α-decay Half-Lives for Pb Isotopes Within Gamow-Like Model

Transcription

1 Raiation Science an Technology 7; 3(4): oi:.648/j.rst.734. α-decay Half-Lives for Pb Isotopes Within Gamow-Like Moel Dashty T. Akrawy, Akre Coputer Institute, Ministry of Eucation, Akre, Iraq Becquereal Institute for Raiation Research an Measurements, Erbil, Iraq aress: To cite this article: Dashty T. Akrawy. α-decay Half-Lives for Pb Isotopes Within Gamow-Like Moel. Raiation Science an Technology. Vol. 3, No. 4, 7, pp oi:.648/j.rst.734. Receive: May, 7; Accepte: August 3, 7; Publishe: September 6, 7 Abstract: We stue alpha ecay half-lives of Pb isotopes in the range 78 A, whithin Gamow-like moel (GLM) which is base on Gamow theory. The empirical formulae like Royer formula, Universal Decay Low (UDL), Viola-Seaborg formula (VSS), Semi-empirical formula for Poenaru (SemFIS) an Denisov & Khuenko formula (DEKH) are use to calculate alpha ecay half-lives. The results are compare with experimental ata, an also with other theoretical moels, the results are in a goo agreement was achieve with experimental ata. Keywors: α-decay, Gamow-Like Moel, Semi-empirical Formula. Introuction The stuy of alpha ecay is one of the important moes to escribe nuclear structure [], it can be successfuly escribe by qunatum theory []. The alpha ecay was escribe in 98 [3] in base of quantum tunneling. A simple formula for α-ecay half-lives was erive using Wentzel Kramers Brillouin (WKB) approximation theory by A. Zeb etal [4] 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. There are more moel that was use to etermine alpha ecay half-lives such as generalize the liqui rop moel (GLDM) which are use by Royer to escribe α-ecay half-lives [5]. By consiering Yukawa proximity potential (YPP) from groun-state to groun-state, Javaimanesh et.al was use to calculate α-ecay half-lives [6]. Preforme cluster moel (PCM) was use by Kumar, to evaluate alpha ecay half-lives for superheavy nuclei [7]. Double-foling with the stanar M3Y nucleon-nucleon interaction (M3YDDIM) was use by Ren to calculate alpha ecay half-lives [8]. New empirical formula was use to calculate alpha ecay half-lives by Akrawy, which is moifie the Royer formula [9]. Akrawy use (GLM) to calculate α-ecay half-lives for Lv isotopes []. In this stuy we calculate the alpha ecay ecay half-lives for Pb isotopes using Gamow-Like Moel (GLM), we compare the results with CPPM theoretical moel [] an with experimental ata. Also we use some empirical formula to calculate alpha ecay half-lives an to compare with GLD, like Royer formula [], Universal Decay Low (UDL) [3], Viola-Seaborg formula (VSS) [4], Semi-empirical formula for Poenaru et. Al. (SemFIS) [5], an finally with Denisov & Khuenko formula (DEKH) [6].. Material Methos A. Zeb et. al [4], evelope an use a simple moel to calculate alpha ecay half-lives, this formula baise on Gamow theory. this moel formula is contain Coulomb potential ajuste parameter. The penetration probability of tunneling alpha particle through barrier potential are calculate within Wentzel- Kramers-Brillouin (WKB) approximation using the following integral [4, 7] b P = exp µ ( V ( r) ) r ħ R where µ is reuce mass, is energy relase. a an b are tunniling point of integral which are given as ()

2 Raiation Science an Technology 7; 3(4): /3 /3 R = r A + A () Where R is the spherical square well raius, A an A are atomic number of parent an aughter nuclei. r value has been taken from Ref. [5], An ZZe b = (3) where e is the electron charge [8], Z an Z are the atomic number of parent an aughter nuclei respectively. The potential energy V(r) is given by the following conitions V ( r) V r R = ZZe (4) r r> R Where Vo is the epth of the potential well which is equal to V = 5A [9]. The α-ecay half-life can be evaluate by using [] T / ln = (4) vp Where v is the assault frequency which is equal to s- [].. Royer empirical formula (Royer) Royer preicte new formula for alpha ecay half-lives [, -3] by fitting 373 nuclei which have transition from groun-state to groun state. /6 cz log [ T/ ( s)] = a ba Z + (5) where a, b an c are coefficinet are etermine by least squre fitting metho, A an Z are mass number of an atomic number of parent nuclei respectively, an represente energy release uring the reaction. Universal ecay law formula (UDL) i et al, was preicte the universal ecay law for cluster ecay from micrscopic mechanism process of the raioactive ecay [3], the relation is foun to be as [4,5] /3 /3 A T s = az Z + b AZ Z A + A + c (6) log [ /] c c c c where A = A A c ( A + A ) c = aχ + bρ + c (7), an the a, 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 + + hlog (8) 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 hlog 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 [4]. 3. 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 [5] where an log [ T ( s)] =.4349K χ.446 (9) / / A Ks =.5956Z arccos r r( r) A α /3 ( A ) s () r =.43 α () Z an the numerical coefficient χ, close to the unity, is a secon orer polynomial χ = B + B y + B z + B y + B yz + B z () The reuce variable y an z, expresse the istance from the closest magic-plus-one neutron an proton numbers Ni an Zi is given as ( N Ni) z =, Ni < N N ( N N ) i ( Z Zi ) y =, Zi < Z Z ( Z Z ) i (3) (4) with Ni =, 5, 83, 7, 85, 9,, an Zi =, 9, 5, 83, 7,.; Bi have been obtaine by using a high quality selecte ata set of α-ecay. 4. Denisov & Khuenko formula (DEKH) From the Guy Royer empirical formula [3], Denisov & Khuenko [6, 3] 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:

