Most quantum mechanics textbooks introduce the atomic photoelectric effect expressing the final continuum state in the high energy limit as a plane wave. This approximation has shown to give clear differences between gauges though. In this work, we show that an approximation based on the asymptotic limit of the exact wave function for the final state leads to better results whether form of the interaction Hamiltonian is used as the photon energy increases. This asymptotic eikonal approximation leads to the exact result in velocity gauge for increasing photon energies, evidencing the relevance of the Coulomb potential even at large distances.Gran parte de los textos de mecánica cuántica introducen el efecto fotoeléctrico expresando la función de onda para el estado final en el límite de altas energías como una onda plana. Sin embargo, esta aproximación conduce a resultados que dependen de la norma utilizada para el calculo de la amplitud de transición. En este trabajo se muestra que una aproximación basada en el limite asintótico de la función de onda exacta para el estado final, conduce a mejores resultados independientemente de la forma del hamiltoniano de interacción utilizado a medida ´ que se incrementa la energía del fotón. Esta aproximación asintótica eikonal conduce al resultado exacto en la norma de velocidad para altas energías del fotón incidente, evidenciando la relevancia del potencial coulombiano a grandes distancias.Fil: Otranto, Sebastián. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Física del Sur. Universidad Nacional del Sur. Departamento de Física. Instituto de Física del Sur; ArgentinaFil: Gasaneo, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Física del Sur. Universidad Nacional del Sur. Departamento de Física. Instituto de Física del Sur; ArgentinaFil: Alberdi, E.D.. Universidad Nacional del Sur. Departamento de Física; Argentin

Alberdi, E.D.

Gasaneo, Gustavo

Otranto, Sebastián

Revista Mexicana de Física E

English

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

ENSEÑANZA REVISTA MEXICANA DE FÍSICA E 50 (1) 47–50 JUNIO 2004An eikonal approach for the atomic photoelectric effect on H-like atomsS. Otrantoa,b, G. Gasaneoa,b, and E.D. Alberdiba Consejo Nacional de Investigaciones Cientı́ficas y T́ecnicas, ArgentinabDepartamento de F́ısica, Universidad Nacional del Sur,8000 Bah́ıa Blanca, Buenos Aires, ArgentinaRecibido el 13 de mayo de 2003; aceptado el 10 de octubre de 2003Most quantum mechanics textbooks introduce the atomic photoelectric effect expressing the final continuum state in the high energy limit asa plane wave. This approximation has shown to give clear differences between gauges though. In this work, we show that an approximationbased on the asymptotic limit of the exact wave function for the final state leads to better results whether form of the interaction Hamiltonianis used as the photon energy increases. This asymptotic eikonal approximation leads to the exact result in velocity gauge for increasingphoton energies, evidencing the relevance of the Coulomb potential even at large distances.Keywords: Photoionization; Coulomb wave functions.Gran parte de los textos de mecánica cúantica introducen el efecto fotoeléctrico expresando la función de onda para el estado final en ellı́mite de altas energı́as como una onda plana. Sin embargo, esta aproximación conduce a resultados que dependen de la norma utilizada parael cálculo de la amplitud de transición. En este trabajo se muestra que una aproximación basada en el lı́mite asint́otico de la funcíon de ondaexacta para el estado final, conduce a mejores resultados independientemente de la forma del hamiltoniano de interacción utiliz do a medidaque se incrementa la energı́a del fot́on. Esta aproximación asint́otica eikonal conduce al resultado exacto en la norma de velocidad para altasenerǵıas del fot́on incidente, evidenciando la relevancia del potencial coulombiano a grandes distancias.Descriptores: Fotoionizacíon; funcíon de onda coulombiana.PACS: 01.40.Fk; 32.80.FbIn the last few years, there