# Exploring the competition between $\alpha$-decay and proton radioactivity: A
comparative study of proximity potential formalisms

A. Adel<EMAIL_ADDRESS>Physics Department, Faculty of Science, Cairo
University, 12613 Giza, Egypt  Karim H. Mahmoud  Physics Department, Faculty
of Science, Cairo University, 12613 Giza, Egypt  Haitham A. Taha  Department
of Physics, Zewail City for Science and Technology, Egypt

###### Abstract

We have conducted a comprehensive and systematic study of the proton
radioactivity and $\alpha$-decay half-lives of neutron-deficient nuclei. This
investigation involved the utilization of various Proximity potentials and
also considered the incorporation of thermal effects. For the half-life
calculations, we employed both temperature-independent and temperature-
dependent interaction potentials. We observed that proton radioactivity serves
as the dominant mode of decay for nuclides situated very close to the proton
drip-line. We explored a universal curve that examines the correlation between
the decimal logarithm of experimental half-lives and the negative decimal
logarithm of the penetrability for both proton radioactivity and
$\alpha$-decay.

##  I Introduction

The investigation of exotic nuclei containing an extreme number of nucleons
has consistently garnered considerable interest, with a primary focus on
$\alpha$-decay and proton radioactivity [1, 2, 3, 4, 5, 6, 7, 8, 9].
$\alpha$-decay offers valuable insights into nuclear structure properties,
providing abundant information on shell and pairing effects [10], neutron skin
thickness [11], isospin asymmetry [12], nuclear deformation [13], and shape
staggering identification [14]. Moreover, $\alpha$-decay plays a crucial role
in identifying recently synthesized superheavy nuclei [15]. On the other hand,
the emission of protons from the nuclear ground state imposes restrictions on
the formation of increasingly exotic nuclei on the proton-rich side of the
beta stability valley [16]. One proton radioactivity can serve as a valuable
tool for obtaining spectroscopic information since the decaying proton is an
unpaired proton that does not fully occupy its orbit. The rates of these
decays are highly sensitive to the $Q$ values and orbital angular momenta,
which, in turn, assist in determining the orbital angular momenta of the
emitted protons [4].

Driplines represent the boundaries of the nuclear landscape, defined by
nuclear binding energies. The proton drip-line marks the border where nuclei
become unbound to proton emission from their ground states [17]. Beyond the
proton drip-lines, the proton separation energy becomes negative, indicating
insufficient binding energy to retain protons within the nucleus. These
proton-rich nuclei have positive $Q$-values for proton emissions, which drives
the spontaneous tendency to shed excess protons. This region of proton
emitters presents a significant challenge for nuclear theory in predicting the
properties of unstable states.

Proton radioactivity was experimentally confirmed in 1970, with evidence of
proton emission from the high-spin isomeric state of ${}^{53}\textrm{Co}^{m}$
to the ground state of ${}^{52}\textrm{Fe}$ [22]. Subsequently, proton
emissions from various ground states (e.g., ${}^{151}\textrm{Lu}$ and
${}^{147}\textrm{Tm}$) and isomeric states have been discovered, with a
combined total of about 28 proton emitters decaying from ground states and 20
from isomeric states [17]. With the advancement of new experimental facilities
and radioactive beams, more new proton emitters are anticipated to be
observed, shedding light on the detailed nuclear structure in the proton
dripline region.

Proton radioactivity can be effectively addressed using the Wentzel-Kramers-
Brillouin (WKB) method, as this process can be treated as a quantum tunneling
effect through a potential barrier, similar to $\alpha$-decay [18]. Several
methods have been employed to investigate proton radioactivity, including the
density-dependent M3Y effective interaction [19, 4], the JLM interaction [19],
the unified fission model [20], the generalized liquid drop model (GLDM) [21],
the cluster model [22], the deformed density-dependent model [3], the Gamow-
like model [23], the Coulomb and proximity potential model for deformed nuclei
(CPPMDN) [24], the covariant density functional (CDF) theory [25], analytic
formulas [26], the distorted-wave Born approximation (DWBA) [27], the two-
potential approach (TPA) [27], the quasiclassical method [27], and others.

Several theoretical approaches have been developed for describing
$\alpha$-decay, including the density-dependent cluster model [28, 29, 30],
the generalized liquid-drop model (GLDM) [31], the modified generalized
liquid-drop model (MGLDM) [32, 33], the effective liquid-drop model (ELDM)
[34, 35], the coupled channel approach [36, 37], the fission-like model [38],
and the Coulomb and Proximity potential model (CPPM) [39, 40]. In addition,
various analytical and empirical formulas have been proposed to predict
$\alpha$-decay half-lives [41, 42, 43, 44, 45]. The key element in most of
these studies is the adopted model for the nuclear part of the potential.

Blocki et al. introduced the proximity potential as a means to address heavy
ion reactions [46]. As a nucleus-nucleus interaction potential, it relies on
the proximity force theorem [46, 47], represented as the product of a factor
dependent on the mean curvature of the interaction surface and a universal
function (dependent on the separation distance) that remains unaffected by the
masses of the colliding nuclei [48]. One of its key advantages lies in its
straightforward and accurate formalism. However, to overcome the limitations
of the original version of the proximity potential (Prox.1977) [46], several
improvements and modifications have been proposed. These developments have
involved either enhancing the form of the surface energy coefficients [49, 50,
51] or introducing an improved universal function or another parameterization
for nuclear radius [47, 48, 52, 53].

Several comparative studies have been conducted by different researchers,
including Yao et al. [54], Ghodsi et al. [55], and Santhosh et al. [56], to
explore various proximity potential formalisms used in describing
$\alpha$-decay and heavy particle radioactivity. Furthermore, these proximity
potential formalisms have found application in investigating heavy ion fusion
[57, 58] and ternary fission [59, 60].