3 38 Dashty T. Akrawy: α-decay Half-Lives for Pb Isotopes Within Gamow-Like Moel ( + ) /6 ba Z cz l l log [ T ( s)] = a + + e ( ) / /6 µ α A l (5) ecay half-lives within GLM, which are compare with experimental ata [6, 33] are given in Table. Here A an Z are the mass number an atomic number of parent nucleus, respectively, lis the orbital moment of /6 emitte α particle, an A / ( A 4) µ =. is the reaction energy value. The α-particle emission from nuclei obeys the spin-parity selection rule [3,3] j for even j an π p = π j + for o j an π p = π l min = (6) j for o j an π p π j + for even j an π p π where j = j j, j, π, j, π are spin an parity values p p p of the parent an aughter nuclei, respectively, anl = l min, the a, b, c,, an e are free parameters have been evaluate by Denisov et al.[3]. 3. Results an Discussion The theoretical α-ecay half-lives of Pb isotopes, which have atomic number Z = 8 within the range A = 78- have been stuie by using Gamow-Like Moel. The - value was taken in Ref. [33] which is evaluate experimentaly are liste in Table. The evaluation of α Neutron Number Figure. The comparison of the evaluate α-ecay half-lives of Pb isotopes. 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 Semi-empirical formula for Poenaru et. Al. (SemFIS)], an finally analytical Denisov & Khuenko formula (DEKH) have also been evaluate an presente in Table The maximum value of logarithm (T/) half-life is for the ecay of the parent nuclei 7Pb, which have N=5, the maximum value of logarithm (T/) half-life which is reference the behavior nearly oubly magic number, this observation inicate the role of neutron shell close in α-ecay. Table. The comparison of the evaluate α-ecay half-lives of Pb isotopes with experimental ata an with other theoretical formula in the range 78 A, Z = 8. A (MeV) GLM Exp. CPPM Royer UDL SemFIS VSS DEKH Log (T / ) (s) GLM CPPM Royer UDL VS SemFIS DEKH

4 Raiation Science an Technology 7; 3(4): Table. Continues. A (MeV) GLM Exp. CPPM Royer UDL SemFIS VSS DEKH Log (T / ) (s) value (MeV) Figure. Correlation between -value vs. logarithm (T/) half-lives of Pb isotopes. From Table, 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 experimental ata an Coulomb proximity potential moel. The correlation between -value an logarithm (T/) α- ecay half-lives was shown in Figure, it show the logarithm (T/) α-ecay half-lives exponential ecrease with increasing -value 4. Conclusion The α-ecay half-lives for Pb isotopes within GLM have been performe. The comparison between GLM an experimental ata have been one, also the our result was compare with CPPM an some empirical formulae such as Royer, UDL, VSS, SemFIS an DEKH, the result is match close to each other. The -value have the main role, which is ecrease exponential with increasing logarithm (T/) halflife. From the result we preict by GLM, the maximum value of logarithm (T/) half-life of Pb isotopes is foune in N = 5, which is influence of the neutron shell close. References [] Shin, E., Lim, Y., Hyun, C. H., & Oh, Y. (6). Nuclear isospin asymmetry in α ecay of heavy nuclei. Physical Review C, 94(), 43. [] Hassanabai, H., Javaimanesh, E., Zarrinkamar, S., & Rahimov, H. (3). An angle-epenent potential an alpha-ecay halflives of eforme nuclei for 67 Z 9. Chinese physics C, 37(4), 44. [3] Gamow, G. (98). Zur quantentheorie es atomkernes. Zeitschrift für Physik, 5(3-4), 4-. [4] Zeb, A., Wara, M., & Pomorski, K. (3). Half-lives for α an cluster raioactivity within a Gamow-like moel. Physical Review C, 87(), 438. [5] Royer, G. (). Alpha emission an spontaneous fission through quasi-molecular shapes. Journal of Physics G: Nuclear an Particle Physics, 6(8), 49.

