Nuclea size coections to the enegy levels of single-electon atoms Babak Nadii Nii a eseach Institute fo Astonomy and Astophysics of Maagha (IAAM IAN P. O. Box: 554-44. Abstact A study is made of nuclea size coections to the enegy levels of single-electon atoms fo the gound state of hydogen like atoms. We conside Femi chage distibution to the nucleus and calculate atomic enegy level shift due to the finite size of the nucleus in the petubation theoy context. The exact elativistic coection based upon the available analytical calculations is compaed to the esult of fist-ode elativistic petubation theoy and the non-elativistic appoximation. We find small discepancies between ou petubative esults and those obtained fom exact elativistic calculation even fo lage nuclea chage numbe. Keywods: Single-electon atoms elativistic coection Fist-Ode Petubation Theoy Nuclea Chage Numbe. Intoduction As we know the unphysical infinity in the potential at the oigin makes it necessay that this potential be modified fo values of inside a egion about the oigin that can be identified with the nucleus of the atom. The emedy is attibuting finite size to the nucleus of the atom. The esulting coection due to the finite size of the nucleus leads to the shift of atomic enegy levels. Fom anothe point of view thee is isotope shift of atomic enegy levels due to this kind of coections. The dependence of the coection to the atomic enegy level on the fom of the potential enegy inside the nucleus necessitates a choice of a model fo the nuclea potential. Fo two common models fo a Autho to whom any coespondence should be addessed E-mail: bnbnadii9@gmail.com Tel: 98 95645495 Fax : 98 4 744
nuclea potential function which espectively simulate eithe a unifom chage distibution o a constant potential inside nucleus the atomic enegy level shift has been calculated []. Calculation of these type of coections have attacted a lot of attention. Fo a eview see ef [-7]. The exact teatment of the poblem is based on a solution of the Diac equation fo all values of. The method educes the computation of the enegies of the electon in inteaction with a finite size nucleus to a bounday value poblem involving a single unknown eigenvalue []. In the pesent pape we adopt anothe two appopiate chage distibution to the nucleus: Femi and distibutions; and calculate the coection fo the gound state of electonic hydogen like atom due to these chage distibutions of nucleus (nuclea size. The main focus is on the compaison of the exact esults with the esults of two appoximate methods. The appoximate methods ae petubation theoy and nonelativistic teatment as descibed in Section. In Section we biefly discuss the exact solution of Diac equation in the pesence of extenal potential. The appoximate methods ae descibed in Section. In Section 4 the numeical calculation in petubation theoy is discussed. Finally ou numeical esults ae compaed with the esults obtained fom petubation theoy using both elativistic and nonelativistic wave functions fo two physical chage distibution models to the nucleus.. EXACT CALCULATION The solution of the Diac equation in the pesence of extenal potential leads to the coupled diffeential equations fo the adial wave functions as [8] d κ g E mc V f d c ( ( ( d κ f E mc V g d c ( ( ( ( (
Whee f ( and g ( ae the uppe and lowe components of the adial eigenfunctions espectively; E is the enegy eigenvalue and κ is the eigenvalue of the opeato ˆ κ ˆ σ Lˆ. Hee fo a given value of j the quantum numbeκ has the possible values ± ( j coesponding to values of l and l equal to j ± and j espectively. Fo values of adial coodinate geate than o equal to a value which defines the nuclea adius we assume that the cental potential has the coulomb fom V e ( ( ( Solution of the adial Diac equation fo this egion leads to the familia fomulae fo the allowed enegy eigenvalues of the electon given by n j -( a E mc n... (4 n j -( a ( a Whee e c is the fine-stuctue constant and n n j is defined fom pincipal 7 quantum numbe n. It can be shown that the functions g ( and f ( have the explicit foms[]: g g( N βe F g ζ E g η E F g ζ E g ( ( ( ( ( g g ( βg F( g ζ( E g ( βgη ( E F( g ζ( E g (5 E f ( N β e F( ζ ( E η ( E F( ζ ( E E ( β F( ζ( E ( βη ( E F( ζ( E Whee N β epesents a nomalization constant and the enegy paamete E must be deived fom the continuity conditions at. Beside we have used the following notation (6