The interplay between $\alpha$-decay and proton emission becomes evident in
the region of neutron-deficient nuclei [61]. Prompted by this observation, we
have been motivated to thoroughly investigate the competition between these
two decay modes, aiming to ascertain which one holds dominance. This
investigation provides a valuable testing ground for theoretical models. Thus,
the main objective of this research is to examine the appropriateness of
various proximity potentials concerning the interplay between $\alpha$-decay
and proton radioactivity in neutron-deficient nuclei. Moreover, we investigate
the impact of thermal effects by examining temperature-independent and
temperature-dependent interaction potentials for half-life calculations.
Subsequently, we compare these calculated values with the available
experimental data.

##  II Theoretical Framework

###  II.1 Half-life time calculation

To determine the half-life time of radioactive nuclei, it is necessary to
calculate the total interaction potential $V(r)$ [29, 18]. It can be
mathematically represented as follows:

$V(r)=V_{N}(r)+V_{C}(r)+V_{\ell}(r),$ (1)

where $V_{C}(r)$,$V_{N}(r)$, and $V_{\ell}(r)$ are the coulomb, nuclear, and
centrifugal potentials, respectively. In this study, we employ the proximity
potential formalism to compute the nuclear potential $V_{N}(r)$. The Coulomb
potential, $V_{C}(r)$, is postulated as the potential arising from a uniformly
charged sphere with a radius of $R$. It can be expressed as follows:

$V_{C}(r)=\begin{cases}\frac{Z_{1}Z_{2}e^{2}}{2R}\left[3-\left(\frac{r}{R}\right)^{2}\right],&r\leq
R,\\\ \frac{Z_{1}Z_{2}e^{2}}{r},&r>R,\end{cases}$ (2)

where $R=R_{1}+R_{2}$ is given by the sum of the radii of daughter nucleus
$R_{1}$ and emitted particle (alpha, proton, or cluster) $R_{2}$. While
$Z_{1}$ and $Z_{2}$ are the proton numbers of the daughter nucleus and emitted
particle, respectively.

We utilize the Langer modified form for the centrifugal potential
$V_{\ell}(r)$ due to the necessity of the correction term
$\ell(\ell+1)\rightarrow(\ell+\frac{1}{2})^{2}$ for one-dimensional problems
[62]. This form can be expressed as follows:

$V_{\ell}(r)=\frac{\hbar^{2}(\ell+\frac{1}{2})^{2}}{2\mu r^{2}},$ (3)

where $\ell$ is the angular momentum carried by the emitted particle. We can
determine the minimum angular momentum $\ell_{min}$ using spin and parity
conservation rules, and $\mu$ is the reduced mass of the emitted particle and
daughter nucleus where $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$, where $m_{1}$ and
$m_{2}$ are the masses of the daughter nucleus and emitted particle,
respectively.

Once the total interaction has been computed, the penetration probability of
the particle can be determined using the following expression:

$P=\exp\left(-2\int_{R_{in}}^{R_{out}}k(r)\,dr\right),$ (4)

Here, the classical turning points $R_{in}$ and $R_{out}$ are determined from
$V(R_{in})=V(R_{out})=Q$. The wave number $k(r)$ of the emitted particle given
as $k(r)=\sqrt{\frac{2\mu}{\hbar^{2}}|V(r)-Q|}$ and $Q$ is the decay energy.

The knocking frequency (or assault frequency) $\nu$ is the number of assaults
on the barrier per second and it is related to oscillation frequency $\omega$.
It is given by [55]:

$\nu=\frac{\omega}{2\pi}=\frac{2E_{\nu}}{h}$ (5)

where $E_{\nu}$ is zero-point vibration energy, which, in the case of a proton
it is related to its $Q$-value ($Q_{p}$) [63] as follows:

$E_{\nu}=\begin{cases}0.1045\,Q_{p},&\text{ for
even-}Z\text{-even-}N\text{-parent nuclei, }\\\ 0.0962\,Q_{p},&\text{ for
odd-}Z\text{-even-}N\text{-parent nuclei, }\\\ 0.0907\,Q_{p},&\text{ for
even-}Z\text{-odd-}N\text{-parent nuclei, }\\\ 0.0767\,Q_{p},&\text{ for
odd-}Z\text{-odd-}N\text{-parent nuclei. }\end{cases}$ (6)

In the case of an emitted alpha particle or cluster, the value of the
empirical zero-point vibration energy $E_{\nu}$ can be obtained from the
following expression [64, 65]:

$\displaystyle
E_{v}{=}Q\left\\{0.056+0.039\;\mathrm{exp}\left[\frac{4-A_{2}}{2.5}\right]\right\\},\hbox{
for }A_{2}\geq 4.$

The half-life time is subsequently calculated using the formula:

$T_{\frac{1}{2}}=\frac{\ln 2}{\lambda}$ (7)

where $\lambda$ is the decay constant given by $\lambda=\nu P$.

###  II.2 The proximity potential

For evaluating the nuclear potential term in the total interaction between the
daughter nucleus and the emitted particle, we will employ six distinct
proximity potentials: CW76 [66], Denisov [67], Guo2013 [68], Prox00 DP [48],
Bass73 [52, 53], and Denisov DP [48].

####  II.2.1 The proximity potential Guo2013

Guo et al. introduced a novel universal function of proximity potential model
labeled as Guo2013 by employing the double folding model with density-
dependent nucleon-nucleon interaction and fitting the universal functions of
numerous reaction systems [68]. The expression for this proximity potential is
as follows:

$V_{N}(r)=4\pi\gamma b\frac{R_{1}R_{2}}{R_{1}+R_{2}}\phi(\xi),$ (8)

In this expression, the parameter $b$ represents the diffuseness of the
nuclear surface and is assigned a value of unity and $\gamma$ is the surface
coefficient and is obtained by the following expression:

$\gamma=0.9517\left[1-1.7826\left(\frac{N-Z}{A}\right)^{2}\right]$ (9)

In this context, $N$, $Z$, and $A$ denote the neutron, proton, and mass
numbers, respectively, of the parent nucleus. The values of $R_{1}$ and
$R_{2}$ can be expressed as follows:

$\begin{array}[]{cc}R_{i}=1.28A_{i}^{\frac{1}{3}}-0.76+0.8A_{i}^{-\frac{1}{3}}&(i=1,2).\end{array}$
(10)

The universal function $\phi(\xi)$ is given by the following expression:

$\phi(\xi)=\frac{p_{1}}{1+exp(\frac{\xi+p_{2}}{p_{3}})}$ (11)

where $p_{1}$, $p_{2}$ and $p_{3}$ are adjustable parameters whose values are
-17.72, 1.30 and 0.854, respectively and $\xi$ is given by
$\xi=\frac{r-R_{1}-R_{2}}{b}$.