has been a notorious increase ofstudies referring many electron ionization of atoms by radi-ation fields. This is mainly consequence, of a new gener-ation of experimental devices, which now provide fully de-tailed information on the dynamic of the charged fragmentsinvolved in the collision of charged particles and photonswith atoms [1, 2]. These advances in experimental studieshave also encouraged theoreticians to improve the theoreticalmodels regularly used in atomic collisions physics to describethe many particles continuum [3–5].In spite of the conceptual advance achieved on com-plex multielectron systems, by the end of quantum mechan-ics courses the student generally does not realize the conse-quences of considering the many particles continuum underdifferent approximations.In this work we consider the atomic photoelectric effectof the hydrogen atom which is usually introduced within thesemiclassical theory of radiation. The equivalence betweenthe different representations of the interaction operator,i.e.,gauges, is usually shown using the Heisenberg equation ofmotion assuming that the exact wave functions for the initialand final states are being employed. Furthermore, the processis commonly presented in a non-relativistic high energy ap-proximation [6–8] where terms in the Hamiltonian that scaleas 1/c2 are neglected. This approximation is usually denotedas dipolar approximation and is valid fora0/λ << 1 be-ing a0 the Bohr radius andλ the photon wavelength. An-other usual approximation consists in considering the photo-electron continuum wave function as a plane wave. This ap-proximation notably simplifies the analytical work and leavesa clear qualitative insight of the differential cross sections.However, the cross sections for the photoelectric effect of hy-drogen calculated in different gauges and using plane wavesto represent the final electron wave function, lead to differ-ent results [9]. The main reason for this discrepancy is givenby the fact that the plane wave is not a solution ofH0 +V (r)(with V (r) = −Z/r) but of the free particle HamiltonianH0.In other words, since the exact initial state is considered, themain reason for the lack of agreement between gauges is con-sequence of an approximated final state wave function whichdoes not properly take account of the Coulomb potential.In this work we propose the use of the asymptotic form ofthe Coulomb wave function in order to represent the photo-electron continuum. We show that the differential cross sec-tions in both gauges lead to a better description of the processwithout incrementing the mathematical complexity. Compar-ison with the exact results obtained when the final state isrepresented with a Coulomb wave function is performed andthe advantages of the here proposed approximation are sum-marized. Atomic units will be used throughout this paper.We describe an initial bound electron in the ground stateof an hydrogenic atom by,ϕi =Z3/2√πe−Zr. (1)We perform our analysis for H-like atoms but for practicalpurposes, we consider the valueZ = 1 restricting ourselvesto the H atom. The solution of the Schrödinger equation forthe two body Coulomb problem in the continuum with in-48 S. OTRANTO, G. GASANEO, AND E.D. ALBERDIcoming boundary condition,[H0 + V (r)]Ψ(−) = EΨ(−) (2)is given by a Coulomb wave function [8,10,11],Ψ(−)C =NC(2π)3/2eik·r 1F1[iα, 1,−ikr − ik · r] (3)where1F1[a, c, z] is the Kummer function [12],NC = e−(πα/2)Γ(1− iα)is the normalization constant andα = −Zµ/k is the Som-merfeld parameter [8,10,11]. The momentum of the emittedelectron isk and the reduced mass of the two particles systemis µ. The usual plane wave approximation for the continuumstate, could be easily recovered by taking the limitα → 0.The differential cross section for the photoionization pro-cess is given by [6,7],dσdΩ=4π2kc ω|MG1s|2 (4)whereω is the photon energy,c = 137 andMG1s is the transi-tion amplitude in gaugeG. Taking into account the Heisen-berg equation of motion[H0 + V (r), r] = p we obtain thefollowing expressions for the transition amplitudes