correspond to the $4\hbar\omega$,
$5\hbar\omega$, $6\hbar\omega$, and $7\hbar\omega$ harmonic-oscillator shells,
respectively.

The condition in Eq. (9) ensures that the bound state between the light
cluster and the heavy daughter nucleus is an allowed state. It gives us the
global quantum number, $G$, that relates the number of nodes, $n$, to the
shell model (in this case within a harmonic oscillator basis) and ensures that
the Pauli exclusion principle is satisfied Ni and Ren (2010). In other words,
it ensures that the $\alpha$ or light cluster is outside the shell occupied by
the daughter. This nonlocality is not the same as that included through the
exchange term or the nonlocal framework of the present work. We refer the
reader to Refs. Seif and Abdelhady and Adel (2020); Amiri and Ghodsi (2020)
for possible ambiguities introduced by the introduction of this condition in
the case of certain kind of potentials.

The Coulomb potential $V_{C}(r)$ is obtained by using a similar DF procedure
with the matter density distributions being replaced by their respective
charge density distributions, $\rho^{C}(r)$. The charge density distributions
are taken to have a similar form as that of the matter densities but they are
normalized to the number of protons, $\int\rho^{C}(\mathbf{r})\,{\rm
d}\mathbf{r}=Z$. The fundamental interaction is the standard proton-proton
Coulomb interaction
$v_{C}\left(\mathbf{s}\right)=\frac{e^{2}}{\left|\mathbf{s}\right|}$. For both
nuclear and Coulomb potentials six-dimensional integrals are involved. These
are solved numerically by working in momentum space. The detailed procedure
can be found in Ref. Satchler and Love (1979).

Figure 2: The nuclear interaction potential between 4He and different isotopes
of thorium (Th) in the $\alpha$ decay of the uranium (U) isotopes. The boxes
show a comparison of the effective potentials, $U^{\rm{PB}}(r)$ (dash-dotted
line), $U_{l}^{\rm{M}}(r)$ (dashed line) and the DF potential $V_{N}(r)$
(solid line) for decays involving different values of the angular momentum
quantum number, $l$. Figure 3: Same as Fig. 2 but for the different cluster
decay modes of the Uranium (U) isotopes.

###  II.2 Nonlocality

The existence of nonlocal interactions is a purely quantum mechanical
phenomenon. This phenomenon arises because of the very nature of nucleons.
When calculating an interaction potential between nuclei, a nonlocality effect
appears. This kind of effect has often been discussed in literature in the
context of nucleon-nucleus scattering Ahmad  _et al._ (1975); Cândido Ribeiro
_et al._ (1997); Rotureau  _et al._ (2017). Taking into account nonlocal
approaches improves the description of nuclear interactions, however,
introducing nonlocality results in an integro-differential Schrödinger
equation which can be, in principle, quite difficult to solve. The general
form of this equation is written as,

$-{\hbar^{2}\over 2\mu}\nabla^{2}\Psi({\bf r})+[V_{\rm L}({\bf r})-E]\Psi({\bf
r})\\\ =-\int\,{\rm d}{\bf r}^{\prime}\,V_{\rm NL}({\bf r},{\bf
r}^{\prime})\Psi({\bf r}^{\prime})\,,$ (10)

where $V_{\rm L}$ can be some isolated local potential and $V_{\rm NL}$ can be
the nonlocal one.

The sources of nonlocalities in literature are globally classified into two
types: the Feshbach and the Pauli nonlocalities Balantekin  _et al._ (1998).
The Feshbach nonlocality is attributed to inelastic intermediate transitions
in scattering processes. In other words, the description of an excitation at a
point r in space followed by an intermediate state which propagates and de-
excites at some point r′ to get back to the elastic channel (virtual
excitations) is contained in the right-hand side of Eq. (10). In this work, we
shall not consider the Feshbach nonlocality. The Pauli nonlocality is
attributed to the exchange effects which require antisymmetrization of the
wave function between the projectile and the target.

In order to solve Eq. (10), it is necessary to know the explicit form of
$V_{\rm NL}({\bf r},{\bf r}^{\prime})$, which as such is not known. However,
we do know that $V_{\rm NL}$ should be such, that in the limit of vanishing
nonlocality (represented by a nonlocality parameter $b$, with $b\to 0$), the
integrodifferential Schrödinger equation reduces to the homogeneous one.
Assuming this, a nonlocal kernel is usually described in literature Perey and
Buck (1962); Chamon  _et al._ (2003); Upadhyay  _et al._ (2017) in terms of a
factorized form of the potential. The latter is then used to obtain an
effective (local) potential which can be used in the homogeneous Schrödinger
equation.

In this article, we use two nonlocality models to construct the effective
local potential and study the effects of the nonlocality in the $\alpha$ and
cluster decay of some heavy nuclei which can decay by both modes. The first
approach we consider in which nonlocality is included was proposed by Perey
and Buck in the early 1960s Perey and Buck (1962). In the Perey-Buck (PB)
model, an energy-independent nonlocal potential for elastic neutron-nucleus
scattering is suggested in order to study how far the energy dependence of
phenomenological local potentials which had been used earlier could be
accounted for by nonlocality. To facilitate calculations, the authors assumed
that the nonlocal kernel can be written using a separable form with a local
potential times a Gaussian part which contains the nonlocality

$V_{NL}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\frac{1}{\pi^{3/2}b^{3}}U_{N}\left(\frac{1}{2}\left|\mathbf{r}+\mathbf{r}^{\prime}\right|\right)\exp\left[-\left(\frac{\mathbf{r}-\mathbf{r}^{\prime}}{b}\right)^{2}\right]\,,$
(11)

where $b$ is the nonlocality range parameter and $U_{N}$ is the energy
independent potential. The nonlocality range parameter in the case of nucleus-
nucleus scattering can be expressed as $b=b_{0}m_{0}/\mu$, where $b_{0}$ is
the nonlocal range of the nucleon-nucleus interaction, $m_{0}$ is the nucleon
mass, and $\mu$ is the $\alpha$-daughter or cluster-daughter reduced mass
Jackson and Johnson (1974). For our calculations, we choose $b_{0}=0.85$ fm.
By using the factorized form of Eq. (11), the nonlocal Schrödinger equation is
given as

$\biggl{[}{\hbar^{2}\over 2\mu}\nabla^{2}+E\biggr{]}\,\Psi_{N}({\bf r})=\\\
\frac{1}{\pi^{3/2}b^{3}}\int
U_{N}\left(\frac{1}{2}\left|\mathbf{r}+\mathbf{r}^{\prime}\right|\right)\exp\left[-\left(\frac{\mathbf{r}-\mathbf{r}^{\prime}}{b}\right)^{2}\right]\,{\rm
d}{\bf r}^{\prime}\,.$ (12)

By assuming that the nonlocal and local wavefunctions are approximately the
same [$\Psi_{N}({\bf r})\approx\Psi_{L}({\bf r})$], and the potential inside
the nucleus is constant, the authors finally obtained a local equivalent
potential given by

$U^{PB}(r)\exp\left\\{\frac{\mu
b^{2}}{2\hbar^{2}}\left[E-U^{PB}(r)\right]\right\\}=U_{N}(r)\,.$ (13)

The $r$ dependence in the equation above is introduced by hand and is
justified a posteriori due to the success of the model in reproducing data.
Finally, the transcendental equation [Eq. (13)] is solved to obtain
$U^{\rm{PB}}(r)$. The approximations used by the authors introduce an
inconsistency at small $r$ Perez Velasquez  _et al._ (2019), which as we will
see later leads to differences in the results between the two nonlocality
models we use.

The second approach in which nonlocality is included was developed in Ref.
Upadhyay  _et al._ (2017). We will refer to this approach as the Mumbai (M)
model. In this approach, the authors use the mean value theorem of integral
calculus to solve the integrodifferential equation (10). Performing a partial
wave expansion of $V_{\rm NL}({\bf r},{\bf r}^{\prime})$ and $\Psi({\bf
r}^{\prime})$ in Eq. (10), they obtained the radial equation, which, in the
absence of the spin-orbit and Coulomb interaction was given as

$\frac{h^{2}}{2\mu}\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}\right)u_{l}(r)+Eu_{l}(r)\\\
=\int_{0}^{\infty}{g_{l}(r,r^{\prime})u_{l}(r^{\prime})\,{\rm
d}r^{\prime}}\,,$ (14)

where

$g_{l}\left(r,r^{\prime}\right)=\left(\frac{2rr^{\prime}}{\sqrt{\pi}b^{3}}\right)\exp\left(-\frac{r^{2}+r^{\prime
2}}{b^{2}}\right)\\\
\times\int_{-1}^{1}U_{N}\left(\frac{\left|\mathbf{r}+\mathbf{r}^{\prime}\right|}{2}\right)\exp\left(\frac{2rr^{\prime}x}{b^{2}}\right)P_{l}(x)\,{\rm
d}x\,.$ (15)

By making use of the mean value theorem to rewrite the integral on the right-
hand side of Eq. (14), and after some simplifications, the authors obtained
the homogenized form of the Schrödinger equation,

$\left[\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}+\frac{2\mu}{\hbar^{2}}\Big{(}E-U_{l}^{M}(r)\Big{)}\right]u_{l}(r)=0\,,$
(16)

with an effective strong potential given by Upadhyay  _et al._ (2017)

$U_{l}^{\rm{M}}(r)=\int_{0}^{r_{m}}\,g_{l}(r,r^{\prime})\,{\rm
d}r^{\prime}\,,$ (17)

where $g_{l}(r,r^{\prime})$ is written as in Eq. (15). The upper limit of the
integration over $g_{l}(r,r^{\prime})$ is numerically chosen to be the
distance at which the integral becomes negligible, which in our case is
essentially the range of the nuclear interaction. In contrast to the PB model,
the effective potential obtained in the Mumbai approach is energy independent.
However, it is dependent on the angular momentum quantum number, $l$. As a
result of these two differences, it shows a different behavior at small
distances with $U_{l}^{\rm{M}}\to 0$ for $r\to 0$. We refer the reader to Ref.
Perez Velasquez  _et al._ (2019) for a detailed discussion of the above
differences in the PB and M models. Here we show a comparison of the DF
potential $V_{N}(r)$ and the effective potentials in Figs. (2) and (3). To
find the PB local equivalent potential $U^{\rm{PB}}(r)$, we replace $U_{N}(r)$
with $V_{N}(r)$ and use Eq. (13). The local effective Mumbai potential
[$U_{l}^{\rm{M}}(r)$] is found using Eq. (17) by doing a similar replacement:
namely, $U_{N}$ in Eq. (15) is replaced by $V_{N}(r)$. We choose different
isotopes of the same nucleus, uranium (U) for comparison in order to see the
effect of the $l$ dependence in the potentials. As expected, the behavior of
the M and PB potentials is quite different at small distances. Due to the
approximation used in the PB model, the potential appears to be almost the
same for the different decays and is insensitive to the change in the $l$
value. This pattern is more evident in $\alpha$ decay than in cluster decay.

