Journal of Physical Science Application 5 (2015) 66-70 oi: 10.17265/2159-5348/2015.01.010 D DAVID PUBLISHING Calculating Bining Energy for O Isotopes of Beryllium (7 A 13) Fahime Mohammazae, Ali Akbar Rajabi Lale Nickhah Department of Physics, University of Shahroo, Shahroo 3619995161, Iran Abstract: In this article, we present exact solution of the Schröinger equation (for an N-ientical boy-force) for o-a isotopes of Beryllium in the presence of Yukawa potential by Nikiforov-Uvarov (NU) metho. The NU metho can be use to solve secon orer ifferential equation. By this metho, we fin the wave equation bining energy. Numerical results of bining energy are presente show that these results are in goo agreement with experimental values. Key wors: Schröinger equation, yukawa potential, NU metho, o isotopes of beryllium, bining energy, wave equation. 1. Introuction The exact solution of the problem has always playe an important role in quantum mechanics, because the important information is hien in solutions of questions [1]. The Schroinger equation is one of the funamental wave equation in physics. Its solutions for some certain potential have important applications in atomic, nuclear, conense high energy physics particle physics [2-14]. So far, several approaches have presente to solve Schröinger equation two-particle systems three-particle systems in the presence of ifferent potentials [15]. In this paper, we solve the N-particle Schröinger equation for o isotopes of Beryllium. Schroinger equation for the N-particles which interact with each other is [16, 17]: ћ 2 1 2 An 33 1 In this equation, D, N enote respectively the space imension, total angular momentum the number of particles, m is one of particle masses, is the hyper raius. If we write the equation for Beryllium 7 (7Be), Corresponing author: Lale Nickhah, Ph.D. stuent, research file: theoretical physics. E-mail: L_Nickhah@yahoo.com. with substitution 2 ћ215217 ћ 8! " 0 2 3"73 where the potential v(r) is taken as Yukawa potential in Eq. (1) [16]: $ % & '( Since the Schroinger with Yukawa potential has no solution [18], we use an approximation for the centrifugal terms as [19]: 1 )2* Or equivalently: 1 )4* The raial Schröinger equation will be as follows: 2 ћ ",2*$ % ћ " 215217 4* 8-0 3 This paper is arrange as follows. In Section 2, we solve the Schroinger equation for o isotopes of Beryllium in the presence of the Yukawa potential, then the energy spectrum the wave equation of the system are obtaine by using the NU metho. The etails of the NU metho are given in Appenix A.
Calculating Bining Energy for O Isotopes of Beryllium (7 A 13) 67 Finally, our conclusion is given in Section 3. 2. Solution of the Schröinger Equation by Nikiforov-Uvarov Metho Several methos are use in the solution of the Schroinger equation, one of them is the Nikiforov-Uvarov metho [19]. We use NU generalize metho to solve the Schröinger equation (see the Appenix A) [20]: 2 ћ ",2*$ % ћ " 215217 4* 8-0 3 Using variables the form below:. 2 ћ,0 Substituting into Eq. (3) we have: 0 0 10 0 s10 0 1 0.10 10, 4* $ % ћ * 010 2152170 4 4 Comparing Eq. (4) Eq. (A11), we can easily obtain the coefficients 2 3 4 1,2,3 analytical expression for 5 3 as follows: 2 1,2 1,2 6 1 5 An 5 7 4* $ % ћ *, 5 7 2* $ % ћ * 2152171 4 6 7 5 6 4* The values of the coefficients 2 3 4 4,5, 9 are foun from Eqs. (A16)-(A18), the specific values of the coefficients 2 3 4 4,5, 9 necessary for the energy eigenvalues eigenfunctions are isplaye as follows: 2 ; 0,2 < 1 2,2 = 1 4 7 4* $ % ћ * 2 > 7 2* $ % ћ * 2152171 4 7 2? 