in the ve-locity and length gauges [9],MV1s = 〈Ψ(−)|eikγ ·rε̂ · 5|ϕi〉 (5)ML1s = ω〈Ψ(−)|eikγ ·rε̂ · r|ϕi〉 (6)where we have noted withkγ the momentum vector of thephoton. In the following, we work within the dipolar approx-imation, which is equivalent to considerkγ = 0 in the aboveexpressions. When the exact wave functions are used to rep-resent the initial and final states, Eq. (5) and Eq. (6) leadidentical results. This means that the obtained cross sectionsare gauge-independent. The transition amplitudes calcula-tions are performed by using Nordsieck-li e integrals whenCoulomb wave functions are used. Nordsieck-like integralsare widely used in atomic physics and nowadays are tabu-lated in several textbooks and articles [7, 11, 13–15]. How-ever, this could not be considered a straightforward calcula-tion in a quantum mechanics course and could probably ex-ceed the time deserved to the subject. The plane wave ap-proximation mainly consist on the Fourier transform of theinitial state which leads toMVPW = iε̂ ·(Z5/2√π(2π)3/2)8πk(k2 + Z2)2. (7)By the other sideMLPW is exactly twiceMVPW leading toa constant factor4 of discrepancy between the velocity andlength gauges for the differential cross section. We now con-sider the eikonal approximation which mainly concerns us.Asymptotically (for larger), the continuum wave functionEq. (3) is given by [8,12]Ψ(−)r →eik·r(2π)3/2{e−iα log(kr+k·r)+Γ(1− iα)Γ(iα)e−i(kr+k·r)(−kr − k · r)1−iα}. (8)The first term of Eq.(8) is given by a plane wave timesan eikonal distortion that explicitly shows that the Coulombpotential of the nucleus is felt by the emitted electron even atlarge distances. The second term is related to the scatteringof the photoelectron by the nucleus [8] and clearly behavesas an spherical incoming wave.As an approximated model we propose the eikonal ap-proach which retains the asymptotic distortion effect of theCoulomb potential:Ψ(−)eik =eik·r(2π)32e−iα log(kr+k·r). (9)The transition amplitude in velocity gauge for the eikonalapproximation is given byMVeik =−Z5/2ε̂·√π(2π)3/2∫dre−ik·r−Zr r̂ (kr + k · r)iα . (10)In order to simplify calculations, we now rewrite this ex-pression as follows:MVeik = − limk′→kε̂ ·(Z5/2√π(2π)3/2)kiαi×∇k′[∫dre−ik′k̂·r−Zr 1r(r + k̂·r)iα], (11)where we have consideredk′ = k′k̂. Introducing theparabolic coordinates commonly used for the two body prob-lem [8]:ξ = r + k̂·r,η = r − k̂·r,φ = arctan(y/x),where z is defined along thek axis. Taking into account thatthe volume element in these coordinates is given bydr =(ξ + η)4dξdηdφ, (12)the integral between brackets in Eq. (11) turns separable:I = π∫ ∞0∫ ∞0dξdηe−(ik′+Z) ξ2 ξiαe(ik′−Z) η2 . (13)The integrals can be easily performed and they giveI =4πΓ(1 + iα)(k′2 + Z2)(ik′ + Z2)−iα. (14)Rev. Mex. F́ıs. E 50 (1) (2004) 47–50AN EIKONAL APPROACH FOR THE ATOMIC PHOTOELECTRIC EFFECT ON H-LIKE ATOMS 49Finally, the transition amplitude in velocity gauge for theeikonal approximation is obtained:MVeik =(ε̂ · k̂) √ 2πZ5/2k−iZk Γ(1−iZk)(ik+Z2)i Zk×(2k2(ik + Z) + Z(k2 + Z2)k(k2 + Z2)2(ik + Z)). (15)Following the same line of reasoning, it could be shownthat the length gauge transition amplitude is given by,MLeik = −(ε̂ · k̂)√ 2πZ3/2ik−iZk Γ(1− iZk)×(2−1−iZk Z(ik + Z)iZk (5k2 + Z2)k2(k − iZ)(k + iZ)2)(16)Since the angular factor that modulates the transition am-plitude is given by(ε̂ · k̂)= cos θ, we could expressdσGdΩ= BG cos2 θ. (17)The factorBG does not depend on angular coordinates.In other words, it contains all the information on the electronemission probability as a function of the photon energy. InFig. 