A few comments regarding the inclusion of antisymmetrization and medium
effects within a nonlocal framework are in order here. The authors in Refs.
Adel and Alharbi (2017, 2020) have studied the cluster and $\alpha$ decays of
several nuclei within a formalism which includes the antisymmetrization
through the finite-range exchange part of the nucleon-nucleon interaction Khoa
(2020). The folding potential in this case becomes energy dependent and
becomes nonlocal through the exchange term. The equivalent local potential is
obtained by using a localized approximation for the nonlocal densities of the
interacting nuclei. One can also include the density dependence of the NN
interaction within such a model Khoa  _et al._ (1997). As mentioned earlier,
in the present work, including antisymmetrization which leads to the so-called
“knock-on exchange” term in the M3Y NN interaction will result in double
counting. The effective nonlocal potentials used in this work include the
exchange term and medium effects of the NN interaction in a phenomenological
way through the nonlocality parameter. In Ref. Chamon  _et al._ (2002), the
authors provided a good description of the nucleus-nucleus scattering data
with the São Paulo potential (which is similar in spirit to the PB potential)
without the exchange term or density dependence included in the NN
interaction. The authors noted explicitly that including these effects amounts
to double counting of the nonlocality effects and does not provide a good
description of data. The nonlocality parameter defined earlier, namely,
$b=b_{0}m_{0}/\mu$, where $b_{0}$ is the nonlocal range of the nucleon-nucleus
interaction, $m_{0}$ is the nucleon mass, and $\mu$ is the $\alpha$-daughter
or cluster-daughter reduced mass, takes the antisymmetrization and medium
effects into account in a phenomenological way.

Finally, we discuss one of the first attempts to evaluate the effects of
nonlocality in $\alpha$ decay by Chaudhury Chaudhury (1963, 1960). The author
considered an interaction potential consisting of the pointlike Coulomb
interaction between the $\alpha$ and the daughter nucleus superimposed by a
nonlocal nuclear potential. The latter was taken to be the Igo potential with
the nonlocal part given by a Gaussian function as is used in the approaches
mentioned above too. The author obtained a radial differential equation
starting from the integro-differential equation (10) with nonlocality and used
it to write the JWKB penetration factor. The equation, however, was not the
standard radial Schrödinger equation, but rather an equation of the form,

$\left[\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}+\frac{2\mu
E\,\epsilon(r)}{\hbar^{2}}-\frac{2\mu
V(r)\,\epsilon(r)}{\hbar^{2}}\right]\,u_{l}(r)\,=0\,,$ (18)

where the factor $\epsilon(r)$ contained the nonlocality parameter $b$. The
standard JWKB penetration factor is derived by starting from the radial
Schrödinger equation for a particle with energy $E$, tunneling a potential
barrier $V(r)$. However, the author mentions using a JWKB solution of the
above equation to obtain the following preformation factor:

$P=\exp\biggl{\\{}-2\sqrt{2\mu\over\hbar^{2}}\int_{r_{i}}^{r_{0}}\biggl{[}{2(Z-2)e^{2}\over
r}\,\epsilon(r)\\\ -\,V_{0}f(r)\,\epsilon(r)\,+\,{\hbar^{2}\over
2\mu}{l(l+1)\over r^{2}}\,-\,E\,\epsilon(r)\biggr{]}^{1/2}{\rm d}r\biggr{\\}}$
(19)

where, $\epsilon(r)=[1+\eta f(r)]^{-1}$ and $\eta=(\mu
b^{2}/2\hbar^{2})V_{0}$. The strong potential, $V_{0}f(r)$ was taken from the
Igo potential. The penetrability was found to increase by about 50% due to
nonlocality for the nuclei studied.

###  II.3 Potentials for deformed nuclei

Figure 4: Schematic representation of the spherical-deformed interacting
nuclei. $\phi$ is the orientation angle of the symmetry axis of the deformed
nucleus with respect to the vector connecting the center of masses of the two
nuclei.

The cluster decay half-lives are calculated using the expression of the decay
width given by Eq. (2) for the case of spherical nuclei. To extend the cluster
decay half-life calculations to deformed nuclei, axial symmetry of nuclei is
considered Rhoades-Brown  _et al._ (1983). An additional simplification of the
problem is done by considering just one of the nuclei to be deformed Maroufi
_et al._ (2019). The latter simplification seems to be justified for the
decays studied in the present work. Since it is reasonable to consider the
$\alpha$ (4He nucleus) to be spherical, in the case of $\alpha$ decay the
deformation of the heavy daughter nucleus is taken into account. The heavier
daughter in case of cluster decay is often (the doubly magic nucleus) 208Pb or
a nucleus close to it. Thus, in case of cluster decay, we take the deformation
of the light cluster into account. This implies the tunneling particle is
spherical for $\alpha$ decay and deformed for cluster decay.

For the spherical-deformed interacting system, we begin by writing the nuclear
potential between the heavy daughter and the light cluster (or $\alpha$
particle) in Eq. (4) with one spherical density, $\rho_{1}(r)$, and the other
density deformed with axial symmetry, $\rho_{2}(r,\theta)$. The latter is
given as,

$\rho_{2}\left(r,\theta\right)=\frac{\rho_{0}}{1+\exp\left[\frac{r-R(\theta)}{a(\theta)}\right]}\,,$
(20)

where the value of $\rho_{0}$ is fixed by normalizing the density distribution
$\rho(r,\theta)$ and the angle-dependent half-density radius $R(\theta)$ and
diffuseness $a(\theta)$ are given by Bohr and Mottelson (1969); Scamps  _et
al._ (2013); Maroufi  _et al._ (2019)

$R(\theta)=R_{0}\big{(}1+\beta_{2}Y_{20}(\theta)+\beta_{4}Y_{40}(\theta)\big{)}$
(21)

and

$a(\theta)=a_{\perp}(\theta)\sqrt{1+\left|\vec{\nabla}R(\theta)_{\mid
r=R(\theta)}\right|^{2}}\,,$ (22)

respectively. In the above expressions $R_{0}=1.07A_{d}^{1/3}$ fm, $\beta_{2}$
and $\beta_{4}$ are quadrupole and hexadecapole deformation parameters of the
deformed nucleus, and

$a_{\perp}(\theta)=a_{0}\big{(}1+\beta_{2}Y_{20}(\theta)+\beta_{4}Y_{40}(\theta)\big{)}\,,$
(23)

where $a_{0}$ is taken to be $0.54$ fm. As mentioned earlier, the potential in
coordinate space is evaluated by converting the six-dimensional integral in
Eq. (4) to a momentum space integral. For the decay problem under
consideration, this reduces to a one-dimensional integral in momentum space
(see Ref. Satchler and Love (1979) for a detailed derivation with spherical
nuclei). When the interacting nuclei are deformed, the shape as well as the
orientation of the two nuclei must be taken into account. Assuming one nucleus
to be deformed but axially symmetric and the other spherical (for the general
case of two deformed nuclei, see Ref. Rhoades-Brown  _et al._ (1983)) the DF
nuclear potential can be evaluated as a sum of different multipole components
Rhoades-Brown  _et al._ (1983); Ismail  _et al._ (2002); Xu and Ren (2006)

$V_{N}(r,\phi)=\sum_{L=0,2,4\dots}V_{N}^{L}(r,\phi)\,.$ (24)

where $\phi$ is the orientation angle of the symmetry axis of the deformed
nucleus with respect to the vector connecting the center of masses of the two
nuclei (Fig. 4). Each of the multipole component is given by the momentum
space integral,

$V_{N}^{L}(r,\phi)=\frac{2}{\pi}\left[\frac{2L+1}{4\pi}\right]^{1/2}\,P_{L}(\cos\phi)\;\times\\\
\int_{0}^{\infty}{\rm
d}k\,k^{2}\,j_{L}(kr)\rho_{1}(k)\,\rho_{2}^{L}(k)\,v_{N}(k)\,,$ (25)

such that $\rho_{2}^{L}(k)$ is the Fourier transform of

$\rho_{2}^{L}(r)=2\pi\int\rho_{2}(r,\theta)\,Y_{L0}(\theta)\,{\rm
d}(\cos\theta)\,.$ (26)

The above is obtained simply from the multipole expansion

$\rho_{2}(r,\theta)=\sum_{L=0,2,4}\rho_{L}(r)Y_{L0}(\theta)\,.$ (27)

The sum in Eqs. (27) and (24) is usually truncated at $L=4$ to take the
dominant form factors into account. The Bohr–Sommerfeld quantization
condition, Eq. (8), which is an important constraint for the $\alpha$ and
cluster decay half-lives calculations (within the JWKB approximation) and
determines the depth of the nuclear potential, is now calculated at each
angle. This condition is rewritten as

$\int_{r_{1}(\phi)}^{r_{2}(\phi)}k(r,\phi)\,{\rm d}r=(G-l+1)\frac{\pi}{2}\,,$
(28)

where the turning points are similarly obtained by
$V\big{(}r_{i}(\phi),\phi\big{)}=Q$. $l$ is the orbital angular momentum
quantum number. The global quantum number $G$ is taken to be the same as in
the spherical case and is determined by using the Wildermuth-Tang condition,
Eq. (9). The deformed Coulomb potential is evaluated in a similar manner by
using deformed charge density distributions as in Eq. (20). Thus, the total
potential is given by

$V(r,\phi)=\lambda\,V_{N}(r,\phi)+V_{C}(r,\phi)+\frac{\hbar^{2}}{2\mu}\frac{\left(l+\frac{1}{2}\right)^{2}}{r^{2}}\,.$
(29)

As a consequence of this orientation angle dependence of the interaction
potential, the wave number is also angle dependent, $k(r,\phi)$. Since this
term is required to obtain the decay width $\Gamma$, it is necessary to
average over all possible orientation angles of the deformed nucleus. Some
authors average the penetration probability and the normalization factor (or
the so-called assault frequency of the tunneling particle related to this) and
then multiply them Xu and Ren (2006); Adel and Alharbi (2017); Maroufi  _et
al._ (2019). However, it is not appropriate since the penetration probability
and the normalization factor are not independent of each other. The suitable
way to obtain the decay width is then calculating an angle-dependent width,

$\Gamma(\phi)=\frac{\hbar^{2}}{2\mu}\left[\int_{r_{1}(\phi)}^{r_{2}(\phi)}\frac{{\rm
d}r}{k(r,\phi)}\right]^{-1}\times\\\
\exp\left[-2\int_{r_{2}(\phi)}^{r_{3}(\phi)}k(r,\phi)\,{\rm d}r\right]\,,$
(30)

and then averaging it over all directions Denisov and Ikezoe (2005); Ni and
Ren (2015); Qian  _et al._ (2016),

$\Gamma=\frac{1}{2}\int_{0}^{\pi}\Gamma(\phi)\sin\phi\;{\rm d}\phi\,.$ (31)

The $\alpha$ and cluster decay half-lives related to the decay width are
calculated using Eqs. (1), (30), and (31). The preformation factor is chosen
to be 1 and will be determined phenomenologically later.