E E ± ζ E ζ ( E η ( E mc ± E κ ζ ( E ( E E (7 E g j ( and β β± a ± ( mc E Fo values of the adial coodinate less than the nuclea adius the adial Diac equations have been calculated analytically fo two common models in [] such as unifomly chaged nucleus and constant potential inside nucleus. The solutions of the Diac equation fo values of exteio and inteio to the nucleus need to be made continuous at the bounday of the nucleus defined by poduces the simultaneous equations:. The continuity equiement at g ( g ( f ( f ( (8 inteio exteio inteio exteio which can be conveniently combined into the matching equation as g f inteio inteio ( ( g f ( ( exteio (9 exteio The equation has the effect of educing the computation of the enegies of the atomic electon in the case of a finite size nucleus to a bounday value poblem involving a single unknown E the solution of which detemines the allowed enegy eigenvalues.. APPOXIMATE METHODS We can compae the enegy eigenvalues deived fom equation (9 with the coected eigenvalues obtained fom the fist-ode petubation theoy unde the assumption that the change in the coulomb potential in the inteio of the nucleus is teated as a petubation to the Hamiltonian as H H V V( ; in which V ( > V e and ( ( 4
Fo spheically symmetic chage distibution (model inside the nucleus the elation out in yields V ( E d E d e ( ( In which we have substituted the well-known fomulas fo electic fields inside and outside the sphee: E ( ε > ε Pefoming simila staightfowad calculations fo the constant potential inside the nucleus (model one gets ( e e ( ( Now we want to obtain the enegy shift of the gound state ( n atomic electon in which we have assumed unifom chage distibution inside the nucleus. Fom fist ode petubation theoy E ψ V ( ψ and using elativistic (Diac fom fo ψ ( ig ( Ω ( jlm j ψ ( f ( jl m Ω j whee Ω is the Diac spino we obtain jlm j d ( d dωˆ ψ ( 4π [ g ( f ( ] V ( ψ V ( 4π ( dω ( ˆ Ω Ω jlm j jlm j (4 (5 On the othe hand f ( and g ( is deived fom elations (5 and (6 fo gound state of atomic electon as g N e κ f N e Whee g q g q ( ( ( ( κ (6 5
6 ( ( state gound c E mc q κ a κ g (7 Using gound-state enegy eigenvalue E ( mc E (8 q may be witten as ( c mc mc q c κ λ (9 Whee c λ is the Compton wavelength of the electon. Fom elations ( and (5 E takes the fom [ ] f g d e E ( ( 4 q e d e N ( Beside fo value of we have q c << λ ; So one can conside q e. Theefoe 4 d e N E ( N e Hee the nomalization constant can be obtained accoding to the pesciption dv ψψ. So one gets fom elations (4 (6 and (9 ( 4 4 Γ λ λ c N e d N C ( Then ( 4 ( c N g g λ Γ
Substituting the expession ( in elation ( we obtain e λc Γ ( Now the following elations (4 e e c mc mc λc λc mc Allow us to wite (4 as []: Γ ( ( ( mc In compaison the non-elativistic calculation gives the esult [9]: E 5 4 ( mc (7 Fo small values of in the appoximation in which ( educes to one the elativistic esult (6 coincides with the non-elativistic esult (7. Using the empiical elation 5 A. m (8 Whee and A ae the nucleus adius and mass numbe espectively the elation (6 ecasts in the fom (.557 6 Γ ( mc ( Fo constant potential inside nucleus the appoach is simila. In the pesent wok fo compaison we adopt anothe two models fo nuclea chage distibution: chage distibution and a Femi chage distibution fo <. With ( defined by eithe of the two fomula A (5 (6 (9 e ( chage distibution ( 7
4p d 4p d e ( e d e d Femi chage distibution c c exp exp k k And using the elation E g ( f ( ( d ( Along with the fom fo ( in equations (- esults in coections to the enegy of gound state given by the espective fomulas ( ( a ( g ( a g E mc chage distibution Γ g g g g. ( g a λ E d V ( Femi ch c g age distibution. Γ (4 ( g Clealy elation ( is being substituted fo ( ( Femi c exp k V ( in the expession (4. Hee we conside The two paametes c and k ae detemined fo instance by fitting to densities deived fom measued fom factos [-]; and the facto ρ is given by nomalization condition (5 ( d e (6 4. CONCLUSIONS The dependence of the coection to the enegy on the fom of the potential enegy inside the nucleus necessitates a choice of a model fo the nuclea potential. Fo two common models fo nuclea potential function which espectively simulate eithe a unifom chage distibution o a constant potential inside nucleus exact and E Pet. have been calculated []. 8