####  II.2.2 The proximity potential CW76

Christensen and Winther analyzed the heavy-ion elastic scattering data and
came up with an empirical nuclear potential[66] which is expressed as follows:

$V_{N}(r)=-50\frac{R_{1}R_{2}}{R_{1}+R_{2}}\phi(r-R_{1}-R_{2})$ (12)

where $R_{1}$ and $R_{2}$ are given by:

$\begin{array}[]{cc}R_{i}=1.233A_{i}^{\frac{1}{3}}-0.978A_{i}^{-\frac{1}{3}}&(i=1,2)\end{array}$
(13)

and the universal function is given by
$\phi(r-R_{1}-R_{2})=exp(-\frac{r-R_{1}-R_{2}}{0.63})$.

####  II.2.3 The proximity potential Bass73

Bass developed a nuclear potential expression in terms of the difference
between finite and infinite separation $\xi$ based on the liquid drop
model[52, 53]. The expression is given as follows:

$\begin{split}V_{N}(r)&=-4\pi\gamma\frac{dR_{1}R_{2}}{R}exp\left(-\frac{\xi}{d}\right)\\\
&=\frac{-da_{s}A_{1}^{\frac{1}{3}}A_{2}^{\frac{1}{3}}}{R}exp\left(-\frac{r-R}{d}\right)\end{split}$
(14)

$\gamma$ represents the specific surface energy of the liquid drop model,
while $d$ is the range parameter whose value is 1.35 fm. $a_{s}$ is the
surface term of the Bethe Weizsacker semi-empirical mass formula, whose value
is 17.0 MeV. R is given by the sum of the half-maximum density radii
$R=R_{1}+R_{2}=r_{0}\left(A_{1}^{\frac{1}{3}}+A_{2}^{\frac{1}{3}}\right)$,
where $r_{0}=1.07$, $R_{1}$, $A_{1}$. $R_{2}$ and $A_{2}$ are the radii and
mass numbers of the daughter nucleus and emitted particle respectively.

####  II.2.4 The proximity potential Denisov

A nuclear potential expression was presented by Denisov, where he worked with
119 spherical or quasi-spherical even-even nuclei around the $\beta$ stability
line in the semi-microscopic approximation between all possible nucleus-
nucleus combinations[67]. The nuclear potential is given by:

$\begin{split}V_{N}(r)=&-1.989843\frac{R_{1}R_{2}}{R_{1}+R_{2}}\phi(\xi)\left[1+0.003525139\right.\\\
&\times\left(\frac{A_{1}}{A_{2}}+\frac{A_{2}}{A_{1}}\right)^{\frac{3}{2}}-0.4113263(I_{1}+I_{2})\left.\right]\end{split}$
(15)

$I_{1}$ and $I_{2}$ represent the isospin asymmetry of the daughter and
emitted particle which is given by $I_{i}=\frac{N_{i}-Z_{i}}{N_{i}+Z_{i}}$
$(i=1,2)$. $Z_{1}$, $N_{1}$, $R_{1}$ and $Z_{2}$, $N_{2}$, $R_{2}$ are the
proton number, neutron number, and the effective nucleus radii of the daughter
and emitted particle respectively. The effective nucleus can be obtained via:

$\begin{split}R_{i}&=R_{ipar}\left(1-\frac{3.413817}{R_{ipar}^{2}}\right)\\\
&+1.284589\left(I_{i}-\frac{0.4A_{i}}{A_{i}+200}\right)(i=1,2)\end{split}$
(16)

where the radius of the emitted particle $R_{ipar}$ is given by:

$R_{ipar}=1.240A_{i}^{\frac{1}{3}}\left(1+\frac{1.646}{A_{i}}-0.191I_{i}\right)$
(17)

The universal function $phi(\xi)$ is given by:

$\phi(\xi)=\left\\{\begin{array}[]{cc}\begin{split}&\left[1-\xi^{2}\left[0.05410106\frac{R_{1}R_{2}}{R_{1}+R_{2}}\exp\left(-\frac{\xi}{1.760580}\right)\right]\right.\\\
&-0.5395420(I_{1}+I_{2})\exp\left(-\frac{\xi}{2.424408}\right)\bigg{]}\\\
&\times\exp\left(-\frac{\xi}{0.7881663}\right)\quad\xi\geq 0,\\\
&1-\frac{\xi}{0.7881663}+1.229218\xi^{2}-0.2234277\xi^{3}\\\
&-0.1038769\xi^{4}-\frac{R_{1}R_{2}}{R_{1}+R_{2}}(0.1844935\xi^{2}\\\
&+0.07570101\xi^{3})+(I_{1}+I_{2})(0.04470645\xi^{2}\\\
&+0.03346870\xi^{3})\quad-5.65\leq\xi\leq 0\end{split}\end{array}\right\\}$
(18)

where $\xi=r-R_{1}-R_{2}-2.65$.