Parent |  Decay mode |  $Q$ [MeV] |  $b$ [fm] | $l_{\rm min}$ |  G |  $\beta_{2}^{(c)}$ |  $\beta_{4}^{(c)}$ |  $\beta_{2}^{(d)}$ | $\beta_{4}^{(d)}$
---|---|---|---|---|---|---|---|---|---
222Ra |  $\alpha$+218Rn |  6.678 |  0.2160 | 0 | 22 |  0 |  0 |  0.040 |  0.029
|  14C+209Pb |  33.049 |  0.0648 | 0 | 68 |  -0.016 |  0.000 |  0 |  0
223Ra |  $\alpha$+219Rn |  5.979 |  0.2160 | 2 | 22 |  0 |  0 |  0.103 |  0.072
|  14C+209Pb |  31.828 |  0.0648 | 4 | 68 |  -0.016 |  0.000 |  0 |  0
228Th |  $\alpha$+224Ra |  5.520 |  0.2160 | 0 | 22 |  0 |  0 |  0.164 |  0.112
|  20O+208Pb |  44.723 |  0.0466 | 0 | 92 |  0.003 |  -0.108 |  0 |  0
231Pa |  $\alpha$+227Ac |  5.150 |  0.2160 | 0 | 22 |  0 |  0 |  0.172 |  0.112
|  23F+208Pb |  51.888 |  0.0411 | 1 | 103 |  -0.202 |  0.110 |  0 |  0
232U |  $\alpha$+228Th |  5.413 |  0.2160 | 0 | 22 |  0 |  0 |  0.182 |  0.112
|  24Ne+208Pb |  62.310 |  0.0395 | 0 | 106 |  -0.215 |  0.155 |  0 |  0
233U |  $\alpha$+229Th |  4.908 |  0.2160 | 0 | 22 |  0 |  0 |  0.190 |  0.114
|  24Ne+209Pb |  60.485 |  0.0395 | 2 | 106 |  -0.215 |  0.155 |  0 |  0
|  28Mg+205Hg |  74.226 |  0.0345 | 3 | 117 |  0.323 |  -0.136 |  0 |  0
235U |  $\alpha$+231Th |  4.678 |  0.2160 | 1 | 22 |  0 |  0 |  0.198 |  0.115
|  25Ne+210Pb |  57.683 |  0.0381 | 3 | 110 |  0 |  0 |  0 |  0
|  28Mg+207Hg |  72.425 |  0.0345 | 1 | 118 |  0.323 |  -0.136 |  0 |  0
237Np |  $\alpha$+233Pa |  4.958 |  0.2160 | 1 | 22 |  0 |  0 |  0.207 |  0.117
|  30Mg+207Tl |  74.790 |  0.0325 | 2 | 127 |  -0.222 |  -0.112 |  0 |  0
236Pu |  $\alpha$+232U |  5.867 |  0.2160 | 0 | 22 |  0 |  0 |  0.207 |  0.117
|  28Mg+208Pb |  79.669 |  0.0345 | 0 | 120 |  0.323 |  -0.136 |  0 |  0
241Am |  $\alpha$+237Np |  5.637 |  0.2160 | 1 | 22 |  0 |  0 |  0.215 |  0.102
|  34Si+207Tl |  93.926 |  0.0291 | 3 | 141 |  0 |  0 |  0 |  0
242Cm |  $\alpha$+238Pu |  6.215 |  0.2160 | 0 | 22 |  0 |  0 |  0.215 |  0.102
|  34Si+208Pb |  96.509 |  0.0291 | 0 | 142 |  0 |  0 |  0 |  0
Table 1: $\alpha$ and cluster decay data. Columns 3 to 6 refer, respectively,
to $Q$-value of the decay from the parent ground state (g.s.) to the daughter
g.s., the nonlocality parameter, the angular momentum carried by the emitted
particle Brookhaven National Laboratory ; Institute of Modern Physics, Chinese
Academy of Sciences , and Global quantum number given by Eq. (9). The last
columns refer to quadrupole $\beta_{2}$ and hexadecapole $\beta_{4}$
deformation parameters of the emitted cluster (c) and daughter nucleus (d),
respectively Moller  _et al._ (1995); Sawhney  _et al._ (2011).

##  III Results and discussions

The nonlocal nuclear interaction has been shown to decrease the half-life of
spherical nuclei decaying by the emission of $\alpha$ particles Perez
Velasquez  _et al._ (2019). Here we study the combined effects of nonlocality
and deformation of the nuclei involved in the decay process. In Table 1,
$Q$-value of the decay from the parent ground state (g.s.) to the daughter
g.s., the nonlocality parameter, the angular momentum carried by the emitted
particle Brookhaven National Laboratory ; Institute of Modern Physics, Chinese
Academy of Sciences , and Global quantum number given by Eq. (9) are listed in
columns 3–6. The last columns refer to quadrupole $\beta_{2}$ and hexadecapole
$\beta_{4}$ deformation parameters of the emitted cluster (c) and daughter
nucleus (d), respectively Moller  _et al._ (1995); Sawhney  _et al._ (2011).

###  III.1 Half-lives

In this section, we shall study the effects of nonlocality and deformation
individually as well as their joint manifestation in the $\alpha$ and cluster
decay half-lives. The calculations are done for two kinds of nonlocal
potentials, namely, the energy-independent, so-called Mumbai potential which
is $l$ dependent and the well-known Perey-Buck potential which is energy
dependent but does not depend on $l$. We define the percentage change (PC) in
the half-life of a nucleus as

$PC=\frac{t_{1/2}^{\rm DF}-t_{1/2}}{t_{1/2}^{\rm DF}}\times 100,$ (32)

where, $t_{1/2}^{\rm DF}$ is the half-life calculated using the DF model only
and $t_{1/2}$ is the half-life evaluated within the DF model including
nonlocality or deformation or both.

|  |  | Double folding | Mumbai | Perey-Buck
---|---|---|---|---|---
Parent |  Decay mode | $l_{\rm min}$ |  $\lambda$ |  $t_{1/2}$ [s] |  $\lambda_{M}$ | $PC$ |  $\lambda_{PB}$ | $PC$
222Ra |  $\alpha$+218Rn | 0 |  2.04 |  3.28 |  2.05 |  3.2 |  2.32 |  35.8
|  14C+208Pb | 0 |  1.61 |  $1.10\times 10^{5}$ |  1.61 |  4.1 |  1.86 |  76.9
223Ra |  $\alpha$+219Rn | 2 |  2.04 |  $7.09\times 10^{3}$ |  2.10 |  12.3 |  2.37 |  37.0
|  14C+209Pb | 4 |  1.61 |  $3.25\times 10^{7}$ |  1.63 |  17.7 |  1.86 |  77.8
228Th |  $\alpha$+224Ra | 0 |  2.04 |  $8.60\times 10^{6}$ |  2.06 |  $3.4$ |  2.37 |  $37.5$
|  20O+208Pb | 0 |  1.49 |  $6.78\times 10^{12}$ |  1.49 |  3.9 |  1.72 |  87.2
231Pa |  $\alpha$+227Ac | 0 |  2.04 |  $4.30\times 10^{9}$ |  2.06 |  $3.2$ |  2.36 |  $38.2$
|  23F+208Pb | 1 |  1.43 |  $2.00\times 10^{14}$ |  1.44 |  6.4 |  1.66 |  90.2
232U |  $\alpha$+228Th | 0 |  2.04 |  $3.61\times 10^{8}$ |  2.05 |  $3.2$ |  2.37 |  $38.2$
|  24Ne+208Pb | 0 |  1.42 |  $1.43\times 10^{10}$ |  1.42 |  3.8 |  1.65 |  91.2
233U |  $\alpha$+229Th | 0 |  2.04 |  $5.82\times 10^{11}$ |  2.06 |  $3.3$ |  2.37 |  $38.9$
|  24Ne+209Pb | 2 |  1.42 |  $7.25\times 10^{12}$ |  1.43 |  10.0 |  1.65 |  91.8
|  28Mg+205Hg | 3 |  1.35 |  $5.80\times 10^{13}$ |  1.36 |  13.6 |  1.57 |  94.6
235U |  $\alpha$+231Th | 1 |  2.02 |  $3.06\times 10^{13}$ |  2.07 |  $12.2$ |  2.34 |  $38.9$
|  25Ne+210Pb | 3 |  1.41 |  $4.57\times 10^{17}$ |  1.42 |  13.6 |  1.63 |  92.6
|  28Mg+207Hg | 1 |  1.36 |  $1.10\times 10^{16}$ |  1.37 |  6.9 |  1.58 |  94.9
237Np |  $\alpha$+233Pa | 1 |  2.01 |  $1.11\times 10^{12}$ |  2.06 |  $12.4$ |  2.33 |  $39.2$
|  30Mg+207Tl | 2 |  1.35 |  $3.11\times 10^{14}$ |  1.35 |  9.6 |  1.56 |  94.9
236Pu |  $\alpha$+232U | 0 |  2.02 |  $9.94\times 10^{6}$ |  2.04 |  $3.3$ |  2.35 |  $38.5$
|  28Mg+208Pb | 0 |  1.38 |  $1.07\times 10^{9}$ |  1.38 |  3.7 |  1.60 |  94.0
241Am |  $\alpha$+237Np | 1 |  1.99 |  $6.55\times 10^{8}$ |  2.04 |  $12.4$ |  2.33 |  $38.9$
|  34Si+207Tl | 3 |  1.32 |  $4.33\times 10^{10}$ |  1.33 |  12.7 |  1.54 |  96.2
242Cm |  $\alpha$+238Pu | 0 |  1.99 |  $1.28\times 10^{6}$ |  2.01 |  $3.4$ |  2.33 |  $39.4$
|  34Si+208Pb | 0 |  1.33 |  $9.54\times 10^{8}$ |  1.33 |  3.5 |  1.55 |  96.1
Table 2: The calculated $\alpha$\- and cluster-decay half-lives using the DF
model (fifth column) and the percentage decrease in the half-lives of
different models of nonlocality compared to those evaluated using the DF model
without nonlocality. $\lambda$, $\lambda_{M}$ and $\lambda_{PB}$ are the
strengths of the nuclear potentials in the DF, M and PB models.

####  III.1.1 Effects of nonlocality

In order to study the change in the half-lives of nuclei due to the nonlocal
interaction potential between the decay products, as mentioned before, we use
two nonlocal approaches Perey and Buck (1962); Upadhyay  _et al._ (2017) from
literature which are formally very different. We shall see that though both
the approaches are equally successful in reproducing the scattering cross-
section data, the manifestation of the nonlocal effects in nuclear decay is
quite different within the two approaches. In Table 2, the half-lives
calculated within the DF model are shown for the $\alpha$ and cluster decay
modes of different nuclei. The percentage decrease in the half-life due to the
inclusion of nonlocality within the Mumbai and Perey-Buck approach is listed
in the next columns. A close look at the numbers reveals the differences in
the two models which are rooted in the nature of the nonlocality model. The
main features to be noted from this table are the following:

  1. (i)

The percentage decrease [listed as PC and evaluated using Eq. (32)] is much
larger in the Perey-Buck approach as compared to the Mumbai model.

  2. (ii)

Whereas the PC remains almost constant (around 38% decrease for $\alpha$ decay
and around 90% decrease in many cases of cluster decay) within the Perey-Buck
approach, this is not the case when the Mumbai model is used.

  3. (iii)

Within the Mumbai approach, the percentage decrease in half-life increases
significantly with increasing value of the relative angular momentum, $l$,
between the cluster (or alpha) and the daughter nucleus.

