solutions of dirac equation for a new improved pseudo coulomb ring shaped potential

+ MODEL Available online at www.sciencedirect.com H O S T E D BY ScienceDirect Karbala International Journal of Modern Science xx (2016) 1e9 http://www.journals.elsevier.com/karbala-international-journal-of-modern-science/ Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential A.N Ikot a,*, M.C Onyeaju a, M.I Ngwueke a, H.P Obong a, I.O Owate a, H Hassanabadi b a Theoretical Physics Group, Department of Physics, University of Port Harcourt, Nigeria b Department of Physics, Shahrood University of Technology, Shahrood, Iran Received August 2016; revised 10 November 2016; accepted 10 November 2016 Abstract We proposed a new solvable novel pseudo-Coulomb ring-shaped potential and investigate its pseudospin symmetry by solving the Dirac equation under the condition of an equal mixing of scalar and vector potentials using the factorization techniques We determine in closed form the energy eigenvalues and eigenfunctions of the bound states of the Dirac equation analytically We also discuss the non-relativistic limits © 2016 The Authors Production and hosting by Elsevier B.V on behalf of University of Kerbala This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) PACSnumbers: 03.65Ge; 03.65Pm; 03.65Db Keywords: Dirac equation; Pseudospin symmetry; Pseudo-Coulomb potential; Bound state; Eigenvalues Introduction The pseudospin symmetry of the Dirac Hamiltonian was discovered many years ago, however, this symmetry has recently been recognized empirically in nuclear and hadronic spectroscopic [1] Within the theory of Dirac equation, the concept of pseudospin symmetry is used in deformed nuclei, superdeformation, and effective shell model [2,3] It was shown that the exact pseudospin symmetry occurs in * Corresponding author E-mail address: ndemikotphysics@gmail.com (A.N Ikot) Peer review under responsibility of University of Kerbala dSðrÞ the Dirac equation when i.e dr ẳ 0, Srị ẳ Vrị ỵ Srị ¼ const, where VðrÞ; SðrÞ are repulsive and attractive scalar potentials, respectively The pseudospin symmetry usually refers to as a quasidegeneracy of single nucleon doublets with the   non- relativistic quantum number n; l; j ẳ l ỵ 12 and   n 1; l ỵ 2; j ẳ l ỵ 32 , where n; l and j are single nucleon radial, orbital and total angular quantum numbers, respectively The total angular momentum is j ẳ l~ỵ s~, where l~ ẳ l ỵ is a pseudo-angular momentum and ~s is pseudospin angular momentum The Dirac equation with different potentials in relativistic http://dx.doi.org/10.1016/j.kijoms.2016.11.002 2405-609X/© 2016 The Authors Production and hosting by Elsevier B.V on behalf of University of Kerbala This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 quantum mechanics with pseudospin symmetry has been investigated in recent years [4e10] In order to fully understand the origin of the concept of pseudospin symmetry, the motion of the nucleons in a relativistic mean field theory has to be taken into consideration which considered the Dirac equation [11] Zhou et al [12] have studied the Dirac equation for Makarov potential with pseudospin symmetry Guo et al [13] studied the bound states of relativistic particles for ring-shaped non-spherical harmonic oscillator potential under pseudospin symmetry The ringshaped molecular potentials are non-central potentials and have many applications in physics and quantum chemistry Non-central potentials have also been studied extensively in the literature