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

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1 International Journal of Energy an Power Engineering 17; 6(1: oi: /j.ijepe ISSN: X (Print; ISSN: 36-96X (Online Theoretical Stuies on the α-ecay Half-Lives of Even-Even Lv Isotopes Dashty T. Akrawy 1, 1 Akre Computer Institute, Ministry of Eucation, Aqrah, Kuristan, Iraq Becquereal Institute for Raiation Research an Measurements, Erbil, Kuristan, Iraq aress: akrawy85@gmail.com To cite this article: Dashty T. Akrawy. Theoretical Stuies on the α-ecay Half-Lives of Even-Even Lv Isotopes. International Journal of Energy an Power Engineering. Vol. 6, No. 1, 17, pp oi: /j.ijepe Receive: February 19, 17; Accepte: March 13, 17; Publishe: March 3, 17 Abstract: The α-ecay half-lives for even-even Lv isotopes within the range 9 A 314 which have Z = 116, have been evaluate using Gamow-like moel (GLM which is base on Gamow theory, the ifferent semi-empirical formula for α-ecay half-lives also was use. The half-lives were evaluate using theoretical -value. The compute α-ecay half-lives were compare with CPPM, an also with empirical formula, the result was achieve in a goo agreement. Keywors: α-ecay; Gamow-Like Moel; Semi-empirical Formula 1. Introuction The α-ecay half-lives is one of the most important ecay moes of nuclei which was escribe in 198 [1] in base of quantum tunneling, who assume the α-particle was ue to the quantum tunneling phenomenon of a charge α-particle through the Coulomb barrier. Violet an Seaborg was preictive a simple formula which is base on Gamow moel []. Poenaru et. al was preicte theoretical moel of cluster raioactivity [3]. The cluster raioactivity phenomenon may be escribe in a similar way as the alpha ecay using Gamow-like theories [, 4] because of the large mass ifference of cluster an aughter nucleons. A simple formula for α-ecay half-lives was erive using Wentzel Kramers Brillouin (WKB approximation theory by A. Zeb et al [] for penetration of the Coulomb barrier with a square well for the nuclear part. GLM epene only one ajustable parameter, the raius constant, which is possible to reprouce with a goo accuracy all existing ata for ecays of eveneven nuclei. To etermine aα-ecay half-lives there are more than moel that was use as shown in the literature such as Royer, he generalize the liqui rop moel to escribe α- ecay half-lives [5], an that formula was use to evaluate the α-ecay half-lives for heavy an superheavy nuclei [6]. Ibrahim et. al etermine the α-ecay preformation probability of superheavy nuclei within the binary cluster moel by using an interaction generate from amalgamation of a phenomenological an microscopic interactions [7]. Yao et. al was use ifferent proximity potentials to evaluate α- ecay half-lives, which was inclue Coulomb proximity potential moel an Coulomb proximity potential moel for ifferent eformation nuclei [8]. Sun et. al stuie α-ecay half-live for even-even nuclei by using two potential approaches with a quasistationary state approximation [9]. Javaimanesh et. al also stuie α-ecay half-lives by consiering a potential moel with Yukawa proximity potential from groun state to groun state [1]. The aim of this work is to etermine the α-ecay half-lives of even-even Lv isotopes within Gamow-Like Moel (GLM an to compare with another theoretical moel such as CPPM [11], Royer formula [1], Universal Decay Low (UDL [13], Viola-Seaborg formula (VSS [14], Semi-empirical formula for Poenaru et. Al. (SemFIS [15], an finally with Denisov & Khuenko formula (DEKH [16].. Gamow-Like Moel for α-ecay Half-Lives A simple phenomenological moel for evaluation of α- ecay half-lives which is base on the Gamow theory was evelope by A. Zeb et. al [], this moel is contain one ajuste parameter which is Coulomb potential.