####  II.2.5 The proximity potential gp77

This proximity potential was used by Santhosh et al.[69] and it is given as
follows:

$V_{N}(r)=4\pi\gamma b\frac{C_{1}C_{2}}{C_{1}+C_{2}}\phi(\xi)$ (19)

which is similar to the Guo2013 potential but instead of using the radii of
the daughter and emitted nuclei, we use the S$\ddot{\text{u}}$ssman central
radii of the fragments, which are related as follows:

$C_{i}=R_{i}-\frac{b^{2}}{R_{i}}$ (20)

where $i=1,2$ representing daughter and emitted particle respectively. The
radii $R_{i}$ are the same as in Guo2013. In this potential, the nuclear
surface tension coefficient is given as follows:

$\gamma=\gamma_{0}\left[1-k_{s}\left(\frac{N-Z}{A}\right)\right]$ (21)

where Z, N, and A are the proton number, neutron number, and mass number of
the parent nucleus respectively, and the coefficients $\gamma_{0}$ and
$k_{s}$, in this case, are given as $\gamma_{0}=0.9517$ and $k_{s}=1.7826$.
The universal function $\phi(\xi)$ is given as:

$\phi(\xi)=\left\\{\begin{array}[]{cc}\begin{split}&-1.7817+0.927\xi+0.0169\xi^{2}-0.05148\xi^{3}\\\
&\quad\xi\leq 1.9475,\\\ &-4.41exp(-\xi/0.7176)\quad\xi\geq
1.9475.\end{split}\end{array}\right\\}$ (22)

###  II.3 Temperature dependent proximity potential

In the previous sections, we worked with nuclei at the ground state(no
temperature dependence). In this section, thermal effects are taken into
account, where temperature-dependent forms of $R$, $\gamma$, and $b$ are
presented[70]. They are expressed as follows:

$\begin{array}[]{cc}R_{i}(T)=R_{i}(T=0)[1+0.0005T^{2}]fm&(i=1,2)\end{array}$
(23) $\gamma(T)=\gamma(T=0)\left[1-\frac{T-T_{b}}{T_{b}}\right]^{\frac{3}{2}}$
(24) $b(T)=b(T=0)\left[1+0.009T^{2}\right]$ (25)

$T_{b}$ represents the temperature associated with near coulomb barrier
energies. In this work b(T = 0) = 1 and $R_{i}(T=0)$ is the normal radius
formula associated with each proximity potential we have worked with. Instead
of the form written in equation $(31)$ we will use a different form of
$\gamma(T)$ given as $\gamma(T)=\gamma(0)(1-0.07T)^{2}$[71]. We can determine
the temperature $T$ (in Mev) using the following expression:

$E^{*}=E_{kin}+Q_{in}=\frac{1}{9}AT^{2}-T$ (26)

$E^{*}$ represents the excitation energy of the parent nucleus, while $A$ is
its mass number. $Q_{in}$ is the entrance channel $Q$ \- value of the system.
To obtain $E_{kin}$, which is the kinetic energy of the emitted particle we
can use the following expression:

$E_{kin}=\left(\frac{A_{d}}{A_{p}}\right)Q$ (27)

##  III RESULTS AND DISCUSSION

We conducted a comprehensive investigation of the half-lives of numerous
neutron-deficient nuclei, spanning the range of atomic numbers from $Z=53$ to
81. These nuclei decay through proton emission and/or $\alpha$-decay. Our
analysis utilized five distinct proximity potentials: Bass73, CW76, Guo2013,
Denisov, and gp77. In Figure 1, the total potential $V(r)$ for the decay of
neutron-deficient ${}^{109}\textrm{I}$ is depicted, utilizing different
proximity potentials. Figure 1(a) showcases the proton decay, while Fig. 1(b)
provides insight into the $\alpha$-decay process. The dashed line represents
the $Q$-value for each decay. In Fig 1(a), the Coulomb barrier height for
proton decay is shown to be highest among various proximity potentials, with
CW76 potential displaying the maximum barrier. Conversely, for $\alpha$-decay,
both GP77 and CW76 exhibit the highest Coulomb barriers. The detailed results
of the half-life calculations are presented in both Tables LABEL:tab:table1
and LABEL:tab:table2. In Tables LABEL:tab:table1 and LABEL:tab:table2, the
first column provides information about the parent nucleus. Additionally, the
symbol "$m$" denotes the first isomeric state for the parent nucleus. Moving
to the second and third columns in both Tables LABEL:tab:table1 and
LABEL:tab:table2, we present the experimental decay energy, denoted as
$Q_{p/\alpha}$, and the transferred angular momentum $\ell_{\textrm{min}}$,
respectively. Subsequent columns in these tables display the logarithmic half-
lives calculated using the aforementioned potential models. Incorporating
considerations for temperature effects, we conducted a comparison between the
temperature-independent half-lives labeled as "T-IND" and the temperature-
dependent half-lives labeled as "T-DEP" in both Tables LABEL:tab:table1 and
LABEL:tab:table2. Finally, the last column contains the available experimental
half-life data for the decaying nucleus.

It is noteworthy that our theoretical calculations for both proton decay and
$\alpha$-decay half-lives across different parent nuclei exhibit remarkable
consistency and robust agreement with the available experimental data. Any
discrepancies between the calculated and experimental half-lives may be
attributed to the uncertainties inherent in the measurements of experimental
$Q$ values, which have a notable impact on the outcomes. This underscores the
effectiveness of our calculations, resulting in a small standard deviation
($\sigma$) between the computed and experimental logarithmic half-lives. In
essence, these findings affirm the robustness and reliability of our
understanding of these fundamental processes. We have calculated the standard
deviation, represented as $\sigma$, between the experimental and computed
values of the logarithmic half-lives as:

$\sigma=\left[\frac{1}{n-1}\sum_{i=1}^{n}(\log_{10}T_{1/2}^{calc.}-\log_{10}T_{1/2}^{expt.})^{2}\right]^{1/2}$
(28)

The standard deviation values ($\sigma$) for different potentials are as
follows: 1.3229, 1.9437, 0.288, and 0.8165 for Bass73, CW76, Guo2013, and
Denisov, respectively, in the case of proton emission. For $\alpha$-decay, the
$\sigma$ values are: 1, 1.25, 2.5249, 2.5981, and 0.4675 for Bass, CW76,
Denisov, Guo2013, and gp77, respectively. These findings highlight that among
the potentials used for $\alpha$-decay half-life calculations, gp77 stands out
as the most effective, while Guo2013 appears to be the most suitable for
proton emission.