since it provides a useful theoretical background for describing the interaction between the ring-shaped molecules and that between the deformed nucleus [14e16] In a similar development, the ring-shaped potentials have found many useful applications in quantum chemistry and are also important in nuclear physics to study rovibrational energy level of molecules, atoms and deformed nucleus and to describe ring-shaped molecular benzene in chemistry [17e20] In recent years, considerable efforts have been made by many authors to obtain the exact analytical solutions of the Schr€ odinger equation with ring-shaped potentials [21e23] Also, different kinds of ring-shaped like potential have been investigated such as the non-spherical harmonic oscillator (NHO) [24], the ring-shaped oscillator (RHO) [25], the double ring-shaped harmonic oscillator (DRHO) [26] among others Dong et al [27] studied a ring-shape non-spherical harmonic (RNHO) potential and obtained the non-relativistic energy spectra and wave function Berkdemir [28] in his paper proposed the novel angle-dependent (NAD) potential where the potential VðrÞ contains a Coulomb potential or a harmonic potential in addition to NAD potential Zhang et al [29] extended the work of Berkdemir and proposed the harmonic novel angle dependent (HNAD) potential by simply replacing sin q by cos q in the numerator of NAD and obtain HNAD Another form of the ring shaped potential investigated is the double ring-shaped oscillator (DRSO) [30] which reduces on special cases to the ring-shaped oscillator (RSO) and spherical oscillator (SO), respectively In addition, Zhang [31] proposed new RNHO and study the energy spectra and wave function for the potential Very recently, Chen et al [32,33] studied the ring-shaped potentials using the universal associated Legendre polynomials and they went further to discussed the super-universal associated Legendre polynomials Even though solutions of the Dirac equation with different potential model have attracted a great attention in recent years, the exact solutions of the Dirac equation are only possible for a few simple systems such as harmonic and hydrogen atom [34,35] Various analytical techniques have been employed to find the bound state solution of the Schrodinger, KleineGordon and Dirac equations such as NikiforoveUvarov method (NU) and formula method [36e38], supersymmetric quantum mechanics (SUSYQM) [39,40], quantization rule [41] and others [42] Consequently, the recent advances in the search for the solutions of Dirac equation with physical motivated potential models will lead to the discovery of a new phenomenon in addition to the spin and pseudospin symmetry discover many years ago in the nuclei of atom in the Dirac theory The investigated potentials include but not limited to Coulomb-like potentials [43], ManningeRosen potential [44], DengeFan potential [45], Mobius potential [6], shifted Hulthen potential [46] and others [47e49] Motivated by the study of the ring-shaped-like potential [50] we proposed the novel pseudo-Coulomb ring-shaped potential of the form, 2  ÀA B D cos2 q ỵ t Vr; qị ẳ ỵ 2ỵCỵ r r r sin q cos q 2  b sin q ỵ g cos2 q ỵ l ỵ : ð1Þ r sin q cos q where A; B and C are the potential depths and b; g; l; t and D are the five dimensionless ring-shaped parameters The plots of the behaviour of this potential are illustrated in Figs 1e3 The purpose of the present paper is to investigate the above potential under the Fig The plot of the behaviour of novel