  4. (iv)

The decrease in half-lives due to nonlocality is always less in the case of
$\alpha$ decay as compared to cluster decay of the same nucleus when the
Perey-Buck model is used. Such a pattern does not exist if the nonlocal
effects are included within the Mumbai model.

The above features can be understood by noting that (i) whereas the Perey-Buck
local equivalent potential, $U^{\rm{PB}}(r)$ is derived starting from the full
nonlocal Schrödinger equation, the effective Mumbai potential,
$U_{l}^{\rm{M}}(r)$, is derived from the radial part of the same equation.
(ii) As a direct consequence of (i), the Mumbai potential becomes $l$
dependent [see Eqs. (15) and (17)]. (iii) As a consequence of starting with
the three dimensional Schrödinger equation and approximating the nonlocal wave
function by the local one, the Perey-Buck potential becomes energy dependent
[see Eq. (13)]. (iv) Finally, it must be noted that the derivation of
$U^{\rm{PB}}$ is done under the assumption of a constant potential and the $r$
dependence is introduced a posteriori. The latter gives rise to the biggest
difference between the two nonlocal potentials: namely, the Mumbai potential
vanishes near the origin whereas the Perey-Buck potential attains a large
constant value.

In Table 2, we also list the values of $\lambda$ which decide the strength of
the nuclear interaction and are obtained by imposing the Bohr-Sommerfeld
condition, Eq. (8). They remain roughly the same for the decay processes that
have the same emitted cluster and do not display any strong $l$ dependence.

|  |  |  $t_{1/2}$ [s] | $PC$
---|---|---|---|---
|  |  |  Spherical |  Deformed with |  Deformed with | Deformed with
Parent |  Decay mode | $l_{\rm min}$ |  Double |  $R(\theta;\beta_{2})$ |  $R(\theta;\beta_{2},\beta_{4})$ | $R(\theta;\beta_{2},\beta_{4})$
|  |  |  folding |  $a_{\perp}(\theta;\beta_{2})$ |  $a_{0}$ | $a_{\perp}(\theta;\beta_{2},\beta_{4})$
222Ra |  $\alpha$+218Rn | 0 |  3.28 |  $5.9$ |  $0.4$ |  $8.8$
|  14C+208Pb | 0 |  $1.10\times 10^{5}$ |  0.4 |  0.1 |  0.4
223Ra |  $\alpha$+219Rn | 2 |  $7.09\times 10^{3}$ |  $8.78$ |  $24.8$ |  $33.4$
|  14C+209Pb | 4 |  $3.25\times 10^{7}$ |  0.4 |  0.1 |  0.4
228Th |  $\alpha$+224Ra | 0 |  $8.62\times 10^{6}$ |  $49.1$ |  $53.3$ |  $66.2$
|  20O+208Pb | 0 |  $6.78\times 10^{12}$ |  0.0 |  4.3 |  17.1
231Pa |  $\alpha$+227Ac | 0 |  $4.30\times 10^{9}$ |  $41.3$ |  $57.4$ |  $68.2$
|  23F+208Pb | 1 |  $2.00\times 10^{14}$ |  65.7 |  52.4 |  78.0
232U |  $\alpha$+228Th | 0 |  $3.63\times 10^{8}$ |  $46.8$ |  $53.9$ |  $62.1$
|  24Ne+208Pb | 0 |  $1.43\times 10^{10}$ |  72.0 |  62.7 |  87.5
233U |  $\alpha$+229Th | 0 |  $5.82\times 10^{11}$ |  $62.5$ |  $48.5$ |  $75.4$
|  24Ne+209Pb | 2 |  $7.25\times 10^{12}$ |  72.2 |  62.9 |  87.5
|  28Mg+205Hg | 3 |  $5.80\times 10^{13}$ |  99.5 |  92.9 |  98.6
235U |  $\alpha$+231Th | 1 |  $3.06\times 10^{13}$ |  $45.1$ |  $66.1$ |  $77.9$
|  28Mg+207Hg | 1 |  $1.10\times 10^{16}$ |  99.5 |  93.1 |  98.6
237Np |  $\alpha$+233Pa | 1 |  $1.11\times 10^{12}$ |  $51.5$ |  $68.8$ |  $81.6$
|  30Mg+207Tl | 2 |  $3.11\times 10^{14}$ |  83.7 |  63.0 |  80.0
236Pu |  $\alpha$+232U | 0 |  $9.94\times 10^{6}$ |  $49.3$ |  $65.9$ |  $76.2$
|  28Mg+208Pb | 0 |  $1.07\times 10^{9}$ |  99.5 |  92.7 |  98.6
241Am |  $\alpha$+237Np | 1 |  $6.55\times 10^{8}$ |  $54.7$ |  $34.5$ |  $77.6$
242Cm |  $\alpha$+238Pu | 0 |  $1.28\times 10^{6}$ |  $51.9$ |  $62.8$ |  $81.2$
Table 3: The calculated $\alpha$\- and cluster-decay half-lives using the DF model (fourth column) and the percentage decrease in the half-lives when different kinds of deformation are included. The fifth column displays the percentage decrease of the half-lives including only the quadrupole deformation parameter $\beta_{2}$ in both $R(\theta)$ and $a_{\perp}(\theta)$. The sixth column displays the percentage decrease including both the quadrupole and hexadecapole deformation parameters in $R(\theta)$. The diffuseness parameter is constant. The seventh column includes the quadrupole and hexadecapole deformation parameters in both the radius and surface diffuseness parameters. |  |  |  $t_{1/2}$ [s] | $PC$
---|---|---|---|---
|  |  |  Spherical | Deformed with | Deformed with | Deformed with
Parent |  Decay mode | $l_{\rm min}$ |  Double |  $R(\theta;\beta_{2})$ and $a_{\perp}(\theta;\beta_{2})$ |  $R(\theta;\beta_{2},\beta_{4})$ and $a_{0}$ |  $R(\theta;\beta_{2},\beta_{4})$ and $a_{\perp}(\theta;\beta_{2},\beta_{4})$
|  |  |  folding | Mumbai | Perey-Buck | Mumbai | Perey-Buck | Mumbai | Perey-Buck
222Ra |  $\alpha$+218Rn | 0 |  3.28 |  9.1 |  39.6 |  4.2 |  35.9 |  12.3 |  40.8
|  14C+209Pb | 0 |  $1.10\times 10^{5}$ |  4.4 |  77.0 |  4.2 |  76.9 |  4.4 |  77.0
223Ra |  $\alpha$+219Rn | 2 |  $7.09\times 10^{3}$ |  28.4 |  42.2 |  33.2 |  52.3 |  40.8 |  57.5
|  14C+209Pb | 4 |  $3.25\times 10^{7}$ |  18.0 |  77.9 |  17.8 |  77.9 |  18.0 |  77.9
228Th |  $\alpha$+224Ra | 0 |  $8.62\times 10^{6}$ |  $50.3$ |  $68.0$ |  $54.2$ |  $70.8$ |  $67.3$ |  $79.8$
|  20O+208Pb | 0 |  $6.78\times 10^{12}$ |  3.9 |  87.2 |  8.0 |  87.8 |  20.4 |  89.5
231Pa |  $\alpha$+227Ac | 0 |  $4.30\times 10^{9}$ |  $41.5$ |  $64.4$ |  $58.8$ |  $73.8$ |  $69.8$ |  $83.4$
|  23F+208Pb | 1 |  $2.00\times 10^{14}$ |  68.0 |  96.8 |  55.5 |  95.5 |  79.4 |  98.0
232U |  $\alpha$+228Th | 0 |  $3.63\times 10^{8}$ |  $54.1$ |  $65.7$ |  $55.5$ |  $71.6$ |  $63.9$ |  $77.0$
|  24Ne+208Pb | 0 |  $1.43\times 10^{10}$ |  73.0 |  97.7 |  64.2 |  96.9 |  88.0 |  99.0
233U |  $\alpha$+229Th | 0 |  $5.82\times 10^{11}$ |  $63.8$ |  $76.9$ |  $50.3$ |  $67.2$ |  $76.8$ |  $86.2$
|  24Ne+209Pb | 2 |  $7.25\times 10^{12}$ |  75.0 |  97.8 |  66.7 |  97.1 |  88.8 |  99.1
|  28Mg+205Hg | 3 |  $5.80\times 10^{13}$ |  99.6 |  100.0 |  93.8 |  99.6 |  98.8 |  99.9
235U |  $\alpha$+231Th | 1 |  $3.06\times 10^{13}$ |  $62.7$ |  $64.5$ |  $70.3$ |  $80.5$ |  $78.3$ |  $86.9$
|  25Ne+210Pb | 3 |  $4.57\times 10^{17}$ |  13.6 |  92.6 |  13.6 |  92.6 |  13.6 |  92.6
|  28Mg+207Hg | 1 |  $1.10\times 10^{16}$ |  99.5 |  100.0 |  93.6 |  99.7 |  98.7 |  99.9
237Np |  $\alpha$+233Pa | 1 |  $1.11\times 10^{12}$ |  $63.8$ |  $67.7$ |  $72.5$ |  $81.8$ |  $82.9$ |  $89.3$
|  30Mg+207Tl | 2 |  $3.11\times 10^{14}$ |  85.3 |  99.2 |  66.5 |  98.1 |  81.9 |  99.0
236Pu |  $\alpha$+232U | 0 |  $9.97\times 10^{6}$ |  $53.3$ |  $69.4$ |  $66.9$ |  $79.8$ |  $77.3$ |  $87.2$
|  28Mg+208Pb | 0 |  $1.07\times 10^{9}$ |  99.5 |  100.0 |  93.0 |  99.6 |  98.6 |  99.9
241Am |  $\alpha$+237Np | 1 |  $6.55\times 10^{8}$ |  $55.8$ |  $73.5$ |  $43.2$ |  $61.1$ |  $78.5$ |  $86.1$
|  34Si+207Tl | 3 |  $4.33\times 10^{10}$ |  12.7 |  96.2 |  12.7 |  96.2 |  12.7 |  96.2
242Cm |  $\alpha$+238Pu | 0 |  $1.28\times 10^{6}$ |  $54.9$ |  $71.0$ |  $65.4$ |  $77.8$ |  $81.9$ |  $87.3$
|  34Si+208Pb | 0 |  $9.54\times 10^{8}$ |  3.5 |  96.1 |  3.5 |  96.1 |  3.5 |  96.1
Table 4: The calculated $\alpha$\- and cluster-decay half-lives using the DF
model and the percentage decrease in half-lives when different models of
nonlocality and different kinds of deformation are included. The last two
columns represent the complete calculation with $\beta_{2}$ and $\beta_{4}$
included in the radius and surface diffuseness parameters. Other columns show
the effect of dropping one or both deformation parameters for each model of
nonlocality.