7 4*,2 @ 1 4 1 4 215217 By using Eq. (A22), we obtaine the energy eigenvalues of the Yukawa potential as: >AB * 2m " 1 2n1D2E 1 4 F1215217GH! " ћ * 1 4 215217 2J1 KL 1 4 F1215217GMN 8 To fin the corresponing wave function, referring to the above parametric constant Eqs. (A18) (A23), we fin the wave function: 0 >AB & (EOP ћ Q RE ; R ; SR<SR> "T U ' Q ћ Q, REOP ' Q ћ Q DE ; R ; SR<SR>HR! " 9 Repeat this proceure for the other isotopes we will have: 0 @AB & (EOP ћ Q RE ; R ; SRSR6 T ' Q ћ Q, REOP ' Q ћ Q DE ; R ; SRSR6HR! U " 10 @AB * 2 " 1 2J1D2E 1 4 1 4 221223H! ћ * 1 4 221223 2J1KL 1 4 1 4 221223MN For 11Be, we will have: 0 AB & (EOP ћ Q RE ; R ; SR>SR@ 11
68 Calculating Bining Energy for O Isotopes of Beryllium (7 A 13) T U ' Q ћ Q, REOP ' Q ћ Q DE ; R ; SR>SR@HR! AB * 2 " 1 2J1D2E 1 4 1 4 227229H! " ћ * 1 4 227229 2J1KL 1 4 1 4 227229MN 12 13 The wave function energy eigenvalue for 13Be become: 0 6AB & (EOP ' Q ћ Q RE ; R ; γr66γr6< " T U ' Q ћ Q, REOP ' Q ћ Q DE ; R ; γr66γr6<hr! 6AB * 2 " " 14 I2J1K2L 1 4 1 4 233235MN ћ * 1 4 233235 2J1KL 1 4 1 4 233235MN 3. Conclusions 15 In this paper, we solve the Schröinger equation for o isotopes Beryllium 7 A 13 by using generalize NU metho choosing Yukawa potential we obtain the wave function bining energy of these isotopes. Comparing the obtaine bining energy with the experimental results, we can see that a goo agreement has been achieve. By putting the values of n,, $ %, ћ1, consiering a = 0.1 fm -1 finally, by comparison with experimental values (Table 1), we will see that a goo agreement is obtaine [21-23]. Table 1 The comparison of the bining energy with the experimental results. O isotopes n V 0 E E (Present (Experimental calculation) value) 7Be 3 1 6 36.5979 37.600 9Be 1 1 9 58.5015 58.165 11Be 1 0 1 65.2684 65.481 13Be 2 0 5 75.9345 68.550 References [1] Maamache, M., Lahoulou, C., Saai, Y. 2009. Influence of Auxiliary Equation on Wave Functions for Time-Depenent Pauli Equation in Presence of Aharonov-Bohm Effect. Communications in Theoretical Physics 51: 803. [2] Cooper, F., Khare, A., Sukhatme, U. 1995. Super Symmetry Quantum Mechanics. Phys. Rep. 251: 267. [3] Chatterjee, A. 1990. Large-N Expansions in Quantum Mechanics, Atomic Physics some O (N) Invariant Systems. Physics Reports 186 (6): 249-370. [4] Sever, R., Tezcan, C. 1987. 1/N Expansion for a more General Screene Coulomb Potential. Phys. Rev. A 36: 1045. [5] Dong, S. H., Lozaa-Casson, M. 2005. Exact Solutions of the Schröinger Equation with the Position-Depenent Mass for a Har-Core Potential. Physics Letters A 337: 313. [6] Panja, M. M., Dutt, R., Varshni, Y. P. 1990. Shifte Large-N Expansion for a Relativistic Spin-1/2 Particle in Screene Coulomb Potentials. Phys. Rev. A 42: 106. [7] Dong, S. H., Morales, D., Garcia-Ravelo, J. 2007. Exact Quantization Rule Its Applications to Physical Potentials. Int. J. Mo. Phys. E 16: 189. [8] Hall, R. L., Saa, N. 1998. Smooth Transformations of Kratzer s Potential in N Dimensions. J. Chem. Phys. 109: 2983. [9] Setare, M. R., Karimi, E. 2007. Algebraic Approach to the Kratzer Potential. Phys. Scr. 75: 90. [10] Bagchi, B., Gorain, P., Quesne, C. 2004. A General Scheme for the Effective-Mass Schröinger Equation the Generation of the Associate Potentials. Phys. Lett. A 19: 2765.