pseudo-Coulomb potential for a fixed r and various values of q ¼ 30; 60 and 90 respectively Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 where E is the relativistic energy of the system, ! ! p ¼ Ài V is the three-dimensional momentum operator and M is the mass of the fermionic particle ! a;b are the  Dirac matrices given as     I 0 ! si ! ;b ¼ ; ð3Þ a ¼ ! ÀI si where I is  unitary matrix and ! s i are the Pauli three-vector matrices:       I Ài I s1 ¼ ; s2 ¼ ; s3 ¼ : ð4Þ I i 0 ÀI Fig The plot of the novel pseudo-Coulomb potential with various values of r ranging from r ¼ 5; 25 and 40 with fixed q and I is the  unitary matrix In addition, we can write the Dirac wave function as  !  r ị jrị ẳ 5ị c! rị where 4ð! r Þ; cð! r Þ represent the upper and lower components of the Dirac wavefunctions and ! s $! p c! r ị ẳ ẵ M Sðrފ4ð! rÞ ð6Þ ! s $! p 4ð! r Þ ẳ ẵ ỵ M Drịc! rị Fig The 3D plot of novel pseudo-Coulomb ring shaped potential as a function of r and q Here we have choose A ¼ B ¼ C ¼ b ¼ g ¼ l ¼ 1; D ¼ and t ¼ pseudospin symmetry limit and study the quantum behaviour arising from it The organization of the paper is as follows In Section 2, we review the Dirac theory under pseudospin symmetry limit Section is devoted to the exact solution of the Dirac equation with pseudo-Coulomb ring-shaped potential Finally, we give a brief conclusion in Section Theory of Dirac equation The Dirac equation for spin- 12 particles moving in an attractive scalar potential SðrÞ and repulsive vector potential VðrÞ in the relativistic unit Z ẳ c ẳ 1ị is [1e10] ẵ! a $! p ỵ bM ỵ Srịịjrị ẳ ẵE Vrịjrị; 2ị 7ị where Srị ẳ Vrị ỵ Srị ¼ Cps ; DðrÞ ¼ VðrÞ À SðrÞ ¼ Cs In the limiting case, that is under the condition of pseudospin symmetry, Eqs (6) and (7) become ! s $! p ! c r ị 8ị 4! r ịẳ M ! s $! p 4! r ị ẳ ẵ þ M À DðrފcðrÞ ð9Þ where εsM i.e only real negative energy states exist when Cps ¼ (exact pseudospin symmetry) and the energy eigenvalues ε depends on the quantum numbers n and L and also on the pseudo-orbital angular momentum quantum number l.~ Combining Eqs (7) and (8) and taking DðrÞ as the pseudo-Coulomb potential, we obtain the Schr€odingerlike equation for the lower component as,  & Á À ÀA B V2 À ε2 À M À ðε Mị ỵ 2ỵC r r 2  D cos q ỵ t ỵ r sin q cos q 2 )#  b sin2 q ỵ g cos2 q ỵ l ỵ cnlm r; q; fị ẳ r sin q cos q 10ị Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 where,     ! v 2v v v v2 V ẳ r ỵ sin q ỵ r vr vr sin q vq vq sin q vf2 ð11Þ Now since the Eq (10) has a decoupling of pseudospin and pseudo-orbital momentum, then the lower component of thespinor has spin up or   wave  function multiplying by the or spin down i.e., component of the spherical co-ordinate, thus the lower component of the wave function can be written as, Rnl~ðrÞHlm ~ ðqÞFm fị cnlm cm ~ r; q; fị ẳ r 12ị where m ¼ ±12 and cm is the spin up or spin down twocomponent spinors Now after substituting Eqs (11) and (12) into Eq (10) and making a separation of variable, we obtain the following sets of second order Schr€ odinger-like differential equations: d2 Rnl~rị ỵ M À ðε À MÞC dr ! Aðε Mị L ỵ B Mị ỵ Rnl~rị ¼ r r2 " d2 Hlm dHlm m2 ~ qị ~ qị ỵ L Mị ỵcot q dq sin q dq   D cos2 qỵt Mị sin q cos q  2 # bsin2 qỵg cos2 qỵl Hlm ~ qị ẳ 0; sinq cos q d2 Ffị ỵ m2 Ffị ẳ df2 Ffị ¼ pffiffiffiffiffiffieimf ; m ¼ 0; ±1; ±2… 2p ð16Þ In the following subsection, we will give the solution of the radial part (Eq (13)) and polar angular part of Eq (14) Solutions of the Dirac equation with the novel pseudo-Coulomb