In Fig. 2(a), we observe the discrepancies in proton decay half-lives among
various proton emitters, as listed in Table LABEL:tab:table1, in relation to
their respective experimental values. These deviations,
$\log_{10}(T_{1/2}^{\textrm{calc.}}/T_{1/2}^{\textrm{expt.}})$, are graphed as
a function of the proton number ($Z_{p}$) of the parent nucleus. Notably,
Figure 2(a) distinctly illustrates that the calculated proton decay half-lives
exhibit deviations from experimental data, typically within an order of
magnitude of 2, across multiple potential scenarios. Our analysis reveals that
the majority of data points cluster around
$\log_{10}(T_{1/2}^{\textrm{calc.}}/T_{1/2}^{\textrm{expt.}})=0$, indicating
that most calculated proton decay half-lives align favorably with experimental
data for different proton emitters. In Fig. 2(b), we illustrate the deviations
between the computed and experimental $\alpha$-decay half-lives. It’s apparent
from the graph that a significant portion of the data points is positioned
near $\log_{10}(T_{\text{calc}}/T_{\text{expt}})=0$, suggesting a notable
alignment between the majority of calculated $\alpha$-decay half-lives and
their experimental counterparts. The conclusions drawn from the Figs. 2(a) and
2(b) are quite clear. They emphasize that, based on the data presented, gp77
stands out as the most effective potential for $\alpha$-decay half-life
calculations, whereas Guo2013 seems to be the most suitable choice for proton
emission calculations.

The universality of the $\alpha$\- and cluster decay phenomena can be
effectively demonstrated by creating a graphical representation that plots the
logarithm of the experimental half-life against the negative logarithm of the
penetration probability through the barrier, as previously explored in Refs.
[72, 73, 74, 75]. It is intriguing to investigate whether this correlation can
also be applied to proton radioactivity of neutron-deficient nuclei as shown
in Fig. 3. Figure 3(a) pertains to proton decay, whereas Fig. 3(b) focuses on
$\alpha$-decay. These figures visually depict the relationship between the
decimal logarithm of experimental half-lives
($\log_{10}T_{1/2}^{\textrm{expt.}}$) and the negative decimal logarithm of
penetrability ($-\log_{10}P$). The penetrability values are computed
theoretically within the framework of the WKB approximation, employing the
Guo2013 potential for proton emission and gp77 for $\alpha$-emission. The
straight-line relationship evident in these plots affirms the robustness of
our calculations and underscores the credibility of our methods.

Qi et al. [76] have introduced a linear universal decay formula (UDL) that
exhibits remarkable accuracy in predicting half-lives across various cluster
types and isotopic series, particularly for ground-state to ground-state
transitions. This formula offers valuable physical insights and is recognized
for its universal applicability. By employing a similar analytical approach as
Qi et al. [77] and accounting for the centrifugal barrier’s impact, we can
discern a distinct and coherent graphical pattern in the proton decay half-
lives [2]. The formulation for the logarithm of the proton decay half-life can
be expressed as follows:

$\log T_{1/2}=a\chi^{\prime}+b\rho^{\prime}+d\,\ell(\ell+1)/\rho^{\prime}+c,$
(29)

where $a,b,c$, and $d$ are constants,
$\chi^{\prime}=A^{1/2}Z_{d}Q_{p}^{-1/2}$,
$\rho^{\prime}=\sqrt{AZ_{d}\left(A_{d}^{1/3}+1\right)}$, and
$A=A_{d}/\left(A_{d}+1\right)$.

The constants $a,b,c$, and $d$ are established through the process of fitting
them to existing experimental data. It’s particularly intriguing to explore
how well this formula aligns with our computed values for proton decay half-
lives using the Densiov Potential. To investigate this, we conducted a
comprehensive search to determine the optimal values for the four free
parameters ($a,b,c$, and $d$) in the context of Densiov Potential
calculations, as well as with respect to the most recent experimental data. In
Figure 4, we present the relationship between the quantity $\log
T_{1/2}^{p}-e$ and $\chi^{\prime}$. Here, $e$ is defined as
$e=b,\rho^{\prime}+d,\ell(\ell+1)/\rho^{\prime}+c$. Figure 4 illustrates this
relationship using computed values for proton decay half-lives using the
Denisov potential, denoted as $T_{1/2}^{\text{p}}$. It’s evident from Figure 4
that the quantity $\log
T_{1/2}^{\text{p}}-\left[b,\rho^{\prime}+d,\ell(\ell+1)/\rho^{\prime}+c\right]$
exhibits a linear correlation with $\chi^{\prime}$ in accordance with Eq.
(29).

##  IV SUMMARY AND CONCLUSIONS

An investigation of the proton emission and alpha decay half-life times of
different nuclei has been done, calculations were done without taking
temperature once as well as taking it into account. The calculated half-life
times agree relatively well with the experimental data. We also see that for
the isotopes ${}^{109}\textrm{I}$, ${}^{160}\textrm{Re}$,
${}^{165}\textrm{Ir}^{m}$, ${}^{185}\textrm{Bi}$, ${}^{161}\textrm{Re}$ and
${}^{112}\textrm{Cs}$, the main decay mode is proton emission. These isotopes
are found close to the proton-drip line. In this study a new set of fitting
parameters are presented for the universal decay law (UDL) on the proton decay
half-life times, using the Data found through the Denisov potential. A
universal curve for both proton emission, as well as alpha decay, has been
investigated, in which a correlation is present between the logarithm of the
experimental half-lives ($\log_{10}T_{1/2}^{exp.}$) vs the negative of the
logarithm of the penetration probability($-\log_{10}P$). We notice that the
points lying in the universal curve for alpha decay are a bit more scattered
than those in the proton radioactivity case. This difference may be due to
using a smaller sample for the alpha decay curve, it could also be due to the
uncertainties in the measurements of different parameters like spin party
assignments, decay energies as well as half-life times.