1 we showB as a function of the emitted electron en-ergy for the different models considered to represent the finalwave function for the photoelectron.Both forms (velocity and length) of the interaction Hamil-tonian are presented. It could be seen that the plane wavemodel in velocity gauge overestimates the exact result givenby the Coulomb wave function of Eq. (3) for electrons emit-ted with more than about 5.5 eV. The length gauge predictionFIGURE 1. B as a function of the emitted electron energy for thedifferent models considered. Theories are: solid-line: the Coulombwave model; dot-dashed line: eikonal wave model in velocitygauge; dashed-line: eikonal wave model in length gauge; dotted-line: plane wave model in velocity gauge; short-dashed-line: planewave model in length gauge.for the same model, increases the overestimation by an exactfactor 4, as already mentioned. The alternative approximatedeikonal model, here suggested, shows a closer agreement be-tween gauges compared to the plane wave model in the in-termediate to high energy range. In this limit, the velocitygauge also tends to the exact result given by the Coulombwave function as the electron energy increases. In the thresh-old region, both approximated models seem to fail to describethe zero energy resonance obtained with the Coulomb wavefunction. This explicitly shows that these approximated mod-els could only be considered consistent in the intermediate tohigh energy limit.In Fig. 2 we present the differential cross section for anelectron emitted withE = 250eV . It could be seen that theeikonal model in both gauges is in better agreement with theCoulomb wave than the plane wave model in velocity gauge.The total cross section for the eikonal model could be easilyobtained in both gauges by an angular integration of Eq. (17),σV,L =16π3k3c ωBV,L. (18)This total cross section represents the photoionizationemission probability as a function of the photon energy.As a conclusion, we have presented an alternative ap-proximation in order to introduce the atomic photoelectriceffect in quantum courses. We have represented the photo-electron with an asymptotically correct eikonal model. Wehave shown that this model reduces the gauges discrepanciestypical of the plane wave approximation used so far. We haveshown that in the intermediate to high energy limit this ap-proximation in velocity gauge tends to the exact value givenby the Coulomb wave. Furthermore, its implementation doesnot require further knowledge on special functions or integralrepresentations. This kind of analysis, could be helpful to es-timulate the student’s intuition on the spurius effects inherentto the use of approximated wave functions in the descriptionof physical processes.FIGURE 2. The angular distribution in atomic units forE =250eV . Results obtained with the Coulomb wave function, theplane wave and the eikonal wave are shown in both length and ve-locity gauges. Theories as in Fig. 1.Rev. Mex. F́ıs. E 50 (1) (2004) 47–5050 S. OTRANTO, G. GASANEO, AND E.D. ALBERDIAcknowledgmentsWe gratefully acknowledge Dr. Walter Cravero for fruitfuldiscussions and a careful reading of the manuscript. Thiswork is supported by PICT 99/03/06249 of the APCYT (Ar-gentina) and PGI 24/F027, Universidad Nacional del Sur (Ar-gentina).1. R. Dörneret al., Phys. Rev. A57 (1998) 1074.2. J. Ullrich et al., J. Phys. B30 (1997) 2917.3. C.R. Garibotti and J.E. Miraglia,Phys. Rev. A21 (1980) 572.4. S. Otranto and C.R. Garibotti,Eur. Phys. J.D.21 (2002) 285.5. G. Gasaneo, F.D. Colavecchia, C.R. Garibotti, J.E. Miraglia,and P. Macri,Phys. Rev. A55 (1997) 2809.6. C. Cohen-Tannoudji, B. Diu, and F. Laloë,Quantum MechanicsVol. II (John Wiley & Sons, New York, 1977).7. B.H. Bransden and C.J. Joachain,Physics of Atoms andMolecules(Longman, London, 1983).8. L. Schiff, Quantum Mechanics(McGraw-Hill, New York,1968).9. C.K. Au, Am. J. Phys50 (1982) 260.10. R. Newton, Scattering Theory of Waves and Particles(McGraw-Hill, New York, 1966).11. L.D. Landau and E.M. Lifshitz, Quantum Mechanics(Butterworth-Heinemann, Oxford, 1977).12. M. Abramowitz and I.A. Stegun,Handbook of MathematicalFunctions(Dover, New York, 1970).13. A. Nordsieck,Phys. Rev.93 (1954) 785.14. M.S. Gravielle and J.E. Miraglia,Com. Phys. Comm.69 (1992)53.15. F.D. Colavecchia, G. Gasaneo, and C.R. Garibotti,J. Math.Phys38 (1997) 6603.Rev. Mex. F́ıs. E 50 (1) (2004) 47–50

An eikonal approach for the atomic photoelectric effect on H-like atoms

CONICET Digital