####  III.1.2 Role of deformation

The decay products in the $\alpha$ and cluster decays considered in the
present work are not necessarily spherical nuclei. In the case of $\alpha$
decay, it is the daughter nucleus which can be deformed whereas in case of the
cluster decay, it is mostly the light cluster which is deformed. The heavier
daughter in this latter case is mostly spherical. The nuclear deformation is
incorporated through a modification of the interaction potential as described
in Sec. II.3. In the previous table, we noticed that for a given parent
nucleus and within a given approach for nonlocality, the manifestation of the
nonlocal effects in the half-life of the nucleus depends on the mode of decay
($\alpha$ or cluster). In the same spirit, we could also ask if a deformed
interaction potential would affect the half-lives of the same parent nucleus
decaying by $\alpha$ or cluster decay, in a different way. In a semiclassical
scheme as that of the present work, the decay is described by the tunneling of
the lighter decay product through the Coulomb barrier created by the
interaction between the decay products. Seeing it from this point of view, in
the case of $\alpha$ decay, the tunneling nucleus is spherical (and the heavy
daughter deformed), whereas in case of cluster decay, it is the deformed light
nucleus which tunnels through the barrier generated due to its interaction
with the spherical heavier daughter nucleus.

Table 3 displays the percentage change in the half-lives of various nuclei
decaying by the two modes, namely, $\alpha$ and cluster decay. The last column
lists the results with a complete inclusion of the deformation effects in the
radius as well as diffuseness parameter in the potential. The fifth and the
sixth columns show the effect of ignoring some of the terms involving the
hexadecapole and quadrupole deformation (these columns are listed for
completeness since such omissions are sometimes found in literature). In spite
of the tunneling picture considered above, deformation manifests itself in a
similar way in both $\alpha$ and cluster decay. The half-lives decrease after
the inclusion of deformation in the interaction potential. The effect is not
sensitive to the sign of the deformation parameters but is greater for larger
magnitudes of the parameters.

####  III.1.3 Half-lives with a nonlocal and deformed interaction potential

In Table 4, we show the effects of including both nonlocality and deformation
in the interaction potential between the daughter nuclei. This table is in
principle meant to display the overlap of the effects shown in Tables 2 and 3.
However, the overlap of the two effects cannot be expected to be linear.
Nonlocality and deformation both lead to a decrease in the $\alpha$ as well as
cluster decay half-lives. Since the decrease due to nonlocality in the Perey-
Buck model is already large, one does not see a significant difference in the
results with nonlocality and deformation together. The Mumbai model, however,
displayed a smaller decrease in half-lives due to nonlocality and including
deformation enhances the effect a lot. The effect is more enhanced for larger
values of the angular momentum quantum number $l$. Once again, this difference
between the results using the two different approaches for nonlocality could
be due to the fact that whereas the Mumbai potential is $l$ dependent but
energy independent, the Perey-Buck potential is energy dependent but not $l$
dependent. It is interesting to note that both potentials can reproduce
scattering data equally well. The latter is understandable since the
differences in the models would not show up in the calculation of total cross
sections (where we sum over all partial waves). In this sense, the $\alpha$ or
the cluster decay of a nucleus which picks up a particular $l$ value can be a
better handle to distinguish between different models of nonlocality and fine
tune them in a better way.

###  III.2 Preformation factors

Figure 5: Preformation factors $P_{i}$ ($i=$ $\alpha$ or $c$) as a function of
the $Q$ value or energy of the emitted particle in $\alpha$ and cluster decay
of different isotopes. The left panel corresponds to the comparison between
the DF model (black symbols) and the Mumbai potential including complete
deformation (colored symbols representing different values of $l_{\rm min}$).
The right panel compares $P_{i}$ calculated using the DF and PB models with
deformation. The inset shows a magnified view of the $\alpha$ preformation
factors which appear to be clubbed together at the small $Q$ values. Most
points appear to lie on a band around a straight line which is shown to guide
the eye.

The phenomenon of the emission of 4He ($\alpha$) and light nuclei such as 14C,
20O, 23F, 24-25Ne, 28-30Mg, and 34Si in the radioactive decay of nuclei can be
explained quite well within a preformed cluster model where the decaying
parent nucleus is a cluster of the two decay products Adel and Alharbi (2017);
Singh  _et al._ (2010); Adel and Alharbi (2020). The probability of formation
of an $\alpha$ particle or a light cluster inside the parent nucleus is
proportional to the overlap of the wave function of the final state with that
of the initial state. However, one can determine this quantity in a
phenomenological way within the preformed cluster model. Within such a model,
one usually defines the decay constant or the width as a product of the
preformation factor, $P_{i}$ ($i=\alpha$ or $c$ here), assault frequency, and
the penetration probability. Calculations are performed by setting $P_{i}$ = 1
and the disagreement with the experimental half-life is then attributed to the
performation probability. Thus, one evaluates the theoretical width $\Gamma$
as in Eq. (2), or Eq. (31), with $P_{i}$ = 1, and then defines the probability
for cluster formation as Singh  _et al._ (2010), $P_{i}=\Gamma^{\rm
exp}/\Gamma$. Here $\Gamma^{\rm exp}=\hbar\ln 2/t_{1/2}^{\rm exp}$ is given by
the experimental half-life. Note that in case the parent nucleus decays into
more than one channel, one must take the branching ratio for the channel under
consideration into account. For decays involving the parents as well as
daughter nuclei with non zero spins, the $\alpha$ or the light cluster can be
emitted with a non-zero angular momentum quantum number. In this case, the
width calculated using Eq. (2) is in principle a partial width, $\Gamma_{l}$.
In order to compare with the experimental (full) width, one would in principle
have to consider the sum $\sum_{l}P_{i}^{(l)}\,\Gamma_{l}=\Gamma^{\rm exp}$.
Indeed, in microscopic calculations, one does consider the so-called “reduced
widths” and evaluates the $l$-dependent preformation factor within a given
theoretical model. One of the first such calculations was performed by Mang
Mang (1960, 1960) in a test of the shell model with the Po and At isotopes.
For a phenomenological determination of the preformation factor within the
preformed cluster model, however, it is customary to define $P_{i}$ as

$P_{i}=\frac{\Gamma^{\rm exp}}{\Gamma_{l_{\rm min}}}$ (33)

with $\Gamma_{l_{\rm min}}$ evaluated for the minimum $l$ value using Eq. (2)
with $P_{i}$ = 1. The above relation is a good approximation if the
preformation factors and widths for all $l>l_{\rm min}$ are much smaller than
those for $l_{\rm min}$. In the absence of any theoretical calculations of
$P_{i}$ confirming the latter, we shall evaluate the preformation factors
using Eq. (33), which is commonly found in literature. In the tables 5 and 6,
we present the effects of nonlocality and deformation on the values of $P_{i}$
for the different decays discussed earlier. The notation is similar to that
used in the tables showing the half-lives and the percentage changes due to
nonlocality and deformation. We note that the preformation factors in the
tables have been evaluated only for the transitions from the parent ground
state (g.s) to the g.s. of the heavy daughter nucleus using Eq. (33). The
branching ratio (B.R.) for the transition from (parent) g.s. $\to$ (daughter)
g.s. is listed in the table in case of $\alpha$ decay and should be multiplied
with the values of $P_{i}$ given in the table in order to determine the
performation probability for this particular transition. Thus, the
preformation factors for transitions from the parent g.s. to the different
excited states of the daughter will differ depending on the B.R. to that
level. Since the focus of the present work is to investigate the effects of
nonlocality and deformation, we leave such an investigation of g.s. to excited
states transitions for the future.

Though the tables give all details of the dependence of the preformation
factors on deformation and nonlocality (individually as well as the combined
effect), in Fig. 5, we plot the values of $P_{i}$ for $\alpha$ and cluster
decay as a function of the $Q$ values in order to visualize these effects. The
two panels show the calculations with two different models of nonlocality with
the inclusion of deformation with all the relevant terms in the radius and the
diffuseness parameters. The insets show a magnified view of the preformation
factors, $P_{i}$ (with $i$ = $\alpha$) for $\alpha$ decay which appear in the
left corner of each panel. Different interesting features emerge from this
figure:

  1. (i)

All values of $P_{i}$ evaluated for different isotopes seem to lie on a band,
displaying a linear dependence as a function of energy.

  2. (ii)

For a given energy, the preformation factors decrease as a function of $l$
(shown by the colored symbols).

  3. (iii)

The effect of nonlocality + deformation is more pronounced in case of the
Perey-Buck (PB) model (right panel) as compared to the Mumbai model (left
panel). The black symbols represent the calculations within the double folding
(DF) model without the inclusion of nonlocality or deformation.

The $Q$ values or the energies carried by the $\alpha$ or light cluster are
related to the binding energies of the parent and daughter nuclei which in
turn also define the formation energy of a cluster Deng  _et al._ (2015). A
bigger binding energy of the daughter therefore implies a higher $Q$ value and
more internal energy available for the two nuclei in the cluster. The latter
explains the decrease in the values of $P_{i}$ with increasing values of $Q$.
The observation of a linear dependence of the phenomenologically evaluated
$P_{i}$ in Fig. 5 can be useful for theoretical investigations of cluster
formation probabilities.