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Table 1: Calculation of proton decay half-lives of various neutron deficient nuclei. The superscript $m$ in the different nuclei denotes the isomeric states. The experimental values of $Q_{p}$ are taken from [78]. Parent |  $Q_{p}$(MeV) | $\ell_{min}$ | Temp(MeV) |  $\log_{10}T_{1/2}^{p}$ ($T_{1/2}$ in s) |  |  |
---|---|---|---|---|---|---|---
|  |  |  | Bass73 | CW76 | Denisov | Guo2013 | Expt.
|  |  |  | T-IND | T-DEP | T-IND | T-DEP | T-IND | T-DEP | T-IND | T-DEP |
${}^{109}\textrm{I}$ | $0.820\pm{0.004}$ | 2 | 0.411 | -3.031 | -2.914 | -2.125 | -2.008 | -4.304 | -4.188 | -3.767 | -3.616 | $-4.032_{-0.004}^{+0.004}$
${}^{112}\textrm{Cs}$ | $0.816\pm{0.004}$ | 2 | 0.404 | -2.115 | -2.000 | -1.220 | -1.102 | -3.436 | -3.318 | -2.876 | -2.725 | $-3.310_{-0.007}^{+0.007}$
${}^{157}\textrm{Ta}$ | $0.935\pm{0.010}$ | 0 | 0.357 | 1.232 | 1.322 | 2.028 | 2.118 | -0.257 | -0.167 | 0.438 | 0.557 | $-0.527_{-0.017}^{+0.017}$
${}^{160}\textrm{Re}$ | $1.267\pm{0.007}$ | 2 | 0.406 | -1.718 | -1.628 | -0.953 | -0.863 | -3.462 | -3.372 | -2.675 | -2.548 | $-3.045_{-0.048}^{+0.079}$
${}^{161}\textrm{Re}$ | $1.197\pm{0.005}$ | 0 | 0.394 | -1.934 | -1.842 | -1.143 | -1.051 | -3.438 | -3.346 | -2.736 | -2.612 | $-3.357_{-0.010}^{+0.010}$
${}^{161}\textrm{Re}^{m}$ | $1.321\pm{0.005}$ | 5 | 0.413 | 1.222 | 1.311 | 1.742 | 1.831 | -1.554 | -1.465 | -0.418 | -0.279 | $-0.678_{-0.009}^{+0.009}$
${}^{165}\textrm{Ir}^{m}$ | $1.727\pm{0.070}$ | 5 | 0.461 | -1.931 | -1.853 | -1.411 | -1.333 | -4.697 | -4.618 | -3.561 | -3.428 | $-3.462_{-0.087}^{+0.087}$
${}^{166}\textrm{Ir}$ | $1.152\pm{0.008}$ | 2 | 0.381 | 0.296 | 0.390 | 1.043 | 1.136 | -1.508 | -1.415 | -0.688 | -0.560 | $-0.824_{-0.091}^{+0.091}$
${}^{166}\textrm{Ir}^{m}$ | $1.324\pm{0.010}$ | 5 | 0.406 | 1.795 | 1.884 | 2.303 | 2.391 | -1.010 | -0.922 | 0.147 | 0.285 | $-0.076_{-0.026}^{+0.026}$
${}^{167}\textrm{Ir}$ | $1.070\pm{0.004}$ | 0 | 0.376 | 0.427 | 0.524 | 1.202 | 1.298 | -1.142 | 1.046 | -0.406 | -0.279 | $-0.959_{-0.025}^{+0.025}$
${}^{167}\textrm{Ir}^{m}$ | $1.245\pm{0.005}$ | 5 | 0.394 | 2.602 | 2.692 | 3.107 | 3.198 | -0.210 | -0.119 | 0.954 | 1.092 | $0.875_{-0.009}^{+0.009}$
${}^{170}\textrm{Au}$ | $1.472\pm{0.012}$ | 2 | 0.422 | -2.767 | -2.684 | -2.024 | -1.941 | -4.577 | -4.493 | -3.754 | -3.633 | $-3.487_{-0.060}^{+0.075}$
${}^{170}\textrm{Au}^{m}$ | $1.757\pm{0.018}$ | 5 | 0.458 | -1.687 | -1.610 | -1.177 | -1.100 | -4.478 | -4.401 | -3.323 | -3.191 | $-2.971_{-0.028}^{+0.035}$
${}^{171}\textrm{Au}^{m}$ | $1.707\pm{0.016}$ | 5 | 0.450 | -1.416 | -1.339 | -0.908 | -0.830 | -4.210 | -4.132 | -3.051 | -2.919 | $-2.585_{-0.013}^{+0.013}$
${}^{177}\textrm{Tl}$ | $1.155\pm{0.019}$ | 0 | 0.369 | 0.478 | 0.570 | 1.231 | 1.323 | -1.175 | -1.084 | -0.396 | -0.273 | $-1.176_{-0.121}^{+0.121}$
${}^{177}\textrm{Tl}^{m}$ | $1.962\pm{0.030}$ | 5 | 0.472 | -2.858 | -2.786 | -2.353 | -2.282 | -5.651 | -5.579 | -4.482 | -4.354 | $-3.346_{-0.076}^{+0.076}$
${}^{185}\textrm{Bi}$ | $1.527\pm{0.081}$ | 0 | 0.416 | -3.697 | -3.620 | -2.950 | -2.872 | -5.345 | -5.269 | -4.563 | -4.451 | $-4.191_{-0.030}^{+0.030}$
${}^{113}\textrm{Cs}$ | $0.973\pm{0.002}$ | 2 | 0.438 | -4.493 | -4.493 | -3.596 | -3.597 | -5.797 | -5.269 | -5.241 | -5.205 | $-4.752_{-0.010}^{+0.010}$
${}^{117}\textrm{La}$ | $0.820\pm{0.003}$ | 2 | 0.395 | -1.547 | -1.430 | -0.668 | -0.551 | -2.918 | -5.797 | -2.331 | -2.181 | $-1.602_{-0.048}^{+0.048}$
${}^{121}\textrm{Pr}$ | $0.890\pm{0.010}$ | 2 | 0.402 | -1.935 | -1.822 | -1.067 | -0.955 | -3.346 | -2.802 | -2.738 | -2.592 | $-2.000_{-0.130}^{+0.261}$
${}^{130}\textrm{Eu}$ | $1.028\pm{0.015}$ | 2 | 0.413 | -2.437 | -2.333 | -1.594 | -1.490 | -3.929 | -3.324 | -3.278 | -3.139 | $-3.046_{-0.140}^{+0.236}$
${}^{131}\textrm{Eu}$ | $0.947\pm{0.005}$ | 2 | 0.396 | -1.417 | -1.309 | -0.577 | -0.470 | -2.919 | -3.825 | -2.261 | -2.120 | $-1.699_{-0.046}^{+0.046}$
${}^{135}\textrm{Tb}$ | $1.188\pm{0.007}$ | 3 | 0.431 | -2.826 | -2.729 | -2.034 | -1.937 | -4.621 | -2.812 | -3.860 | -3.723 | $-3.027_{-0.102}^{+0.152}$
${}^{140}\textrm{Ho}$ | $1.094\pm{0.010}$ | 3 | 0.408 | -1.026 | -0.927 | -0.252 | -0.153 | -2.876 | -4.525 | -2.085 | -1.947 | $-2.222_{-0.217}^{+0.217}$
${}^{141}\textrm{Ho}$ | $1.177\pm{0.007}$ | 3 | 0.420 | -2.143 | -2.048 | -1.368 | -1.272 | -3.983 | -2.776 | -3.194 | -3.059 | $-2.387_{-0.011}^{+0.011}$
${}^{141}\textrm{Ho}^{m}$ | $1.243\pm{0.014}$ | 0 | 0.431 | -4.863 | -4.771 | -4.023 | -3.931 | -6.162 | -3.887 | -5.564 | -5.439 | $-5.137_{-0.018}^{+0.018}$
${}^{145}\textrm{Tm}$ | $1.736\pm{0.007}$ | 5 | 0.496 | -3.712 | -3.632 | -3.144 | -3.064 | -6.361 | -6.070 | -5.314 | -5.174 | $-5.499_{-0.027}^{+0.027}$
${}^{147}\textrm{Tm}$ | $1.059\pm{0.003}$ | 5 | 0.391 | 2.812 | 2.912 | 3.358 | 3.458 | 0.097 | 0.197 | 1.180 | 1.328 | $0.587_{-0.022}^{+0.022}$
${}^{147}\textrm{Tm}^{m}$ | $1.127\pm{0.007}$ | 2 | 0.403 | -1.947 | -1.851 | -1.147 | -1.051 | -3.570 | -3.475 | -2.844 | -2.713 | $-3.444_{-0.048}^{+0.048}$
${}^{150}\textrm{Lu}$ | $1.270\pm{0.002}$ | 5 | 0.421 | 0.881 | 0.974 | 1.422 | 1.515 | -1.845 | -1.753 | -0.754 | -0.610 | $-1.197_{-0.029}^{+0.029}$
${}^{151}\textrm{Lu}$ | $1.241\pm{0.002}$ | 5 | 0.415 | 1.088 | 1.181 | 1.629 | 1.722 | -1.639 | -1.546 | -0.543 | -0.400 | $-0.896_{-0.011}^{+0.011}$
${}^{151}\textrm{Lu}^{m}$ | $1.319\pm{0.010}$ | 2 | 0.427 | -3.521 | -3.432 | -2.729 | -2.640 | -5.167 | -5.078 | -4.429 | -4.303 | $-4.796_{-0.027}^{+0.027}$
${}^{155}\textrm{Ta}$ | $1.453\pm{0.015}$ | 5 | 0.440 | -0.573 | -0.487 | -0.037 | 0.049 | -3.309 | -3.223 | -2.203 | -2.064 | $-2.538_{-0.165}^{+0.225}$
${}^{156}\textrm{Ta}$ | $1.020\pm{0.004}$ | 2 | 0.373 | 0.923 | 1.024 | 1.694 | 1.794 | -0.806 | -0.706 | -0.026 | 0.107 | $-0.826_{-0.016}^{+0.016}$
${}^{156}\textrm{Ta}^{m}$ | $1.122\pm{0.008}$ | 5 | 0.389 | 3.202 | 3.299 | 3.727 | 3.824 | 0.434 | 0.531 | 1.557 | 1.701 | $0.933_{-0.005}^{+0.005}$
${}^{159}\textrm{Re}^{m}$ | $1.600\pm{0.050}$ | 5 | 0.454 | -1.386 | -1.304 | -0.858 | -0.776 | -4.138 | -4.056 | -3.020 | -2.883 | $-4.665_{-0.087}^{+0.087}$
${}^{176}\textrm{Tl}$ | $1.265\pm{0.018}$ | 0 | 0.386 | -0.865 | -0.777 | -0.109 | -0.021 | -2.504 | -2.416 | -1.734 | 1.613 | $-2.284_{-0.012}^{+0.025}$