ENSE ˜NANZA REVISTA MEXICANA DE F´ISICA E 50 (1) 47–50 JUNIO 2004An eikonal approach for the atomic photoelectric effect on H-like atomsS. Otrantoa,b, G. Gasaneoa,b, and E.D. Alberdiba Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, ArgentinabDepartamento de Fı´sica, Universidad Nacional del Sur,8000 Bahı´a Blanca, Buenos Aires, ArgentinaRecibido el 13 de mayo de 2003; aceptado el 10 de octubre de 2003Most quantum mechanics textbooks introduce the atomic photoelectric effect expressing the final continuum state in the high energy limit asa plane wave. This approximation has shown to give clear differences between gauges though. In this work, we show that an approximationbased on the asymptotic limit of the exact wave function for the final state leads to better results whether form of the interaction Hamiltonianis used as the photon energy increases. This asymptotic eikonal approximation leads to the exact result in velocity gauge for increasingphoton energies, evidencing the relevance of the Coulomb potential even at large distances.Keywords: Photoionization; Coulomb wave functions.Gran parte de los textos de meca´nica cua´ntica introducen el efecto fotoele´ctrico expresando la funcio´n de onda para el estado final en ellı´mite de altas energı´as como una onda plana. Sin embargo, esta aproximacio´n conduce a resultados que dependen de la norma utilizada parael ca´lculo de la amplitud de transicio´n. En este trabajo se muestra que una aproximacio´n basada en el lı´mite asinto´tico de la funcio´n de ondaexacta para el estado final, conduce a mejores resultados independientemente de la forma del hamiltoniano de interaccio´n utilizado a medidaque se incrementa la energı´a del foto´n. Esta aproximacio´n asinto´tica eikonal conduce al resultado exacto en la norma de velocidad para altasenergı´as del foto´n incidente, evidenciando la relevancia del potencial coulombiano a grandes distancias.Descriptores: Fotoionizacio´n; funcio´n de onda coulombiana.PACS: 01.40.Fk; 32.80.FbIn the last few years, there has been a notorious increase ofstudies referring many electron ionization of atoms by radi-ation fields. This is mainly consequence, of a new gener-ation of experimental devices, which now provide fully de-tailed information on the dynamic of the charged fragmentsinvolved in the collision of charged particles and photonswith atoms [1, 2]. These advances in experimental studieshave also encouraged theoreticians to improve the theoreticalmodels regularly used in atomic collisions physics to describethe many particles continuum [3–5].In spite of the conceptual advance achieved on com-plex multielectron systems, by the end of quantum mechan-ics courses the student generally does not realize the conse-quences of considering the many particles continuum underdifferent approximations.In this work we consider the atomic photoelectric effectof the hydrogen atom which is usually introduced within thesemiclassical theory of radiation. The equivalence betweenthe different representations of the interaction operator, i.e.,gauges, is usually shown using the Heisenberg equation ofmotion assuming that the exact wave functions for the initialand final states are being employed. Furthermore, the processis commonly presented in a non-relativistic high energy ap-proximation [6–8] where terms in the Hamiltonian that scaleas 1/c2 are neglected. This approximation is usually denotedas dipolar approximation and is valid for a0/λ << 1 be-ing a0 the Bohr radius and λ the photon wavelength. An-other usual approximation consists in considering the photo-electron continuum wave function as a plane wave. This ap-proximation notably simplifies the analytical work and leavesa clear qualitative insight of the differential cross sections.However, the cross sections for the photoelectric effect of hy-drogen calculated in different gauges and using plane wavesto represent the final electron wave function, lead to differ-ent results [9]. The main reason for this discrepancy is givenby the fact that the plane wave is not a solution of H0 +V (r)(with V (r) = −Z/r) but of the free particle HamiltonianH0.In other