ring-shaped potential 3.1 Solution of the angular part In order to obtain the energy eigenvalues and the corresponding wave function for the angular part of Dirac equation under the pseudospin symmetry limit, we take a new variable transformation of the form, x ¼ cos2 q in Eq (13) and after a little algebra we obtain,   d Hlm 13x dHlm ~ xị ~ xị ỵ þ xð1ÀxÞ xð1ÀxÞ ð17Þ dx2 dx À Á Àu1 x2 ỵu2 xu3 Hlm ~ xị ẳ where, 13ị L ỵ Mị D2 ỵ g ỵ bị u1 ẳ ; Lm2 2DtMịỵ2bMịbgỵlị Mị b ỵ lị2 ỵ t2 u3 ¼ u2 ¼ ð14Þ where L and m2 are the separation constants The solutions of Eq (15) satisfy the boundary Ff ỵ 2pị ẳ Ffị whose solution is given as, ð19Þ ð20Þ We take the physically acceptable ansatz for the wave function as, q p Hlm ~ xị ẳ x ð1 À xÞ f ðxÞ ð15Þ ð18Þ ð21Þ and substituting it into Eq (17) yields,   ! 00 x1 xịf xị ỵ 2p ỵ 2p ỵ 2q ỵ x f xị 2    1 p ỵ q ỵ s p ỵ q ỵ ỵ s f xị ¼ 4   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À À ÁÁ 1 ỵ ỵ Mị b þ lÞ2 þ t2 ; ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À À 2 Mị D2 ỵ g þ bÞ þ ðε À MÞ ðb þ lÞ þ t2 ỵ m2 ỵ 2Dt Mị 2b Mịb g ỵ lị; qẳ s L ỵ Mị D2 ỵ g ỵ bị ỵ sẳ 16 22ị pẳ 23ị Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 where, Now comparing Eq (22) with the standard form of the second order differential of hypergeometric function [51] 00 z1 zị4 zị ỵ ẵc a ỵ b ỵ 1ịx40 zị ab4zị ẳ ð24Þ we get the parameters a; b; c as follows, a ẳ p ỵ q ỵ s; b ẳ p ỵ q ỵ ỵ s; c ẳ ỵ 2p 25ị The solutions of Eq (22) can be expressed in terms of the Gauss's hypergeometric function as À Áp À Áq Hlm sin q ~ qị ẳ cos q   1 2 F1 nr ; nr ỵ 2p þ 2q þ ; 2p þ ; cos q 2 26ị However, when p ỵ q ỵ 14 ỵ s ẳ nr or p ỵ q ỵ À s ¼ Ànr for nr ¼ 0; 1; 2:::, then the hypergeometric function in Eq (26) reduces to a polynomial of degree nr In order to obtain the relationship between the separation constant L and the nonnegative integer nr , we used the quantization condition p ỵ q þ 14 À s ¼ Ànr with Eq (24), we obtain is the orbital angular momentum is defined as l~ ẳ 2nr ỵ ỵ jmj; m ẳ 0; 1; 2::: It can be observed that the angular part of the Pseudo-Coulomb ring-shaped potential has singularities at q ¼ tpt ẳ 0; 1; 2:::ị when D; t; b; g; lÞ are taken as positive values and also at very small and very large values of r However, it is known that the angular wave function Hlm ðqÞ exist as an odd and even function [52e55] So in order to remove this singularities then the combination of the ring shaped parameters D; t; b; g; lị ẳ n4n 2ị or D; t; b; g; lị ẳ n4n ỵ 2ị For instance when g ẳ n4n 2ị and similarly for other terms, the rest of the parameters being zero would reduced the Hlm ðqÞ to the associated legendre polynomials Pm l ðqÞ which is an odd or an even function [52e55] thus removing the singularities The complete angular wave function can be written as,  À Áp q cos sin Hlm qị ẳ N q q F n ~ À nr ; nr þ 2p þ 2q  1 þ ; 2p ỵ ; cos q 2 28ị where Nn is the normalization constant of the angular wave function Hlm ðqÞ In order to determine the normalized angular wave function, we used the following normalization conditions [51e54], Z2p Z1 sinðqÞjHlm ~ qịj dq ẳ jHlm ~ qịj dx ẳ 2 ð29Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Á À Á 12 À   u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ðε À MÞ D2 ỵ g ỵ bị2 ỵ Mị b þ lÞ2 þ t2 ÁÁ À À u C B1 ỵt C B ỵ ỵ Mị b ỵ lị ỵ t2 C B LẳB C ỵm2 ỵ 2Dt Mị 2b Mịb g ỵ lị C B A @ þ2 þ 2nr À 2Á Àðε À MÞ D2 ỵ g ỵ bị 27ị Eq (26) is the contribution of the angle-dependent part of pseudo-Coulomb plus ring-shaped like potential