|  |  |  | $P_{i}$
---|---|---|---|---
|  |  |  |  Spherical | Nonlocality | Deformation
Parent |  Decay mode |  B.R. | $l_{\rm min}$ |  Double | Mumbai | Perey-Buck |  $R(\theta;\beta_{2})$ |  $R(\theta;\beta_{2},\beta_{4})$ | $R(\theta;\beta_{2},\beta_{4})$
|  |  |  |  Folding |  |  |  $a_{\perp}(\theta;\beta_{2})$ |  $a_{0}$ | $a_{\perp}(\theta;\beta_{2},\beta_{4})$
222Ra |  $\alpha$+218Rn |  0.969 | 0 |  8.63$\times 10^{-2}$ |  8.35$\times 10^{-2}$ |  5.54$\times 10^{-2}$ |  8.11$\times 10^{-2}$ |  8.59$\times 10^{-2}$ |  7.82$\times 10^{-2}$
|  14C+208Pb |  - | 0 |  8.64$\times 10^{-7}$ |  8.29$\times 10^{-7}$ |  2.00$\times 10^{-7}$ |  8.61$\times 10^{-7}$ |  8.63$\times 10^{-7}$ |  8.61$\times 10^{-7}$
223Ra |  $\alpha$+219Rn |  0.01 | 2 |  7.18$\times 10^{-3}$ |  6.30$\times 10^{-3}$ |  5.42$\times 10^{-3}$ |  6.55$\times 10^{-3}$ |  5.40$\times 10^{-3}$ |  4.82$\times 10^{-3}$
|  14C+209Pb |  - | 4 |  2.93$\times 10^{-8}$ |  2.41$\times 10^{-8}$ |  6.50$\times 10^{-9}$ |  2.92$\times 10^{-8}$ |  2.92$\times 10^{-8}$ |  2.92$\times 10^{-8}$
228Th |  $\alpha$+224Ra |  0.734 | 0 |  1.40$\times 10^{-1}$ |  1.37$\times 10^{-1}$ |  8.37$\times 10^{-2}$ |  7.28$\times 10^{-2}$ |  6.68$\times 10^{-2}$ |  5.23$\times 10^{-2}$
|  20O+208Pb |  - | 0 |  1.12$\times 10^{-8}$ |  1.08$\times 10^{-8}$ |  1.44$\times 10^{-9}$ |  1.12$\times 10^{-8}$ |  1.08$\times 10^{-8}$ |  9.31$\times 10^{-9}$
231Pa |  $\alpha$+227Ac |  0.110 | 0 |  4.16$\times 10^{-3}$ |  4.03$\times 10^{-3}$ |  2.57$\times 10^{-3}$ |  2.44$\times 10^{-3}$ |  1.78$\times 10^{-3}$ |  1.38$\times 10^{-3}$
|  23F+208Pb |  - | 1 |  1.91$\times 10^{-12}$ |  1.79$\times 10^{-12}$ |  1.87$\times 10^{-13}$ |  6.56$\times 10^{-13}$ |  9.11$\times 10^{-13}$ |  4.21$\times 10^{-13}$
232U |  $\alpha$+228Th |  0.685 | 0 |  1.67$\times 10^{-1}$ |  1.61$\times 10^{-1}$ |  1.02$\times 10^{-1}$ |  8.88$\times 10^{-2}$ |  7.69$\times 10^{-2}$ |  6.53$\times 10^{-2}$
|  24Ne+208Pb |  - | 0 |  5.91$\times 10^{-11}$ |  1.65$\times 10^{-11}$ |  5.20$\times 10^{-12}$ |  1.48$\times 10^{-10}$ |  2.20$\times 10^{-11}$ |  7.39$\times 10^{-12}$
233U |  $\alpha$+229Th |  0.843 | 0 |  1.15$\times 10^{-1}$ |  1.12$\times 10^{-1}$ |  7.08$\times 10^{-2}$ |  4.42$\times 10^{-2}$ |  5.95$\times 10^{-2}$ |  3.08$\times 10^{-2}$
|  24Ne+209Pb |  - | 2 |  1.04$\times 10^{-12}$ |  9.36$\times 10^{-13}$ |  9.57$\times 10^{-14}$ |  2.89$\times 10^{-13}$ |  1.99$\times 10^{-12}$ |  1.30$\times 10^{-13}$
|  28Mg+205Hg |  - | 3 |  1.50$\times 10^{-14}$ |  1.30$\times 10^{-14}$ |  8.12$\times 10^{-16}$ |  7.78$\times 10^{-17}$ |  1.07$\times 10^{-15}$ |  2.12$\times 10^{-16}$
235U |  $\alpha$+231Th |  0.0477 | 1 |  1.37$\times 10^{-3}$ |  1.21$\times 10^{-3}$ |  8.42$\times 10^{-4}$ |  7.56$\times 10^{-4}$ |  4.67$\times 10^{-4}$ |  3.23$\times 10^{-4}$
|  25Ne+210Pb |  - | 3 |  1.65$\times 10^{-10}$ |  1.42$\times 10^{-10}$ |  1.22$\times 10^{-11}$ |  1.65$\times 10^{-10}$ |  1.65$\times 10^{-10}$ |  1.65$\times 10^{-10}$
|  28Mg+207Hg |  - | 1 |  3.97$\times 10^{-12}$ |  3.70$\times 10^{-12}$ |  2.03$\times 10^{-13}$ |  1.98$\times 10^{-14}$ |  2.74$\times 10^{-13}$ |  5.41$\times 10^{-14}$
237Np |  $\alpha$+233Pa |  0.0239 | 1 |  1.64$\times 10^{-2}$ |  1.44$\times 10^{-2}$ |  1.00$\times 10^{-2}$ |  8.00$\times 10^{-3}$ |  5.14$\times 10^{-2}$ |  3.71$\times 10^{-2}$
|  30Mg+207Tl |  - | 2 |  1.84$\times 10^{-13}$ |  1.66$\times 10^{-13}$ |  9.37$\times 10^{-15}$ |  2.99$\times 10^{-14}$ |  6.80$\times 10^{-14}$ |  3.68$\times 10^{-14}$
236Pu |  $\alpha$+232U |  0.691 | 0 |  1.10$\times 10^{-1}$ |  1.06$\times 10^{-1}$ |  6.80$\times 10^{-2}$ |  5.60$\times 10^{-2}$ |  3.77$\times 10^{-2}$ |  2.78$\times 10^{-2}$
|  28Mg+208Pb |  - | 0 |  2.38$\times 10^{-13}$ |  2.29$\times 10^{-13}$ |  1.43$\times 10^{-14}$ |  1.17$\times 10^{-15}$ |  1.74$\times 10^{-14}$ |  4.02$\times 10^{-13}$
241Am |  $\alpha$+237Np |  0.0037 | 1 |  4.80$\times 10^{-2}$ |  4.20$\times 10^{-2}$ |  2.93$\times 10^{-2}$ |  2.22$\times 10^{-2}$ |  3.09$\times 10^{-2}$ |  1.18$\times 10^{-2}$
|  34Si+207Tl |  - | 3 |  2.35$\times 10^{-15}$ |  2.05$\times 10^{-15}$ |  8.92$\times 10^{-17}$ |  2.35$\times 10^{-15}$ |  2.35$\times 10^{-15}$ |  2.35$\times 10^{-15}$
242Cm |  $\alpha$+238Pu |  0.748 | 0 |  9.16$\times 10^{-2}$ |  8.85$\times 10^{-2}$ |  5.52$\times 10^{-2}$ |  4.40$\times 10^{-2}$ |  3.40$\times 10^{-2}$ |  1.88$\times 10^{-2}$
|  34Si+208Pb |  - | 0 |  6.78$\times 10^{-15}$ |  6.55$\times 10^{-15}$ |  2.65$\times 10^{-16}$ |  6.78$\times 10^{-15}$ |  6.78$\times 10^{-15}$ |  6.78$\times 10^{-15}$
Table 5: Preformation factor $P_{i}$ ($i$ = $\alpha$ or $c$) using the DF model (fourth column), different models of nonlocality (fifth and sixth columns), and including different kinds of deformation (seventh, eighth, and ninth columns). The last column represents the calculation with $\beta_{2}$ and $\beta_{4}$ included in the radius and diffuseness parameters. Since the calculations are performed for transitions from the parent to the daughter ground state (g.s), the branching ratio (B.R.) for the g.s. $\to$ g.s. transition, whenever available from experiment is given. |  |  |  | $P_{i}$
---|---|---|---|---
|  |  |  |  Spherical | Deformed with | Deformed with | Deformed with
Parent |  Decay mode |  B.R. | $l_{\rm min}$ | Double |  $R(\theta;\beta_{2})$ and $a_{\perp}(\theta;\beta_{2})$ |  $R(\theta;\beta_{2},\beta_{4})$ and $a_{0}$ |  $R(\theta;\beta_{2},\beta_{4})$ and $a_{\perp}(\theta;\beta_{2},\beta_{4})$
|  |  |  | Folding | Mumbai | Perey-Buck | Mumbai | Perey-Buck | Mumbai | Perey-Buck
222Ra |  $\alpha$+218Rn |  0.969 | 0 |  8.63$\times 10^{-2}$ |  7.84$\times 10^{-2}$ |  5.20$\times 10^{-2}$ |  8.26$\times 10^{-2}$ |  5.52$\times 10^{-2}$ |  7.58$\times 10^{-2}$ |  5.10$\times 10^{-2}$
|  14C+208Pb |  - | 0 |  8.64$\times 10^{-7}$ |  8.26$\times 10^{-7}$ |  1.99$\times 10^{-7}$ |  8.28$\times 10^{-7}$ |  2.00$\times 10^{-7}$ |  8.26$\times 10^{-7}$ |  1.99$\times 10^{-7}$
223Ra |  $\alpha$+219Rn |  0.01 | 2 |  7.18$\times 10^{-3}$ |  5.14$\times 10^{-3}$ |  4.15$\times 10^{-3}$ |  4.80$\times 10^{-3}$ |  3.42$\times 10^{-3}$ |  4.25$\times 10^{-3}$ |  3.05$\times 10^{-3}$
|  14C+209Pb |  - | 4 |  2.93$\times 10^{-8}$ |  2.40$\times 10^{-8}$ |  6.46$\times 10^{-9}$ |  2.41$\times 10^{-8}$ |  6.48$\times 10^{-9}$ |  2.40$\times 10^{-8}$ |  6.46$\times 10^{-9}$
228Th |  $\alpha$+224Ra |  0.734 | 0 |  1.40$\times 10^{-1}$ |  7.21$\times 10^{-2}$ |  4.57$\times 10^{-2}$ |  6.54$\times 10^{-2}$ |  4.61$\times 10^{-2}$ |  4.99$\times 10^{-2}$ |  2.46$\times 10^{-2}$
|  20O+208Pb |  - | 0 |  1.12$\times 10^{-8}$ |  1.08$\times 10^{-8}$ |  1.44$\times 10^{-9}$ |  1.03$\times 10^{-8}$ |  1.38$\times 10^{-9}$ |  8.95$\times 10^{-9}$ |  1.19$\times 10^{-9}$
231Pa |  $\alpha$+227Ac |  0.110 | 0 |  4.16$\times 10^{-3}$ |  2.43$\times 10^{-3}$ |  1.48$\times 10^{-3}$ |  1.71$\times 10^{-3}$ |  1.09$\times 10^{-3}$ |  1.34$\times 10^{-3}$ |  8.30$\times 10^{-4}$
|  23F+208Pb |  - | 1 |  1.91$\times 10^{-12}$ |  6.13$\times 10^{-13}$ |  6.18$\times 10^{-14}$ |  8.51$\times 10^{-13}$ |  8.65$\times 10^{-14}$ |  3.93$\times 10^{-13}$ |  3.84$\times 10^{-14}$
232U |  $\alpha$+228Th |  0.685 | 0 |  1.67$\times 10^{-1}$ |  7.66$\times 10^{-2}$ |  5.73$\times 10^{-2}$ |  7.61$\times 10^{-2}$ |  4.73$\times 10^{-2}$ |  6.31$\times 10^{-2}$ |  4.25$\times 10^{-2}$
|  24Ne+208Pb |  - | 0 |  5.91$\times 10^{-11}$ |  1.59$\times 10^{-11}$ |  1.38$\times 10^{-12}$ |  2.12$\times 10^{-11}$ |  1.84$\times 10^{-12}$ |  7.11$\times 10^{-12}$ |  5.86$\times 10^{-13}$
233U |  $\alpha$+229Th |  0.843 | 0 |  1.15$\times 10^{-1}$ |  5.68$\times 10^{-2}$ |  2.67$\times 10^{-2}$ |  5.76$\times 10^{-2}$ |  3.80$\times 10^{-2}$ |  2.98$\times 10^{-2}$ |  1.81$\times 10^{-2}$
|  24Ne+209Pb |  - | 2 |  1.04$\times 10^{-12}$ |  2.60$\times 10^{-13}$ |  2.27$\times 10^{-14}$ |  3.46$\times 10^{-13}$ |  3.03$\times 10^{-14}$ |  1.17$\times 10^{-13}$ |  9.69$\times 10^{-15}$
|  28Mg+205Hg |  - | 3 |  1.50$\times 10^{-14}$ |  6.71$\times 10^{-14}$ |  3.49$\times 10^{-15}$ |  9.29$\times 10^{-16}$ |  5.64$\times 10^{-17}$ |  1.84$\times 10^{-16}$ |  1.08$\times 10^{-17}$
235U |  $\alpha$+231Th |  0.0477 | 1 |  1.37$\times 10^{-3}$ |  5.14$\times 10^{-4}$ |  4.88$\times 10^{-4}$ |  4.09$\times 10^{-4}$ |  2.67$\times 10^{-4}$ |  2.99$\times 10^{-4}$ |  1.80$\times 10^{-4}$
|  25Ne+210Pb |  - | 3 |  1.65$\times 10^{-10}$ |  1.42$\times 10^{-10}$ |  1.22$\times 10^{-11}$ |  1.42$\times 10^{-10}$ |  1.22$\times 10^{-11}$ |  1.42$\times 10^{-10}$ |  1.22$\times 10^{-11}$
|  28Mg+207Hg |  - | 1 |  3.97$\times 10^{-12}$ |  1.84$\times 10^{-14}$ |  8.50$\times 10^{-16}$ |  2.56$\times 10^{-13}$ |  1.38$\times 10^{-14}$ |  5.05$\times 10^{-14}$ |  2.63$\times 10^{-15}$
237Np |  $\alpha$+233Pa |  0.0239 | 1 |  1.64$\times 10^{-2}$ |  5.94$\times 10^{-3}$ |  4.31$\times 10^{-3}$ |  4.52$\times 10^{-3}$ |  3.00$\times 10^{-3}$ |  3.24$\times 10^{-3}$ |  2.09$\times 10^{-3}$
|  30Mg+207Tl |  - | 2 |  1.84$\times 10^{-13}$ |  2.70$\times 10^{-14}$ |  1.45$\times 10^{-15}$ |  6.16$\times 10^{-14}$ |  3.46$\times 10^{-15}$ |  3.33$\times 10^{-14}$ |  1.85$\times 10^{-15}$
236Pu |  $\alpha$+232U |  0.691 | 0 |  1.10$\times 10^{-1}$ |  6.27$\times 10^{-2}$ |  3.38$\times 10^{-2}$ |  3.65$\times 10^{-2}$ |  2.28$\times 10^{-2}$ |  2.62$\times 10^{-2}$ |  1.56$\times 10^{-2}$
|  28Mg+208Pb |  - | 0 |  2.38$\times 10^{-13}$ |  1.13$\times 10^{-15}$ |  5.94$\times 10^{-17}$ |  1.67$\times 10^{-14}$ |  1.03$\times 10^{-15}$ |  3.88$\times 10^{-13}$ |  2.81$\times 10^{-14}$
241Am |  $\alpha$+237Np |  0.0037 | 1 |  4.80$\times 10^{-2}$ |  2.24$\times 10^{-2}$ |  1.27$\times 10^{-2}$ |  2.69$\times 10^{-2}$ |  1.86$\times 10^{-2}$ |  1.03$\times 10^{-2}$ |  6.65$\times 10^{-3}$
|  34Si+207Tl |  - | 3 |  2.35$\times 10^{-15}$ |  2.05$\times 10^{-15}$ |  8.92$\times 10^{-17}$ |  2.05$\times 10^{-15}$ |  8.92$\times 10^{-17}$ |  2.05$\times 10^{-15}$ |  8.92$\times 10^{-17}$
242Cm |  $\alpha$+238Pu |  0.748 | 0 |  9.16$\times 10^{-2}$ |  3.90$\times 10^{-2}$ |  2.64$\times 10^{-2}$ |  3.16$\times 10^{-2}$ |  2.03$\times 10^{-2}$ |  2.27$\times 10^{-2}$ |  1.37$\times 10^{-2}$
|  34Si+208Pb |  - | 0 |  6.78$\times 10^{-15}$ |  6.55$\times 10^{-15}$ |  2.65$\times 10^{-16}$ |  6.55$\times 10^{-15}$ |  2.65$\times 10^{-16}$ |  6.55$\times 10^{-15}$ |  2.65$\times 10^{-16}$
Table 6: The calculated $\alpha$\- and cluster-decay preformation factor
$P_{i}$ ($i$ = $\alpha$ or $c$) using the DF model and different models of
nonlocality (including deformation partly and fully). The last two columns
represent the complete calculation with $\beta_{2}$ and $\beta_{4}$ included
in the radius and diffuseness parameters. Other columns show the effect of
dropping one or both deformation parameters. B.R. is the branching ratio for
the transition from the parent ground state to the daughter ground state as in
5.

