Theoretical Studies on the α-decay Half-Lives of Even-Even

The α-decay half-lives for even-even Lv isotopes within the range 290 ≤ A ≥ 314 which have Z = 116, have been evaluated using Gamow-like model (GLM) which is based on Gamow theory, the different semi-empirical formula for α-decay half-lives also was used. The half-lives were evaluated using theoretical Q-value. The computed α-decay half-lives were compared with CPPM, and also with empirical formula, the result was achieved in a good agreement.


Introduction
The α-decay half-lives is one of the most important decay modes of nuclei which was described in 1928 [1] in base of quantum tunneling, who assumed the α-particle was due to the quantum tunneling phenomenon of a charged α-particle through the Coulomb barrier. Violet and Seaborg was predictive a simple formula which is based on Gamow model [2]. Poenaru et. al was predicted theoretical model of cluster radioactivity [3]. The cluster radioactivity phenomenon may be described in a similar way as the alpha decay using Gamow-like theories [2,4] because of the large mass difference of cluster and daughter nucleons. A simple formula for α-decay half-lives was derived using Wentzel-Kramers-Brillouin (WKB) approximation theory by A. Zdeb et al [2] for penetration of the Coulomb barrier with a square well for the nuclear part. GLM depended only one adjustable parameter, the radius constant, which is possible to reproduce with a good accuracy all existing data for decays of eveneven nuclei. To determine aα-decay half-lives there are more than model that was used as shown in the literature such as Royer, he generalized the liquid drop model to describe αdecay half-lives [5], and that formula was used to evaluate the α-decay half-lives for heavy and superheavy nuclei [6]. Ibrahim et. al determined the α-decay preformation probability of superheavy nuclei within the binary cluster model by using an interaction generated from amalgamation of a phenomenological and microscopic interactions [7]. Yao et. al was used different proximity potentials to evaluate αdecay half-lives, which was include Coulomb proximity potential model and Coulomb proximity potential model for different deformation nuclei [8]. Sun et. al studied α-decay half-live for even-even nuclei by using two potential approaches with a quasistationary state approximation [9]. Javadimanesh et. al also studied α-decay half-lives by considering a potential model with Yukawa proximity potential from ground state to ground state [10]. The aim of this work is to determine the α-decay half-lives of even-even Lv isotopes within Gamow-Like Model (GLM) and to compared with another theoretical model such as CPPM [11], Royer formula [12], Universal Decay Low (UDL) [13], Viola-Seaborg formula (VSS) [14], Semi-empirical formula for Poenaru et. Al. (SemFIS) [15], and finally with Denisov & Khudenko formula (DEKH) [16].
The probability of tunneling α-particle through the potential barrier was given within (WKB) approximation theory by the following integral [2,17] ( ) Where Q, is kinetic energy of the emitted particle, µ is the reduced mass. The spherical square well radius R, which is equal to sum of the radii of parent and daughter nuclei, 1 A and 2 A are atomic number of parent and daughter nuclei. 0 r has been taken from Ref. [5] The exit point from the barrier b corresponding to point, where the Coulomb potential is equal to the kinetic energy Q: where Z 1 and Z 2 are the atomic number of parent and daughter nuclei respectively, e 2 is the electron charge given by e 2 = 1.43998 MeV.fm [18]. The Q value was taken from theoretical value [11]. The potential energy V(r) is given by (4) Where the depth of the potential well V o is one of the model parameters which is equal to 0 A [19]. The α-decay half-life can be calculated is given by the following [20] 1/ 2 the decay constant is simply defined as = vP λ , v is the assault frequency which is equal to 10 20 s -1 [21], the penetration probability P was calculated by using integral of Eq. (1).

Empirical Formulas for α-decay Half-Lives
Some empirical formulas of α-decay half-lives for Pb isotopes were evaluated in this study for comparison of our calculation with GLM model by using the following empirical formulae.

Royer Empirical Formula (Royer)
The α-decay half-lives have been studied by using Royer formula [12,[22][23], Royer developed formula by fitting procedure using 373 nuclei transition from ground state to ground state which is given as where A, Z and Q represented mass number, atomic number and energy released during the reaction. The Eq.(6) can be assumed for set even-even nuclei, where the coefficients a, b, and c is obtained by least square fitting methods which is -25.31, -1.169 and 1.5864 respectively for even-even nuclei.