2 Dashty T. Akrawy: Theoretical Stuies on the α-ecay Half-Lives of Even-Even Lv Isotopes The probability of tunneling α-particle through the potential barrier was given within (WKB approximation theory by the following integral [, 17] b P = exp ( ( ħ µ V r r (1 R Where, is kinetic energy of the emitte particle, µ is the reuce mass. The spherical square well raius R, which is equal to sum of the raii of parent an aughter nuclei, 1/3 1/3 ( R = r A + A ( 1 A 1 an A are atomic number of parent an aughter nuclei. r has been taken from Ref. [5] The exit point from the barrier b corresponing to point, where the Coulomb potential is equal to the kinetic energy : 1 b = ZZ e where Z 1 an Z are the atomic number of parent an aughter nuclei respectively, e is the electron charge given by e = MeV.fm [18]. The value was taken from theoretical value [11]. The potential energy V(r is given by V r R = 1 V( r ZZ e r r> R Where the epth of the potential well V o is one of the moel parameters which is equal to V =5A 1 [19]. The α-ecay half-life can be calculate is given by the following [] T 1/ ln = λ the ecay constant is simply efine as λ = vp, v is the assault frequency which is equal to 1 s -1 [1], the penetration probability P was calculate by using integral of Eq. (1. 3. Empirical Formulas for α-ecay Half-Lives Some empirical formulas of α-ecay half-lives for Pb isotopes were evaluate in this stuy for comparison of our calculation with GLM moel by using the following empirical formulae Royer Empirical Formula (Royer The α-ecay half-lives have been stuie by using Royer formula [1, -3], Royer evelope formula by fitting proceure using 373 nuclei transition from groun state to groun state which is given as (3 (4 (5 1/6 log 1[ T1/( s] = a ba Z + cz where A, Z an represente mass number, atomic number an energy release uring the reaction. The Eq.(6 can be assume for set even-even nuclei, where the coefficients a, b, an c is obtaine by least square fitting methos which is , an respectively for even-even nuclei. 3.. Universal Decay Law Formula (UDL From the microscopic mechanism of the raioactive ecay charge particle emission, the universal ecay law for alpha ecay an cluster was preicte by i et al [13], It connecte the half-lives of monopole raioactive ecays with the kinetic energy ( values of the outgoing particles as well as the atomic number an mass number of the nuclei implicate in the ecay [4, 5]. The relation is foun to be as A 1/3 1/3 log 1[ T1/( s] = azcz + b AZcZ ( A + Ac + c where A = c (6 (7 = aχ + bρ + c (8 A A c ( A + A c, an the constant a =.3949, b = an c = are constants which are use was etermine by fitting to the experimental ata of α-ecay an cluster [6]. bρ + c is the effects term that inclue the cluster in the parent nucleus The Viola-Seaborg Semi-empirical Formula (VSS From the Geiger-Nuttall law, the VSS formula was etermine by Sobiczewski et al. [7], is given as ( az + b log [ T ( s] = + cz + + h 1 1/ log Here Z is the parent atomic number, is the energy release uring the reaction in unit MeV, half-life in secons, the coefficients a, b, c, an are aaptable parameters an the parameter h log evience the hinrance factor relate with o proton an o neutron numbers [8], as given by Viola et al. [9]. More recent value was etermine by Tiekuang Dong et. Al [14], new ata for even-even nuclei is taken into, have been use to etermine the constant which is given as a = 1.646, b = , c = , = , an h log = for even even nuclei Semi-empirical Formula for Poenaru et al. (SemFIS Poenaru et al. moifie semi-empirical formula for α- ecay half-life which is base on fission theory for α-ecay yiel which is given as [15, 3] 1 1/ (9 log [ T ( s] =.4349K χ.446 (1 s

3 International Journal of Energy an Power Engineering 17; 6(1: where an 1/ A K =.5956 arccos (1 s Z r r r (11 A α 1/3 ( A r =.43 (1 α Z an the numerical coefficient χ, close to the unity, is a secon orer polynomial χ = B + B y + Bz + B y + B yz + Bz ( The reuce variable y an z, expresse the istance from the closest magic-plus-one neutron an proton numbers N i an Z i is given as ( N Ni z =, Ni < N N ( N N i ( Z Zi y =, Zi < Z Z ( Z Z i (14 (15 with N i =, 51, 83, 17, 185, 9,, an Z i =, 9, 51, 83, 17,.; Bi have been obtaine by using a high quality selecte ata set of α-ecay which is equal to B 1 = , B = -.467, B 3 =.171, B 4 = 1.453, B 5 =.181, an B 6 = Denisov & Khuenko Formula (DEKH From the Guy Royer empirical formula [3], Denisov & Khuenko [16, 31] evelope empirical formula for α-ecay half-life between groun state to groun state α-transition of parent an aughter nuclei, the formula is evaluate of α- ecay half-lives for even-even, even-o, o-even an oo nuclei: ( + 1 l l log [ T ( s] = a + + e ( 1 1 1/6 ba Z cz 1 1/ 1/6 µ α A l ( (16 Here A an Z are the mass number an atomic number of parent nucleus, respectively, l is the orbital moment of emitte α particle, an = ( A/ ( A 4 1/6 µ. is the reaction energy value. The α-particle emission from nuclei obeys the spin-parity selection rule [3, 33] l min j for even j = π j + 1 for o j = π = j for o j π j + 1 for even j π (17 where j = j j, j, π, j, π are spin an parity values p p p of the parent an aughter nuclei, respectively, an l = l min, the a, b, c,, an e are free parameters have been evaluate by Denisov et al. [31] which are 6.171, , 1.688,, an respectively. 4. Results an Discussions The theoretical α-ecay half-lives of even-even Lv isotopes, which have atomic number Z = 118 within the range A = 178- have been stuie by using Gamow-Like Moel. The -value was taken in Ref. [11] which is evaluate theoretically are liste in Table 1. The evaluation of α-ecay half-lives within GLM are given in Table 1. The semi-empirical formulae was also been one using the analytical Royer formula (Royer, the universal ecay law (UDL, the Viola-Seaborg formula (VSS, the Semiempirical formula for Poenaru et. Al. (SemFIS, an finally analytical Denisov & Khuenko formula (DEKH have also been evaluate an presente in Table 1. Table 1. The comparison of evaluate logarithm α-ecay half-lives of even-even Lv isotopes. A Log 1 (T 1/ (s GLM CPPM Royer UDL VSS SemFIS DEKH The comparison between GLM, CPPM, an some empirical formulae such as Royer, UDL, VSS, SemFIS an DEKH have been one an presente in Table 1, it can be foun that the results that the α-ecay half-lives evaluate using GLM are in goo agreement with other theoretical moel an empirical formulae.