However, when the ring-shaped term potential vanishes, that is g ¼ b ¼ l ¼ D ¼ t ẳ 0, then the ~ l~ỵ 1ị, where l~ constant of separation becomes L ¼ lð Z1 zgÀ1 ð1 zị sg ẵ2 F1 n; n ỵ s; g; zị dz n! Ggị Gn ỵ s g ỵ 1ị s ỵ 2nị Gn ỵ sịGs ỵ gị ẳ 30ị Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 Now substituting Eq (28) into Eq (29), we obtain,   2 Z1  1  2Nn xp 1xịq 2 F1 nr ;nr ỵ2pỵ2qỵ ;2pỵ ;x  dx 2 00 r4 rị ỵ ẵ2L þ 1Þ À 2hrŠ40 ðrÞ þ ðq À 2hðL þ 1ịị4rị ẳ0 36ị ẳ1 31ị Thus, we obtain the normalization constant as, s 2nr ỵ p ỵ q ỵ 1ị Gnr ỵ p ỵ q ỵ 1ịGp ỵ 2q ỵ 2ị Nn ẳ 2nr ! G1 ỵ pị2 Gnr þ q þ 1Þ ð32Þ 3.2 Solutions of the radial part Now for the radial equation, we let the following parameters rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 k ¼ ε M ;L ẳ ỵ L ỵ B Mị À ; rffiffiffiffiffiffiffiffiffiffiffiffi ð33Þ A εÀM C qẳ ; h ẳ ỵM 4 ỵ Mị and using the new coordinate transformation r ẳ 2kr, Eq (13) becomes ! d2 Rrị q LL ỵ 1ị ỵ h ỵ Rrị ẳ 34ị dr2 r r2 where L is given in Eq (27) Based on the behaviour of the wave function at the origin and at infinity, we define Rrị ẳ rLỵ2 ehr 4ðrÞ ð35Þ By substituting Eq (35) into Eq (33), we find that the wave function 4ðrÞ satisfies the following second order differential equation, If we let z ¼ 2hr in Eq (36), we obtain   q 00 z4 zị ỵ ẵ2L ỵ 1ị z4 L ỵ 4zị ẳ 2h 37ị The solution of Eq (37) isnothing but the confluent  q ; 2L ỵ 1ị; z hypergeometric function F L ỵ À 2h However, for the bound states the confluent function are terminated by a polynomial [52] such that, q 38ị L ỵ ẳ n; n ẳ 0; 1; 2… 2h where n is the node of the radial wave function Using Eq (33), we obtain the energy eigenvalues as, ẵ ỵ Mị C2 4A2 ẳ 39ị ðε2 À M Þ ðn0 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where n0 ẳ ỵ n ỵ 14 ỵ L ỵ B À MÞ As a special case in the non-relativistic limit, if we set ỵ M/ Enl ; À M/2m ; C ¼ B ¼ 0, we obtain Z2 the eigenvalues for the Coulomb potential plus a ring shaped potential as, Enl~ ẳ 8mA2 40aị þ n þ L0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 where L0 ẳ 12 ỵ 4L0 , L ẳ l~ l~ ỵ 1ị with l~ representing the non-relativistic orbital angular momentum quantum number and 12 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á u 2m À 2Á B 2m C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B   ! u Z2 D ỵ g ỵ bị ỵ Z2 b ỵ lị ỵ t C u B1 C u B ỵ ỵ 2m2 b ỵ lị2 ỵ t2 C ỵ t B C Z B C L ¼B 4m 4mb C ỵm ỵ Z2 Dt Z2 b g ỵ lị B C B C B C @ ỵ1 ỵ 2nr A 40bị 2m 2 D ỵ g ỵ bị Z Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 One can observe that the angular orbital momentum quantum in the non-relativistic limits differ from the relativistic case by the factor ε À M/2m Z2 This is a new result which has not been reported before to the best of our knowledge However, if we choose D ¼ t ¼ B ¼ C ¼ b ¼ l ¼ then the potential (1) turns to the Coulomb potential plus a new ringshaped potential [53], A cos2 q ỵa 2 r r sin q Vr; qị ẳ 41ị where BnL is the normalization constant Using the orthogonality of radial wave function, Z 44ị jRnL j r dr ẳ and the following known relations [53,54], Z∞ Gðm þ n þ 1Þ dn;n0 zm eÀz Lmn ðzÞLmn0 ðzÞ ¼ n! ð45Þ where a ¼ g : Making the corresponding replacement of parameters in Eq (40), we obtain the energy spectra and the corresponding wave function for the Coulomb plus new ring-shaped potential in the non-relativistic limit as, Lmn zị ẳ Gn ỵ m ỵ 1ị Fn; m ỵ 1; zị n!Gm ỵ 1ị 46ị and we obtain the normalized radial wave function as, 8mA2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2ma 2ma 3 ỵ m2 2ma ỵ 14 ỵ n ỵ ỵ 2nr ỵ ỵ Z2 ỵ m ỵ Z2 ỵ 2nr ỵ ỵ 2ma Z2 Z2 Enl ẳ  42aị !12  ! !  