Proceeding of the Third Eurasian Conference "Nuclear Science and its Application”, October 5 - 8 , 2004.
APPLICATIONS OF THE DUFFIN-KEMMER-PETIAU
EQUATION FOR THE DEFORMED WOODS-SAXON
POTENTIAL IN NUCLEAR PHYSICS
Yaşuk F., Berkdemir C., Berkdemir A.
Erciyes University, Kayseri, Turkey
ABSTRACT
There has been a recent revival of interest in the Duffm-Kemmer-Petiau (DKP) equation, and its relevance to some problems in nuclear and particle physics. DKP equation is the Dirac- like first order relativistic equation and it can define both spin-zero and spin-one particles.
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Proceeding o f the Third Eurasian Conference “Nuclear Science and its Application”, October 5 - 8 , 2004.
Recently the first-order relativistic DKP equation has been used to study the interactions of spinless mesons with nuclei. In this study, energy eigenvalues are obtained for DKP equation with the deformed Woods-Saxon potential, defined by three parameters: the depth, the radius and the surface diffuseness, for spin zero representation by using Nikiforov-Uvarov method. This method gives the solution of second order linear differential equations whose solutions leads to hypergeometric type equations. The energy spectrum numerically calculated for different values of surface diffuseness parameter of the potential at some discrete levels.
INTRODUCTION
The problem of a relativistic particle with an arbitrary spin has a long history. Following the Dirac equation [1], the search began for similar equations for higher spins. Apart from spin one- half, none of the other spins obeys a single relativistic wave equation [2], For instance, it was generally believed that for spins zero and one, the Klein-Gordon [3, 4] and Proca [5] equations were unique. But, after that the Dirac-like first order relativistic Duffin-Kemmer-Petiau equations were found and it can define both spin-zero and spin-one particles [6-8],
Great deals of important problems in relativistic quantum mechanics as well as non relativistic quantum mechanics have been solved by using Nikiforov-Uvarov method (NU) [9], This method depends on the solution of second order linear differential equations whose solutions leads to hypergeometric type equations.
The goal of this study is to obtain energy eigenvalues of Duffin-Kemmer-Petiau equation in the presence of the deformed Woods-Saxon potential for spin zero representation by using Nikiforov-Uvarov method. Firstly, we talk about Duffin-Kemmer-Petiau formalism briefly and derive second order radial differential equations with general vector potential. After that, NU method is given in terms of hypergeometric polynomials. Finally, deformed Woods-Saxon potential is substituted in DKP equation and then energy eigenvalues are obtained.
DUFFIN-KEMMER-PETIAU FORMALISM
In this section, the Duffin-Kemmer-Petiau formalism is given briefly. The solution of the DKP equation for a particle in a central field needs consideration since earlier work [10], It is convenient to recall some general properties of the solution of the DKP equation in a central interaction for spin zero particles. The central interaction consists of two parts: a Lorentz scalar (Us) and a time-like vector (Uv) potential. The stationary states of the DKP particle in this case are determined by solving
( J3.pc + mc2 +US +J30U°V )y/ (r) = j3°Ey/ (r) (1) The most general solution of Equation (1) is
V j M (r ) = SnJ( r )Y jM ( Q )
/X /, ^nJLM il'ilO ) ,
(2)
where YJM (Q) are the spherical harmonics, Y ^ f i ) are the normalized vector spherical harmonics, and fnJ(r), gnJ(r)and hnJL(r)are radial wave functions. Inserting v j/^ r) as given in Eqn.(2) into Eqn.(l) by using the properties of vector spherical harmonics one gets the following set of first-order coupled relativistic differential radial equations
[E-U°v )F{r) = {mc2 +Us )j(r) (5a)
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Proceeding o f the Third Eurasian Conference “Nuclear Science and its Application”, October 5 - 8 , 2004. he he dF(r) J + l 1 ( 2 t j \ u / \ , Fir) = \mcz + Us JHx(r) dr r J ocj (5b) +1 f (r)l = J - ( m e 2 + U , ) H . t(r) dr r J C j (5c) \ dr r ) \ dr r j =2-((mc2 + { 7 ^ ( 0 - ( £ - { / ; );(/■)) (5d)
with the definition of a j = ^/(j + \)l(lJ + l ) , C j - /(2J + l) and f n j ( r ) = ~^~>
, \ Gir ) A t / \ tf±i(r) g'nj ('•) = --- and h„jj±x (r)
=---CALCULATIONS WITH NIKIFOROV-UVAROV METHOD
For the DKP equation with a spherical deformed Woods-Saxon potential can be used
V(r) = - Vr0
r - R , (6)
1 + q e y a
where V0 is the potential depth, R0 is the width of the potential, q is the deformation parameter and a is the surface thickness. At the presence of vector potential and while scalar potential is zero, radial DKP equation is given as follows
f ^2d 2 J{J + l) 1
dr‘ h2e 2 (
e - U ° J - nr2c4 F(r) = 0 (?)
by letting u° as the deformed Woods-Saxon potential and choosing J=0 (s-wave case)
F a i r y + E 2 + 2 VnE —
- 2 a r - A a r
° 1 + qe~2ar + V2(1 + qe~2ar)2
- m Fq(r) = 0, ( 8)
is obtained. The Nikiforov-Uvarov method [9] is reduced to a generalized equation of hypergeometric type with an appropriate - s = e 2<xr coordinate transformation
s2( / - q 2s 2 - q P 2) + s(j32 + 2qs2) - s 2\ q(s) = 0 (9)
F q( s y + - ^ - F cl( s y + 1
5(1 - sq) [.S'O - .SY/ )]2 where dimensionless parameters are :
* 2 = - 1 4h2c2a 2 [£2 - ( ™ 2)2]>0, = 4 h2c2cc2 Vo 4h2c2a 2
Using NU method, Eqn.(9) is solved and the exact energy eigenvalues of DKP equation with deformed Woods-Saxon potential are derived as
\2
\mc V0'
E = ~ — ± C
nq 2 q \ 4 y 2 + C2 16 q2y 2 (10)
r
Section II. Basic problems o f nuclear physics
Proceeding o f the Third Eurasian Conference “Nuclear Science and its Application”, October 5 - 8 , 2004.
where C = -yjq2 - 4y 2 - q(2n+ 1).
CONCLUSION
In this study, we obtained the energy eigenvalues for DKP equation with deformed Woods- Saxon potential by the method of Nikiforov-Uvarov. It is seen that exact solutions of first order Dirac-like DKP equation with deformed Woods-Saxon potential contain the solutions of Klein- Gordon equation for spin zero. Since DKP equation holds for spin zero and one particles unlike KG equation, it is interesting to solve DKP equation in the case of spin one by applying NU method.
REFERENCES
1. P.A.M. Dirac, Proc. R. Soc. London, Ser. A 155, 447 (1936) 2. V.K. Mishra, S. Hama, B.C. Clark, Phys. Rev. C, 43, 2 (1991) 3. O. Klein, Z. Phys. 37, 895 (1926)
4. W. Gordon, Z. Phys. 40, 117 (1926)
A. Proca, C. R. Acad. Sci. Paris 202, 1420 (1936) 5. N. Kemmer, Proc. R. Soc. London 173, 91 (1939) 6. R. J. Duffin, Phys. Rev. 54, 1114 (1938)
7. G. Petiau, Ph.D Thesis, University of Paris, 1936; Acad. R. Belg. Cl. Sci. Mem. Collect. 8, 16 1936
8. A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Basel, 1988.
9. Y. Nedjadi and R. C. Barrett, J. Physics G, 19, 87 (1993)
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