4 4 Dashty T. Akrawy: Theoretical Stuies on the α-ecay Half-Lives of Even-Even Lv Isotopes The plot for logarithm (T 1/ 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 note that, the maximum value of logarithm (T 1/ half-life is for the ecay of the parent nuclei 314 Lv. Log 1 (T 1/ (s Figure 1. Comparison of the evaluate α-ecay half-lives of even-even Lv isotopes. Log 1 (T 1/ (s Lv Neutron Number R =.993 GLM CPPM Royer Figure. Correlation between value versus Logarithm (T 1/ α-ecay halflives of Lv isotopes. The minimum value of logarithm (T 1/ half-life is for the ecay of the parent nuclei 3 Lv. The minimum value of logarithm (T 1/ half-life which is reference the behavior magic number N = 184, this observation inicate the role of neutron shell in α-ecay. From the Figure 1, it can be seen clearly the GLM, CPPM, Royer, UDL, VSS, SemFIS an DEKH are the same orientation. It shoul be taken into account that GLM values match close to Royer formula up to N = 184 rather than other moel an formulae but after through this magic number the GLM is match close to SemFIS an Royer formulae. The correlation between -value an logarithm (T 1/ α- ecay half-lives was shown in Figure, it show the logarithm (T 1/ α-ecay half-lives linearly ecrease with increasing - value with.993 square correlation coefficient (R. UDL VSS SemFIS DEKH value Linear fit value (MeV 5. Conclusion The α-ecay half-lives for even-even Lv isotopes within GLM have been performe. The comparison between GLM, CPPM an some empirical formulae such as Royer, UDL, VSS, SemFIS an DEKH have been one, the result is match close to each other. From the result we preict by GLM, the minimum value of logarithm (T 1/ half-life of even-even isotopes is foune in N = 184, which is influence of the neutron magic number. Also we foun 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. References [1] Gamow, G. (198. Zur quantentheorie es atomkernes. Zeitschrift für Physik, 51 (3-4, 4-1. [] Zeb, A., Wara, M., & Pomorski, K. (13. Half-lives for α an cluster raioactivity within a Gamow-like moel. Physical Review C, 87 (, 438. [3] Sanulescu, A., Poenaru, D. N., & Greiner, W. (198. New type of ecay of heavy nuclei intermeiate between fission an. cap alpha. ecay. Sov. J. Particles Nucl.(Engl. Transl.;(Unite States, 11 (6. [4] Wara, M., & Robleo, L. M. (11. Microscopic escription of cluster raioactivity in actinie nuclei. Physical Review C, 84 (4, [5] Royer, G. (. Alpha emission an spontaneous fission through quasi-molecular shapes. Journal of Physics G: Nuclear an Particle Physics, 6 (8, [6] Royer, G., & Gherghescu, R. A. (. On the formation an alpha ecay of superheavy elements. Nuclear Physics A, 699 (3, [7] Ibrahim, T. T., & Wyngaart, S. M. (14. Preformation factor an alpha ecay half-lives of superheavy nuclei. Journal of Physics G: Nuclear an Particle Physics, 41 (5, [8] Yao, Y. J., Zhang, G. L., u, W. W., & ian, J.. (15. Comparative stuies for ifferent proximity potentials applie to α ecay. The European Physical Journal A, 51 (9, 1-9. [9] Sun, X. D., Guo, P., & Li, X. H. (16. Systematic stuy of α ecay half-lives for even-even nuclei within a two-potential approach. Physical Review C, 93 (3, [1] Javaimanesh, E., Hassanabai, H., Rajabi, A. A., Rahimov, H., & Zarrinkamar, S. (1. Alpha Decay Half-Lives of Some Nuclei from Groun State to Groun State with Yukawa Proximity Potential. Communications in Theoretical Physics, 58 (1, 146. [11] Santhosh, K. P., & Priyanka, B. (14. Heavy particle raioactivity from superheavy nuclei leaing to aughter nuclei. Nuclear Physics A, 99, -37. [1] Royer, G., & Zhang, H. F. (8. Recent α ecay half-lives an analytic expression preictions incluing superheavy nuclei. Physical Review C, 77 (3, 376. [13] i, C., Xu, F. R., Liotta, R. J., & Wyss, R. (9. Universal ecay law in charge-particle.

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