###  III.3 Comparison of penetration probabilities

Parent |  Decay mode |  $Q$ [MeV] | $l_{\rm min}$ |  $P^{(DF)}$ |  $PC^{(M)}$ | $PC^{(Ch)}$
---|---|---|---|---|---|---
222Ra |  $\alpha$+218Rn |  6.678 | 0 |  $7.40\times 10^{-23}$ |  3.9 |  7.1
|  14C+208Pb |  33.049 | 0 |  $3.23\times 10^{-27}$ |  4.1 |  7.9
223Ra |  $\alpha$+219Rn |  5.979 | 2 |  $3.41\times 10^{-26}$ |  12.2 |  7.4
|  14C+209Pb |  31.828 | 4 |  $1.08\times 10^{-29}$ |  21.1 |  8.3
228Th |  $\alpha$+224Ra |  5.520 | 0 |  $2.78\times 10^{-29}$ |  4.1 |  5.3
|  20O+208Pb |  44.723 | 0 |  $5.56\times 10^{-35}$ |  4.0 |  10.2
231Pa |  $\alpha$+227Ac |  5.150 | 0 |  $5.55\times 10^{-32}$ |  4.2 |  8.1
|  23F+208Pb |  51.888 | 1 |  $1.95\times 10^{-36}$ |  6.8 |  10.9
232U |  $\alpha$+228Th |  5.413 | 0 |  $6.60\times 10^{-31}$ |  4.2 |  8.1
|  24Ne+208Pb |  62.310 | 0 |  $2.78\times 10^{-32}$ |  3.9 |  10.9
233U |  $\alpha$+229Th |  4.908 | 0 |  $4.13\times 10^{-34}$ |  4.1 |  8.4
|  24Ne+209Pb |  60.485 | 2 |  $5.41\times 10^{-35}$ |  11.0 |  11.7
|  28Mg+205Hg |  74.226 | 3 |  $7.00\times 10^{-36}$ |  15.7 |  13.1
235U |  $\alpha$+231Th |  4.678 | 1 |  $7.94\times 10^{-36}$ |  11.8 |  8.4
|  25Ne+210Pb |  57.683 | 3 |  $8.57\times 10^{-40}$ |  15.6 |  12.5
|  28Mg+207Hg |  72.425 | 1 |  $3.65\times 10^{-38}$ |  7.2 |  13.8
237Np |  $\alpha$+233Pa |  4.958 | 1 |  $2.19\times 10^{-34}$ |  11.8 |  8.0
|  30Mg+207Tl |  74.790 | 2 |  $1.34\times 10^{-36}$ |  10.5 |  12.3
236Pu |  $\alpha$+232U |  5.867 | 0 |  $2.41\times 10^{-29}$ |  4.0 |  8.1
|  28Mg+208Pb |  79.669 | 0 |  $3.86\times 10^{-31}$ |  3.8 |  11.6
241Am |  $\alpha$+237Np |  5.637 | 1 |  $3.75\times 10^{-31}$ |  11.9 |  8.2
|  34Si+207Tl |  93.926 | 3 |  $1.00\times 10^{-32}$ |  14.4 |  12.0
242Cm |  $\alpha$+238Pu |  6.215 | 0 |  $1.88\times 10^{-28}$ |  4.1 |  8.1
|  34Si+208Pb |  96.510 | 0 |  $4.59\times 10^{-31}$ |  3.6 |  11.5
Table 7: Comparison of the penetration probabilities calculated within the
Mumbai ($M$) and Chaudhury ($Ch$) approach. The effect of nonlocality is
expressed in terms of a percentage increase in the penetration factor within
the $M$ or $Ch$ model as compared to the double folding model and is given by
$PC^{(NL)}=\big{(}P^{(NL)}-P^{(DF)}\big{)}\times 100/P^{(DF)}$.

In the section discussing the different approaches for nonlocality, we
mentioned one of the earliest works Chaudhury (1963, 1960) studying the
effects of the nonlocal strong interaction in $\alpha$ decay. The author
evaluated the penetration probabilities for the $\alpha$ decay of several
nuclei using Eq. (19). In Table 7, we compare the penetration factors obtained
in the Mumbai approach studied in this work with that of Ref. Chaudhury
(1963). However, in order to perform a consistent comparison we replace the
Igo potential used in Ref. Chaudhury (1963) by the DF one used here and
similarly the point-like Coulomb interaction in Eq. (19) is replaced by that
obtained using the DF procedure too. Apart from this, recalling that applying
the JWKB procedure to the radial equation leads to an improper behavior of the
solution near the origin unless one replaces the centrifugal term by the
Langer modified term mentioned earlier Langer (1937), we replace $l(l+1)$ in
Eq. (19) by $(l+1/2)^{2}$. The Bohr-Sommerfeld quantization, which was not
considered in Ref. Chaudhury (1963), is also taken into account to obtain a
modified version of Eq. (19) given by

$P^{\rm
Ch}=\exp\biggl{\\{}-2\sqrt{2\mu\over\hbar^{2}}\int_{r_{2}}^{r_{3}}\biggl{[}\Big{(}\lambda\,V_{N}(r)\\\
+V_{C}(r)-E\Big{)}\epsilon(r)+\frac{\hbar^{2}(l+1/2)^{2}}{2\mu
r^{2}}\biggr{]}^{1/2}{\rm d}r\biggr{\\}}$ (34)

where $V_{N}$ is the attractive strong potential whose depth is modified by
the factor $\lambda$ due to the Bohr-Sommerfeld condition. The turning points
$r_{2}$ and $r_{3}$ are given by $V(r)=E$ as before with $E$ taken to be the
$Q$ value of the decay.

The penetration factors in both models are found to increase due to the
inclusion of nonlocality in the nuclear potential. This is consistent with the
fact that the half-lives in general decrease due to nonlocality. The effect is
expressed in terms of a percentage change in the penetration probability given
by $PC^{(NL)}=\big{(}P^{(NL)}-P^{(DF)}\big{)}\times 100/P^{(DF)}$. The order
of magnitude of the changes are similar in both the models considered.
However, the results within the Mumbai approach are sensitive to the $l$ value
in the decay. This is expected since the potential defined by Eq. (17),
obtained within the Mumbai approach is $l$ dependent.

##  IV Summary

In an earlier work Perez Velasquez  _et al._ (2019), the effect of including
nonlocality in the nuclear interaction potential was found to decrease the
$s$-wave ($l$ = 0) $\alpha$ decay half-lives of spherical nuclei. Results in
Ref. Perez Velasquez  _et al._ (2019) indicated that whereas the energy-
dependent Perey-Buck (PB) model produced a decrease of 20–40 % in the half-
lives, $t_{1/2}$, the energy-independent Mumbai (M) potential decreased the
half-lives by only 2–4 %. Since both the Perey-Buck and Mumbai potentials
reproduce the scattering data quite well, this large difference in the
decrease of $t_{1/2}$ was surprising. The Mumbai potential is $l$ dependent
whereas the Perey-Buck potential does not depend on $l$. In the calculation of
cross sections, where one usually sums over all possible $l$ values to compare
with data, the differences between the Mumbai and Perey-Buck potentials do not
become quite evident. However, if one considers a nuclear decay, one is
actually picking up a particular value of $l$. Hence, in the present work,
different nuclear decays involving non-zero values of $l$ were investigated.
Considering nuclei with $l\neq$ 0 to be spherical is not always a good
approximation. Hence, the above studies were carried out by using a deformed
potential whenever necessary. The latter allowed us to also study the effect
of deformation on nonlocality.