Table 2: Calculation of $\alpha$-decay half-lives of various neutron deficient nuclei. The superscript $m$ in the different nuclei denotes the isomeric states. The experimental values of $Q_{\alpha}$ are taken from [78]. Parent |  $Q_{\alpha}$(MeV) | $\ell_{min}$ | Temp(MeV) |  $\log_{10}T_{1/2}^{\alpha}$ ($T_{1/2}$ in s)
---|---|---|---|---
|  |  |  | Bass73 | CW76 | Denisov | Guo2013 | gp77 | Expt.
|  |  |  | T-IND | T-DEP | T-IND | T-DEP | T-IND | T-DEP | T-IND | T-DEP | T-IND | T-DEP |
${}^{109}\textrm{I}$ | $3.918\pm{0.021}$ | 2 | 0.839 | -1.655 | -0.836 | -1.459 | -0.652 | -2.327 | -1.516 | -2.696 | -1.736 | -0.498 | 0.339 | $-0.179_{-0.004}^{+0.004}$
${}^{112}\textrm{Cs}$ | $3.934\pm{0.059}$ | 0 | 0.829 | -0.922 | -0.098 | -0.744 | 0.070 | -1.629 | -0.812 | -1.979 | -1.014 | 0.227 | 1.075 | $-0.725_{-0.027}^{+0.027}$
${}^{157}\textrm{Ta}$ | $6.355\pm{0.006}$ | 0 | 0.877 | -3.087 | -2.48 | -3.257 | -2.661 | -4.409 | -3.810 | -4.514 | -3.743 | -2.067 | -1.424 | $-1.981_{-0.017}^{+0.017}$
${}^{160}\textrm{Re}$ | $6.698\pm{0.004}$ | 2 | 0.891 | -3.141 | -2.542 | -3.385 | -2.800 | -4.568 | -3.979 | -4.655 | -3.888 | -2.105 | -1.470 | $-2.040_{-0.048}^{+0.079}$
${}^{161}\textrm{Re}$ | $6.328\pm{0.007}$ | 0 | 0.864 | -2.101 | -1.489 | -2.335 | -1.736 | -3.517 | -2.914 | -3.598 | -2.823 | -1.084 | -0.437 | $-1.503_{-0.010}^{+0.010}$
${}^{161}\textrm{Re}^{m}$ | $6.452\pm{0.007}$ | 0 | 0.872 | -2.573 | -1.967 | -2.796 | -2.203 | -3.977 | -3.380 | -4.058 | -3.289 | -1.559 | -0.919 | $-1.801_{-0.009}^{+0.009}$
${}^{165}\textrm{Ir}^{m}$ | $7.003\pm{0.070}$ | 0 | 0.896 | -3.691 | -3.11 | -3.925 | -3.358 | -5.130 | -4.560 | -5.190 | -4.441 | -2.694 | -2.078 | $-2.637_{-0.087}^{+0.087}$
${}^{166}\textrm{Ir}$ | $6.722\pm{0.006}$ | 0 | 0.876 | -2.729 | -2.139 | -2.988 | -2.411 | -4.199 | -3.619 | -4.252 | -3.497 | -1.728 | -1.103 | $-1.947_{-0.091}^{+0.091}$
${}^{166}\textrm{Ir}^{m}$ | $6.894\pm{0.008}$ | 0 | 0.887 | -3.340 | -2.758 | -3.583 | -3.015 | -4.792 | -4.220 | -4.846 | -4.098 | -2.343 | -1.726 | $-1.813_{-0.026}^{+0.026}$
${}^{167}\textrm{Ir}$ | $6.505\pm{0.003}$ | 0 | 0.860 | -1.948 | -1.350 | -2.226 | -1.642 | -3.442 | -2.854 | -3.489 | -2.728 | -0.943 | -0.311 | $-1.135_{-0.025}^{+0.025}$
${}^{167}\textrm{Ir}^{m}$ | $6.680\pm{0.003}$ | 0 | 0.871 | -2.601 | -2.012 | -2.863 | -2.288 | -4.077 | -3.498 | -4.125 | -3.371 | -1.600 | -0.977 | $-1.426_{-0.009}^{+0.009}$
${}^{170}\textrm{Au}$ | $7.177\pm{0.015}$ | 0 | 0.893 | -3.484 | -2.913 | -3.763 | -3.206 | -5.000 | -4.438 | -5.030 | -4.290 | -2.497 | -1.891 | $-2.579_{-0.060}^{+0.075}$
${}^{170}\textrm{Au}^{m}$ | $7.462\pm{0.020}$ | 0 | 0.910 | -4.415 | -3.859 | -4.667 | -4.122 | -5.900 | -5.352 | -5.933 | -5.204 | -3.434 | -2.841 | $-2.831_{-0.028}^{+0.035}$
${}^{171}\textrm{Au}^{m}$ | $7.344\pm{0.017}$ | 0 | 0.901 | -4.060 | -3.500 | -4.322 | -3.776 | -5.560 | -5.010 | -5.586 | -4.857 | -3.079 | -2.484 | $-2.761_{-0.013}^{+0.013}$
${}^{177}\textrm{Tl}$ | $7.067\pm{0.007}$ | 0 | 0.869 | -2.355 | -1.787 | -2.703 | -2.149 | -3.978 | -3.421 | -3.969 | -3.234 | -1.373 | -0.771 | $-1.608_{-0.121}^{+0.121}$
${}^{177}\textrm{Tl}^{m}$ | $7.874\pm{0.024}$ | 0 | 0.916 | -4.980 | -4.448 | -5.253 | -4.733 | -6.519 | -6.000 | -6.515 | -5.809 | -4.019 | -3.451 | $-3.328_{-0.076}^{+0.076}$
${}^{185}\textrm{Bi}$ | $8.138\pm{0.081}$ | 0 | 0.910 | -5.101 | -4.586 | -5.406 | -4.907 | -6.709 | -6.206 | -6.662 | -5.977 | -4.154 | -3.607 | $-3.327_{-0.030}^{+0.030}$