5 4 Dashty T. Akrawy: α-decay Half-Lives for Pb Isotopes Within Gamow-Like Moel [6] Javaimanesh, E., Hassanabai, H., Rajabi, A. A., Rahimov, H., & Zarrinkamar, S. (). Alpha Decay Half-Lives of Some Nuclei from Groun State to Groun State with Yukawa Proximity Potential. Communications in Theoretical Physics, 58(), 46. [7] Kumar, S., Thakur, S., & Kumar, R. (9). Decay stuies of alpha-ecay chains. Journal of Physics G: Nuclear an Particle Physics, 36(), 54. [8] Ren, Z., & Xu, C. (8). Alpha ecay half-lives of o-z superheavy elements Z= 5 3. In Journal of Physics: Conference Series (Vol., No., p. 4). IOP Publishing. [9] Akrawy, D. T., & Poenaru, D. N. (7). Alpha ecay calculations with a new formula. arxiv preprint arxiv: [] Akrawy, T. (7). Theoretical Stuies on the α-ecay Half- Lives of Even-Even Lv Isotopes. International Journal of Energy an Power Engineering, 6(), -5. [] Santhosh, K. P., & Priyanka, B. (4). Heavy particle raioactivity from superheavy nuclei leaing to 984 aughter nuclei. Nuclear Physics A, 99, -37. [] Royer, G., & Zhang, H. F. (8). Recent α ecay half-lives an analytic expression preictions incluing superheavy nuclei. Physical Review C, 77(3), 376. [3] i, C., Xu, F. R., Liotta, R. J., & Wyss, R. (9). Universal ecay law in charge-particle. [4] Dong, T., & Ren, Z. (5). New calculations of α-ecay halflives by the Viola-Seaborg formula. The European Physical Journal A-Harons an Nuclei, 6(), [5] Poenaru, D. N., Plonski, I. H., & Greiner, W. (6). α-ecay half-lives of superheavy nuclei. Physical Review C, 74(), 43. [6] Denisov, V. Y., & Khuenko, A. A. (9). α-ecay half-lives: Empirical relations. Physical Review C, 79(5), [7] Krappe, H. J., & Pomorski, K. (). Theory of Nuclear Fission: A Textbook (Vol. 838). Springer Science & Business Meia. [8] [8] Poenaru, D. N., Gherghescu, R. A., & Greiner, W. (). Single universal curve for cluster raioactivities an α ecay. Physical Review C, 83(), 46. [9] Blenowske, R., & Walliser, H. (988). Systematics of cluster-raioactivity-ecay constants as suggeste by microscopic calculations. Physical review letters, 6(7), 93. [] Hassanabai, H., Javaimanesh, E., & Zarrinkamar, S. (3). A new barrier potential an alpha-ecay half-lives of even even nuclei in the 8 Z 9 regime. Nuclear Physics A, 96, [] Royer, G., & Moustabchir, R. (). Light nucleus emission within a generalize liqui-rop moel an quasimolecular shapes. Nuclear Physics A, 683(), 8-6. [] Royer, G. (). Analytic expressions for alpha-ecay halflives an potential barriers. Nuclear Physics A, 848(3), [3] Santhosh, K. P., & Priyanka, B. (4). Preictions for the α- ecay chains of Z= superheavy nuclei in the range 7 A 39. Physical Review C, 9(5), [4] [4] i, C., Xu, F. R., Liotta, R. J., Wyss, R., Zhang, M. Y., Asawatangtrakulee, C., & Hu, D. (9). Microscopic mechanism of charge-particle raioactivity an generalization of the Geiger-Nuttall law. Physical Review C, 8(4), [5] Bao, X. J., Zhang, H. F., Dong, J. M., Li, J.., & Zhang, H. F. (4). Competition between α ecay an cluster raioactivity for superheavy nuclei with a universal ecay-law formula. Physical Review C, 89(6), 673. [6] Santhosh, K. P., Sukumaran, I., & Priyanka, B. (5). Theoretical stuies on the alpha ecay of 78 Pb isotopes. Nuclear Physics A, 935, 8-4. [7] Sobiczewski, A., Patyk, Z., & Ćwiok, S. (989). Deforme superheavy nuclei. Physics Letters B, 4(-), -4. [8] Santhosh, K. P., & Priyanka, B. (4). Preictions for the α- ecay chains of Z= superheavy nuclei in the range 7 A 39. Physical Review C, 9(5), [9] Viola, V. E., & Seaborg, G. T. (966). Nuclear systematics of the heavy elements II Lifetimes for alpha, beta an spontaneous fission ecay. Journal of Inorganic an Nuclear Chemistry, 8(3), [3] Denisov, V. Y., & Khuenko, A. A. (). Erratum: α-ecay half-lives: Empirical relations [Phys. Rev. C 79, 5464 (9)]. Physical Review C, 8(5), 599. [3] Denisov, V. Y., & Khuenko, A. A. (9). α-decay halflives, α-capture, an α-nucleus potential. Atomic Data an Nuclear Data Tables, 95(6), [3] Denisov, V. Y., Daviovskaya, O. I., & Seykh, I. Y. (5). Improve parametrization of the unifie moel for α-ecay an α capture. Physical Review C, 9(), 46. [33] Aui, G., Konev, F. G., Wang, M., Pfeiffer, B., Sun, X., Blachot, J., & MacCormick, M. (). The NUBASE evaluation of nuclear properties. Chinese Physics C, 36(), 57.