When a nucleus has the possibility of decaying by emitting an $\alpha$
particle (4He nucleus) or a light cluster (such as 14C, 24Ne, 28Mg, etc.), one
can consider the decaying nucleus to be a preformed cluster of an $\alpha$ and
a heavy daughter or a light nucleus and a not-so-heavy daughter (232U
considered as a preformed cluster of 4He + 228Th or 24Ne + 208Pb, for
example). Would the effects of nonlocality (and deformation) show up
differently in the two scenarios? To answer this question, the half-lives were
calculated for those nuclei which can decay by emitting both an $\alpha$ or a
light nucleus. The exercise was rewarding not only in the sense of finding the
different manifestation of nonlocality in the two kinds of decays of the same
parent nucleus but also in noting the differences in the two nonlocality
models studied here.

Since the detailed observations are many, we summarize here the global
conclusions drawn from the results:

  1. (i)

In the tunneling decay of a nucleus which is considered to be a preformed
cluster of the tunneling light nucleus and a heavy daughter, it is important
to consider the nonlocality in the interaction potential between the two
nuclei forming the cluster.

  2. (ii)

Including the effects of deformation of the light nucleus in cluster decay or
the heavy daughter in $\alpha$ decay, both lead to a decrease in the half-
lives of the nuclei. The amount of decrease is decided by the magnitude of the
deformation parameters but not affected by the sign of the parameters.

  3. (iii)

Sensitivity of the results to the $l$ value (angular momentum) in the decay is
manifested within the Mumbai model for nonlocality.

  4. (iv)

The combined effect of including deformation and nonlocality (on the half-
lives of nuclei) is in general quite large within both models of nonlocality
studied here.

  5. (v)

Phenomenologically determined cluster preformation factors display a linear
dependence on the $Q$-value or the energy carried by the emitted $\alpha$ or
light cluster.

Finally, we add that given the nature of the present work with the many
ingredients entering into the calculations of half-lives, there is surely
scope for better calculations. For example, one could improve the tunneling
model by including the intrinsic degrees of freedom of the composite tunneling
object as shown in Ref. Bertulani  _et al._ (2007). Though the nonlocality in
the two body interaction has been taken into account, there remains the
possibility of including the nonlocality inside the interacting nuclei.
Finally, one could replace the semiclassical approach used here by a fully
quantum mechanical one or study the nuclear decay within different approaches
as in Ref. Ni and Ren (2015). Future investigations in this direction can
prove useful for a better understanding of nuclear structure as well as
nonlocal nuclear interactions.

###### Acknowledgements.

D.F.R-G. thanks the Faculty of Science, Universidad de los Andes, Colombia,
for financial support through Grant No. INV-2020-103-2053. N.J.U acknowledges
financial support from SERB, Government of India (Grant No. YSS/2015/000900).

## References

  * Rutherford (1899) E. Rutherford, Phil. Mag. 47, 109 (1899).
  * Ni and Ren (2009) D. Ni and Z. Ren, Phys. Rev. C 80, 051303 (2009).
  * Qian  _et al._ (2011) Y. Qian, Z. Ren, and D. Ni, Nucl. Phys. A 866, 1 (2011).
  * Mohr (2017) P. Mohr, Phys. Rev. C 95, 011302 (2017).
  * Seif  _et al._ (2021) W. M. Seif, A. R. Abdulghany, and Z. N. Hussein, J. Phys. G 48, 025111 (2021).
  * Sandulescu  _et al._ (1980) A. Sandulescu, D. Poenaru, and W. Greiner, Sov. J. Particles Nucl. 11:6, (1980).
  * Rose and Jones (1984) H. J. Rose and G. A. Jones, Nature 307, 245 (1984).
  * Zhang and Wang (2018) Y. L. Zhang and Y. Z. Wang, Phys. Rev. C 97, 014318 (2018).
  * Perez Velasquez  _et al._ (2019) J. E. Perez Velasquez, N. G. Kelkar, and N. J. Upadhyay, Phys. Rev. C 99, 024308 (2019).
  * Gurvitz and Kalbermann (1987) S. A. Gurvitz and G. Kalbermann, Phys. Rev. Lett. 59, 262 (1987).
  * Kelkar and Castañeda (2007) N. G. Kelkar and H. M. Castañeda, Phys. Rev. C 76, 064605 (2007).
  * Kelkar and Nowakowski (2016) N. G. Kelkar and M. Nowakowski, J. Phys. G: 43, 105102 (2016).
  * Xu  _et al._ (2016) C. Xu, Z. Ren, G. Röpke, P. Schuck, Y. Funaki, H. Horiuchi, A. Tohsaki, T. Yamada, and B. Zhou, Phys. Rev. C 93, 011306 (2016).
  * Xu  _et al._ (2017) C. Xu, G. Röpke, P. Schuck, Z. Ren, Y. Funaki, H. Horiuchi, A. Tohsaki, T. Yamada, and B. Zhou, Phys. Rev. C 95, 061306 (2017).
  * Langer (1937) R. E. Langer, Phys. Rev. 51, 669 (1937).
  * Satchler and Love (1979) G. Satchler and W. Love, Phys. Rep. 55, 183 (1979).
  * Wildermuth (2013) K. Wildermuth and Y. C. Tang,  _A unified theory of the nucleus_ (Springer-Verlag, 2013).
  * Ni and Ren (2010) D. Ni and Z. Ren, Phys. Rev. C 82, 024311 (2010).
  * Seif and Abdelhady and Adel (2020) W. M. Seif, A. M. H. Abdelhady, and A. Adel, Phys. Rev. C 101, 064305 (2020).
  * Amiri and Ghodsi (2020) M. Moghaddari Amiri, and O. N. Ghodsi, Phys. Rev. C 102, 054602 (2020).
  * Ahmad  _et al._ (1975) A. A. Z. Ahmad, S. Ali, N. Ferdous, and M. Ahmed, Nuovo Cimento 30, 385 (1975).
  * Cândido Ribeiro  _et al._ (1997) M. A. Cândido Ribeiro, L. C. Chamon, D. Pereira, M. S. Hussein, and D. Galetti, Phys. Rev. Lett. 78, 3270 (1997).
  * Rotureau  _et al._ (2017) J. Rotureau, P. Danielewicz, G. Hagen, F. M. Nunes, and T. Papenbrock, Phys. Rev. C 95, 024315 (2017).
  * Balantekin  _et al._ (1998) A. B. Balantekin, J. F. Beacom, and M. A. C. Ribeiro, J. Phys. G 24, 2087 (1998).
  * Perey and Buck (1962) F. Perey and B. Buck, Nucl. Phys. 32, 353 (1962).
  * Chamon  _et al._ (2003) L. C. Chamon, B. V. Carlson, L. R. Gasques, D. Pereira, C. De Conti, M. A. G. Alvarez, M. S. Hussein, M. A. Cãndido Ribeiro, E. S. Rossi Jr., and C. P. Silva, Braz. J. Phys. 33, 238 (2003).
  * Upadhyay  _et al._ (2017) N. J. Upadhyay, A. Bhagwat, and B. K. Jain, J. Phys. G 45, 015106 (2018).
  * Jackson and Johnson (1974) D. Jackson and R. Johnson, Phys. Lett. B 49, 249 (1974).
  * Adel and Alharbi (2017) A. Adel and T. Alharbi, Nucl. Phys. A 958, 187 (2017).
  * Adel and Alharbi (2020) A. Adel and T. Alharbi, Braz. J. Phys. 50, 454 (2020).
  * Khoa (2020) D. T. Khoa, Phys. Rev. C 63, 034007 (2001).
  * Khoa  _et al._ (1997) D. T. Khoa, G. R. Satchler, and W. von Oertzen, Phys. Rev. C 56, 954 (1997).
  * Chamon  _et al._ (2002) L. C. Chamon, B. V. Carlson, L. R. Gasques, D. Pereira, C. De Conti, M. A. G. Alvarez, M. S. Hussein, M. A. Cãndido Ribeiro, E. S. Rossi Jr., and C. P. Silva, Phys. Rev. C 66, 014610 (2002).
  * Chaudhury (1960) M. L. Chaudhury , Phys. Rev. Lett. 5, 205–206 (1960).
  * Chaudhury (1963) M. L. Chaudhury , Phys. Rev. 130, 2339–2341 (1963).
  * Rhoades-Brown  _et al._ (1983) M. J. Rhoades-Brown, V. E. Oberacker, M. Seiwert, and W. Greiner, Z. Phys. A 310, 287 (1983).
  * Maroufi  _et al._ (2019) N. Maroufi, V. Dehghani, and S. Alavi, Nucl. Phys. A 983, 77 (2019).
  * Bohr and Mottelson (1969) A. Bohr and B. Mottelson,  _Nuclear Structure: Nuclear deformations_, Nuclear Structure (W. A. Benjamin, 1969).
  * Scamps  _et al._ (2013) G. Scamps, D. Lacroix, G. G. Adamian, and N. V. Antonenko, Phys. Rev. C 88, 064327 (2013).
  * Ismail  _et al._ (2002) M. Ismail, A. Y. Ellithi, and F. Salah, Phys. Rev. C 66, 017601 (2002).
  * Xu and Ren (2006) C. Xu and Z. Ren, Phys. Rev. C 74, 014304 (2006).
  * Denisov and Ikezoe (2005) V. Y. Denisov and H. Ikezoe, Phys. Rev. C 72, 064613 (2005).
  * Ni and Ren (2015) D. Ni and Z. Ren, Ann. Phys. 358, 108 (2015).
  * Qian  _et al._ (2016) Y. Qian, Z. Ren, and D. Ni, Phys. Rev. C 94, 024315 (2016).
  * (45) Brookhaven National Laboratory, National nuclear data center, available at http://www.nndc.bnl.gov.
  * (46) Institute of Modern Physics, Chinese Academy of Sciences, Atomic mass data center, available at https://www-nds.iaea.org/amdc/.
  * Moller  _et al._ (1995) P. Moller, J. Nix, W. Myers, and W. Swiatecki, Atomic Data and Nuclear Data Tables 59, 185 (1995).
  * Sawhney  _et al._ (2011) G. Sawhney, M. K. Sharma, and R. K. Gupta, Phys. Rev. C 83, 064610 (2011).
  * Singh  _et al._ (2010) B. Singh, S. K. Patra, and R. K. Gupta, Phys. Rev. C 82, 014607 (2010).
  * Mang (1960) H. J. Mang, Phys. Rev. 119, 1069 (1960).
  * Mang (1960) H. J. Mang, Z. Phys. 148, 582 (1957).
  * Deng  _et al._ (2015) D. Deng, Z. Ren, D. Ni, and Y. Qian, J. Phys. G 42, 075106 (2015).
  * Bertulani  _et al._ (2007) C. A. Bertulani, V. V. Flambaum, and V. G. Zelevinsky, J. Phys. G 34, 2289 (2007).