Calculating Bining Energy for O Isotopes of Beryllium (7 A 13) 69 [11] Ikhair, S. M., Sever, R. 2007. Exact Solution of the Klein-Goron Equation for the PT-Symmetric Generalize Woos-Saxon Potential by the Nikiforov-Uvarov Metho. Ann. Phys. 16: 218. [12] Ikhair, S. M., Sever, R. 2007. Approximate Eigenvalue Eigenfunction Solutions for the Generalize Hulthen Potential with any Angular Momentum. J. Math. Chem. 41: 461. [13] Ikhair, S. M., Sever, R. 2006. A Perturbative Treatment for the Energy Levels of Neutral Atoms. Int. J. Mo. Phys. A 21: 6465. [14] Bagchi, B., Cannata, F., Quesne, C. 2000. PT-Symmetric Sextic Potentials. Phys. Lett. A 269: 79. [15] Salehi, N., Rajabi, A. A., Ghalenovi, Z. 2012. Calculation of Nonstrange Baryon Spectra Base on Symmetry Group Theory the Hypercentral Potential. Chinese Journal of Physics 50: 28. [16] Rajabi, A. A. 2005. Exact Analytical Solution of the Schröinger Equation for an N-Ientical Boy-Force System. Few-Boy Systems 37: 197. [17] Hassanabai, H., Zarrinkamar, S., Rajabi, A. A. 2011. Exact Solutions of D-Dimensional Schröinger Equation for an Energy-Depenent Potential by NU Metho. Commun. Theor. Phys. 55: 541. [18] Hamzavi, M., Movahei, M., Thylwe, K. E., Rajabi, A. A. 2012. Approximate Analytical Solution of the Yukawa Potential with Arbitrary Angular Momenta. Chin. Phys. Lett. 29 (8): 080302. [19] Nikiforov, A., Uvarov, V. B. 1988. A Unifie Introuction with Applications Special Functions of Mathematical Physics. Special Functions of Mathematical Physics 427. [20] Tezcan, C., Sever, R. 2009. A General Approach for the Exact Solution of the Schröinger Equation. Int. J. Theor. Phys. 48: 337. [21] Ikhair, S. 2012. Approximate κ-state Solutions to the Dirac-Yukawa Problem Base on the Spin Pseuo Spin Symmetry. Central European Journal of Physics 10: 361. [22] Navratil, P., Barrett, B. R. 1998. Large-Basis Shell-Moel Calculations for p-shell Nuclei. Phys. Rev. C 57: 3119. [23] Gönül, B., Köksal, K., Bakir, E. 2006. An Alternative Treatment for Yukawa-Type Potentials. Phys. Scr. 73: 279. Appenix A: Nikiforov-Uvarov Metho The Schroinger equation is transforme into a secon orer ifferential equation of the form with an appropriate coorinate transformation [22]. The Nikiforov-Uvarov metho [21] can be use to solve secon orer ifferential equations with an appropriate coorinate transformation s = s(r): U 0 V 0 X0 U XY0 X 0 U 00 A1 where, σ(s) XY0 are polynomials, at most of secon-egree, V 0 is a first-egree polynomial. To fin particular solution Eq. (A1) by separating of variables, one eals with the transformation as follows: U 0 Z 0 [ U 0 A2 An it reuces to an equation of hyper geometric type: X0[ U 0V0[ U 0\[ U 0 0 Z0 is efine as logarithmic erivative: Z 0 Z0 ]0 X0 [ U 0 is the hyper geometric-type function whose polynomials solutions are given by Roriguez relation: [ U 0 ^U _0`0 U,XU 0_0 - where, ^U is the normalization constant the weight function _0 must satisfy the conition: X0_0 V0_0,X 0_0X0 _ 0 _0 V0 The function ]0 the parameter \ require for this metho are efine as follows: `U A3 A4 A5 A6 ]0 X V 2 al X V 2 XYbX A7 \ b] 0 A8 V0V 02]0,V, 0 c 0 A9
70 Calculating Bining Energy for O Isotopes of Beryllium (7 A 13) Since _ 0 0 X 0 0, hence the erivative of V shoul be negative [10]. This leas to the choice of the solution. If \ in Eq. (A8) is: \ \ U JV 0 UU X " 0, J 0,1,2... The following equation is a general form of the Schroinger equation written for any potential [22]: α 0 ` ` `0 α s1α 6 0s 5 0 5 05 6,s1α 6 0-!"00 We may solve this as follows. When Eq. (A11) is compare with Eq. (A1), we get V 2 2 0 X 012 6 0 also Substituting these into Eq. (A7): where XY 5 0 5 05 6 ]02 ; 2 < 0ag2 = b2 6 0 2 > b02? A10 A11 A12 A13 A14 (A15) α ; 1 2 1α, α < 1 2 α 2α 6,2 = 2 < 5 A16 2 > 22 ; 2 < 5, α? α ; 5 6 A17 α @ α 6 α > α 6 α? α =, α % α α > 2α ; 2gα? α α 2α < 2gα @ α 6 gα? A18 α α ; gα? α 6 α < gα @ α 6 gα? in Eq. (A15), the function uner square root must be the square of polynomial accoring to the NU metho [22], so that ] V becomes: ]02 ; 2 < 0,Fg2 @ 2 6 g2? G-0g2? V 2 22 ; 2 22 < 02hFg2 @ 2 6 g2? G0g2? i V 2 22 < 2hFg2 @ 2 6 g2? Gg2? i V c 0 is obtaine. When Eq. A8 is use with Eqs. A20 A21 the following equation is erive: A19 A20 A21 2 J2J12 < 2J1Fg2 @ 2 6 g2? GJJ12 6 2 > 22 6 2? 2g2? 2 @ 0 A22 This Eq. (A21) gives the energy spectrum. An the wave function will be as follows: T U α }, α }} α~ α } are Jacobi polynomials. 0 s α }Q 1α6 0 α }Q α }~ α α ~ T }, α }} U α ~ α } "12α6 0 A23