Universal Decay Law Formula (UDL)
From the microscopic mechanism of the radioactive decay charge particle emission, the universal decay law for alpha decay and cluster was predicted by Qi et al [13], It connected the half-lives of monopole radioactive decays with the kinetic energy (Q) values of the outgoing particles as well as the atomic number and mass number of the nuclei implicated in the decay [24,25]. The relation is found to be as ( ) , and the constant a = 0.3949, b = -0.3693 and c = -23.7615 are constants which are used was determined by fitting to the experimental data of α-decay and cluster [26]. ′ + b c ρ is the effects term that include the cluster in the parent nucleus.

The Viola-Seaborg Semi-empirical Formula (VSS)
From the Geiger-Nuttall law, the VSS formula was determined by Sobiczewski et al. [27], is given as ( ) 10 1/ 2 log log [ ( )] + = + + + aZ b T s cZ d h Q (9) Here Z is the parent atomic number, Q is the energy released during the reaction in unit MeV, half-life in seconds, the coefficients a, b, c, and d are adaptable parameters and the parameter h log evidence the hindrance factor related with odd proton and odd neutron numbers [28], as given by Viola et al. [29]. More recent value was determined by Tiekuang Dong et. Al [14], new data for even-even nuclei is taken into, have been used to determine the constant which is given as a = 1.64062, b = -8.54399, c = -0.19430, d = -33.9054, and h log = 0 for even even nuclei.

Semi-empirical Formula for Poenaru et al. (SemFIS)
Poenaru et al. modified semi-empirical formula for αdecay half-life which is based on fission theory for α-decay yield which is given as [15,30]   The reduced variable y and z, expressed the distance from the closest magic-plus-one neutron and proton numbers N i and Z i is given as (15) with

Denisov & Khudenko Formula (DEKH)
From the Guy Royer empirical formula [23], Denisov & Khudenko [16,31] developed empirical formula for α-decay half-life between ground state to ground state α-transition of parent and daughter nuclei, the formula is evaluated of αdecay half-lives for even-even, even-odd, odd-even and oddodd nuclei: Here A and Z are the mass number and atomic number of parent nucleus, respectively, ℓ is the orbital moment of emitted α particle, and . Q is the reaction energy value. The α-particle emission from nuclei obeys the spin-parity selection rule [32,33] min for even and 1 for odd and for odd and 1 for even and

Results and Discussions
The theoretical α-decay half-lives of even-even Lv isotopes, which have atomic number Z = 118 within the range A = 178-220 have been studied by using Gamow-Like Model. The Q-value was taken in Ref. [11] which is evaluated theoretically are listed in Table 1. The evaluation of α-decay half-lives within GLM are given in Table 1.
The semi-empirical formulae was also been done using the analytical Royer formula (Royer), the universal decay law (UDL), the Viola-Seaborg formula (VSS), the Semiempirical formula for Poenaru et. Al. (SemFIS), and finally analytical Denisov & Khudenko formula (DEKH) have also been evaluated and presented in Table 1. The comparison between GLM, CPPM, and some empirical formulae such as Royer, UDL, VSS, SemFIS and DEKH have been done and presented in Table 1, it can be found that the results that the α-decay half-lives evaluated using GLM are in good agreement with other theoretical model and empirical formulae.
The plot for logarithm (T 1/2 ) versus the neutron number of the parent nuclei of the α-emissions from even-even Lv isotopes are shown in Figure 1.
From Figure 1 it's noted that, the maximum value of logarithm (T 1/2 ) half-life is for the decay of the parent nuclei 314 Lv.  The minimum value of logarithm (T 1/2 ) half-life is for the decay of the parent nuclei 302 Lv. The minimum value of logarithm (T 1/2 ) half-life which is reference the behavior magic number N = 184, this observation indicate the role of neutron shell in α-decay.
From the Figure 1, it can be seen clearly the GLM, CPPM, Royer, UDL, VSS, SemFIS and DEKH are the same orientation. It should be taken into account that GLM values match close to Royer formula up to N = 184 rather than other model and formulae but after through this magic number the GLM is match close to SemFIS and Royer formulae.
The correlation between Q-value and logarithm (T 1/2 ) αdecay half-lives was shown in Figure 2, it show the logarithm (T 1/2 ) α-decay half-lives linearly decrease with increasing Qvalue with 0.993 square correlation coefficient (R 2 ).

Conclusion
The α-decay half-lives for even-even Lv isotopes within GLM have been performed. The comparison between GLM, CPPM and some empirical formulae such as Royer, UDL, VSS, SemFIS and DEKH have been done, the result is match close to each other. From the result we predict by GLM, the minimum value of logarithm (T 1/2 ) half-life of even-even isotopes is founded in N = 184, which is influence of the neutron magic number. Also we found the GLM is match close to Royer formula up to influence magic number, after this influence the GLM is also be so close to SemFIS formula.