Lỵ1   s $ p 2hn! 4khr 4khr ! 2Lỵ1 4khr n ~m c j r ị ẳ p e Ln p r n0 ị2 n0 ỵ L ỵ n0 n0 2p ε À M      Á1 À Ám À 1  Nn cos2 q sin2 q 2 F1 À nr ; ỵ m ỵ l0 ỵ ỵ ỵ s; ; cos2 q eim4 2 ð42bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q Where l0 ẳ 2nr ỵ 32 ỵ 2ma ỵ m2 2ma ỵ 14 and Z2 Z2 when a ¼ 0, then the angular momentum quantum number becomes l0 ẳ 2nr ỵ ỵ jmj This result is consistent with that given in Ref [53] when m ¼ Z ¼ The corresponding radial eigen functions are expressed as,   q Lỵ1 2khr RnL rịẳBnL 2krị e F Lỵ1 ;2Lỵ1ị;2hkr 2h !12  Lỵ1 2hn! 4khr Rn;L rị ẳ p n0 n0 ị2 n0 ỵ L þ   4khr 4khr eÀ n0 L2Lþ1 n n0 ð47Þ Finally, using Eqs (5) and (8), we obtain the spinor wave function as, ð43Þ !12  ! !  Lỵ1   s $ p 2hn! 4khr 4khr 2Lỵ1 4khr ! n ~m c j r ị ẳ p e Ln p r n0 ị2 n0 ỵ L ỵ n0 n0 2p M   À Áp À Áq 1  Nn cos2 q sin2 q F1 À nr ; p ỵ q ỵ ỵ s; 2p ỵ ; cos2 q eim4 ð48Þ Please cite this article in press as: A.N Ikot et al., Solutions of Dirac equation for a new improved pseudo-Coulomb ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 + MODEL A.N Ikot et al / Karbala International Journal of Modern Science xx (2016) 1e9 The positive energy solutions for the spin symmetry can be obtain directly from the pseudospin symmetric solutions by using the following mapping, ~ l/l; Vðr; qÞ/ À Vðr; qÞ; Cps / À Cs ; ε/ À ~ε Conclusions In this article, we have proposed a novel PseudoCoulomb ring-shaped potential and investigate its exact solutions completely In the limit of the pseudospin symmetry, we solved the Dirac equation and obtain the energy eigenvalues and eigenfunction using the traditional factorization method Special cases of this potential have been discussed Now rescaling the potential parameter of our ring shaped like parameter as follows: a/b2 ; b/g2 ; c/l2 ; p/2bg; 2bl/ q; 2gl/f ; a0 ẳ q ỵ c; b0 ẳ f ỵ c, then the novel pseudo-Coulomb plus ring shaped like potential becomes,  A B Z2 ða sin2 q ỵ a0 ị Vr; qị ẳ ỵ þ C þ r r cos2 q 2Mr  b cos q ỵ b ị ỵp 49ị þ sin2 q Many useful ring-shaped potentials can be deduced from Eq (49) as special cases It is important for us to point out at this point that these results will have many applications in nuclear physics and quantum chemistry [56] Acknowledgement The authors wish to thank the kind reviewers for their positive comments on the manuscript References [1] J.N Ginocchio, Phys Rep 414 (2005) 165 [2] J.N Ginocchio, A Leviathan, Phys Lett Scr B 425 (1998) [3] P.R Page, T Goldman, J.N Ginocchio, Phys Rev Lett 86 (2001) 204 [4] E Maghsoodi, H Hassanabadi, S Zarrinkamar, H Rahimov, Phys Scr 85 (2012) 055007 [5] H Hassanabadi, B.H Yazarloo, M 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ring-shaped potential, Karbala International Journal of Modern Science (2016), http://dx.doi.org/10.1016/j.kijoms.2016.11.002 ... respectively Please cite this article in press as: A. N Ikot et al., Solutions of Dirac equation for a new improved pseudo- Coulomb ring- shaped potential, Karbala International Journal of Modern Science... cos q ð10Þ Please cite this article in press as: A. N Ikot et al., Solutions of Dirac equation for a new improved pseudo- Coulomb ring- shaped potential, Karbala International Journal of Modern Science... 22ị pẳ 23ị Please cite this article in press as: A. N Ikot et al., Solutions of Dirac equation for a new improved pseudo- Coulomb ring- shaped potential, Karbala International Journal of Modern Science