A E exact (ev E Petubation (ev (constant potential E Petubation (ev (unifom chage dist. E code (ev E non elativistic 5.6 9 9. 9 5.6 9 5.6 9 5.6 9 8.89 9 4.8 9 8.89 9 8.89 9 4.8 9 (ev 47 7.6755.598.665.676.6558 47 9.888.48.7854.879.6 6 5 8.694 5.7594 9.894 8.648.56 6 5 8.6766 5.875 9.9757 8.674.58 8 6.8.54 78.88 6.4 8. 8 5 6.698.77 78.8 6.658 8.77 9 5 9.8 4.7 8.57 9.66 5.7 9 8 94.76 4.6 8.6 94.6 5.4 Table : Values deived fom the pesent calculation and fom elativistic and non-elativistic petubation theoy fo coection to the gound state enegy of an hydogenic atom poduced by the finite size of a nucleus of chage. Models and assume ( a unifomly chaged nucleus and ( a constant potential inside the nucleus espectively []. Gound state : n k. Figue : Gaphs vesus of the nuclea size coection to the gound state enegy of a hydogenic atom obtained fom matching condition in equation (8 (Cuve A and elativistic petubation theoy (Cuve B using model [] Fo these two models Table lists calculated values of the enegy coection to the gound states of single-electon atoms coesponding to stable isotopes of the five elements H U Ag Eu andtl. In 9
paticula the table compaes the coections E deived fom the matching condition in equation (9 with the values of obtained fo the same model of the nuclea potential using fist-ode petubation theoy based on both elativistic E pet. and non-elativistic wave functions E non el.. The diffeent values fo as a function of pedicted by the petubation theoy and exact theoy fo these two models ae summaized by the gaphs in figue. It is useful to compae the esults deived fom the matching condition in equation (9 with the esults extacted fom an atomic stuctue code fo the same model. To do this we include in table the values of obtained fom geneal pupose elativistic atomic stuctue pogam GASP [] fo the case of a unifomly chaged nucleus. Compaison of these values denoted by code with the values obtained fom equation (9 shows that the two sets of values ae in excellent ageement. In analogy with the esults listed in table ( we list in table ( the calculated values of fo the gound states of electonic atoms with Femi and chage distibution. As expected esults of these two models ae in excellent ageement with and E pet.. In compaison with pevious models Ecode we find bette esults fo E pet.. The diffeent values fo as a function of pedicted by the petubation theoy and exact theoy (fo and Femi chage distibution ae summaized by the gaphs in figue (. In spite of that the chage distibution is not of much physical inteest the elated esults is in good ageement with and Eexact. E code In summay fo values of geate than 4 in the case of electonic atoms we find lage discepancies between ou esults and those obtained fom fist-ode petubation theoy using elativistic wave functions. But with consideing physical models (Femi chage distibution to the nucleus we find small discepancies between petubative and exact esults even fo lage nuclea chage numbe.
A E exact (ev E Petubation (ev E Petubation (ev E Petubation (ev E code (ev (Unifom (Unifom chage dist. -chage dist. (Femi-chage dist. 5.6 9 5.6 9 4.6 9 5.6 9 5.6 9 8.89 9 8.89 9 7.4 9 8.89 9 8.89 9 47 7.6755.665.447.5.676 47 9.888.7854.576.6.879 6 5 8.694 9.894 8.57759 8.9 8.648 6 5 8.6766 9.9757 8.46 9. 8.674 8 6.8 78.88 47.77 7.6 6.4 8 5 6.698 78.8 48. 7.78 6.658 9 5 9.8 8.57 5. 4.5 9.66 9 8 94.76 8.6 5.6 44. 94.6 Table : Values deived fom the pesent calculation and fom elativistic and non-elativistic petubation theoy fo coection to the gound state enegy of a hydogenic atom poduced by the finite size of a nucleus of chage. Assuming ( chaged nucleus and ( a Femi chage distibution inside the nucleus espectively. Gound state : n k. Figue : Gaphs vesus of the nuclea size coection to the gound state enegy of a hydogenic atom obtained fom matching condition in equation (9 (Cuve A and elativistic petubation theoy using model (Cuve B Cuve C and Cuve D fo and Femi chage distibution models espectively.
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