(a) (b)

Figure 1:  The distributions of total emitted proton-core interaction
potential $V(r)$ including the four versions of proximity potentials in case
of proton emission and five versions of proximity potentials in case of
$\alpha$ decay for the decay of ${}^{109}\textrm{I}$.

(a) (b)

Figure 2:  (a)Deviation of the calculated proton emission half-lives,
$T_{1/2}^{\textrm{Bass73}}$, $T_{1/2}^{\textrm{CW76}}$,
$T_{1/2}^{\textrm{Guo2013}}$, and $T_{1/2}^{\textrm{Denisov}}$ with the
corresponding experimental data for the neutron-deficient proton emitters
listed in Tables LABEL:tab:table1 and LABEL:tab:table2 (b) Deviation of the
calculated $\alpha$ emission half-lives, with the corresponding experimental
data for the neutron-deficient $\alpha$ emitters listed in Tables
LABEL:tab:table1 and LABEL:tab:table2.

(a) (b)

Figure 3:  A universal curve based on the WKB penetration probability, $P$,
using the Guo2013 potential for proton emission and gp77 for $\alpha$
emission. The logarithm of the experimental half-life times are plotted versus
the $-\log_{10}P$ (a) for proton decay (b) for $\alpha$-decay listed in Tables
LABEL:tab:table1 and LABEL:tab:table2. The fitted solid line for each case are
shown in the corresponding figure. Figure 4:  UDL plots for the proton decay
half-lives illustrating the variation of the quantity $\log_{10}T_{1/2}^{p}-e$
versus $\chi^{\prime}$, where $e$ is given by $e=b\hskip 2.84544ptp+d\hskip
2.84544ptl(l+1)/p+c$. Dots are the calculated values of the proton decay half-
lives using the Denisov potential.