words, since the exact initial state is considered, themain reason for the lack of agreement between gauges is con-sequence of an approximated final state wave function whichdoes not properly take account of the Coulomb potential.In this work we propose the use of the asymptotic form ofthe Coulomb wave function in order to represent the photo-electron continuum. We show that the differential cross sec-tions in both gauges lead to a better description of the processwithout incrementing the mathematical complexity. Compar-ison with the exact results obtained when the final state isrepresented with a Coulomb wave function is performed andthe advantages of the here proposed approximation are sum-marized. Atomic units will be used throughout this paper.We describe an initial bound electron in the ground stateof an hydrogenic atom by,ϕi =Z3/2√pie−Zr. (1)We perform our analysis for H-like atoms but for practicalpurposes, we consider the value Z = 1 restricting ourselvesto the H atom. The solution of the Schro¨dinger equation forthe two body Coulomb problem in the continuum with in-48 S. OTRANTO, G. GASANEO, AND E.D. ALBERDIcoming boundary condition,[H0 + V (r)]Ψ(−) = EΨ(−) (2)is given by a Coulomb wave function [8, 10, 11],Ψ(−)C =NC(2pi)3/2eik·r 1F1[iα, 1,−ikr − ik · r] (3)where 1F1[a, c, z] is the Kummer function [12],NC = e−(piα/2)Γ(1− iα)is the normalization constant and α = −Zµ/k is the Som-merfeld parameter [8, 10, 11]. The momentum of the emittedelectron is k and the reduced mass of the two particles systemis µ. The usual plane wave approximation for the continuumstate, could be easily recovered by taking the limit α→ 0.The differential cross section for the photoionization pro-cess is given by [6, 7],dσdΩ=4pi2kc ω|MG1s|2 (4)where ω is the photon energy, c = 137 and MG1s is the transi-tion amplitude in gauge G. Taking into account the Heisen-berg equation of motion [H0 + V (r), r] = p we obtain thefollowing expressions for the transition amplitudes in the ve-locity and length gauges [9],MV1s = 〈Ψ(−)|eikγ ·rε̂ · 5|ϕi〉 (5)ML1s = ω〈Ψ(−)|eikγ ·rε̂ · r|ϕi〉 (6)where we have noted with kγ the momentum vector of thephoton. In the following, we work within the dipolar approx-imation, which is equivalent to consider kγ = 0 in the aboveexpressions. When the exact wave functions are used to rep-resent the initial and final states, Eq. (5) and Eq. (6) leadidentical results. This means that the obtained cross sectionsare gauge-independent. The transition amplitudes calcula-tions are performed by using Nordsieck-like integrals whenCoulomb wave functions are used. Nordsieck-like integralsare widely used in atomic physics and nowadays are tabu-lated in several textbooks and articles [7, 11, 13–15]. How-ever, this could not be considered a straightforward calcula-tion in a quantum mechanics course and could probably ex-ceed the time deserved to the subject. The plane wave ap-proximation mainly consist on the Fourier transform of theinitial state which leads toMVPW = iε̂ ·(Z5/2√pi(2pi)3/2)8pik(k2 + Z2)2. (7)By the other side MLPW is exactly twice MVPW leading toa constant factor 4 of discrepancy between the velocity andlength gauges for the differential cross section. We now con-sider the eikonal approximation which mainly concerns us.Asymptotically (for large r), the continuum wave functionEq. (3) is given by [8, 12]Ψ(−)r →eik·r(2pi)3/2{e−iα log(kr+k·r)+Γ(1− iα)Γ(iα)e−i(kr+k·r)(−kr − k · r)1−iα}. (8)The first term of Eq.(8) is given by a plane wave timesan eikonal distortion that explicitly shows that the Coulombpotential of the nucleus is felt by the emitted electron even atlarge distances. The second term is related to the scatteringof the photoelectron by the nucleus [8] and clearly behavesas an spherical incoming wave.As an approximated model we propose the eikonal ap-proach which retains the asymptotic distortion effect of theCoulomb potential:Ψ(−)eik =eik·r(2pi)32e−iα log(kr+k·r). (9)The transition amplitude in velocity gauge for the eikonalapproximation is given byMVeik =−Z5/2ε̂·√pi(2pi)3/2∫dre−ik·r−Zr r̂ (kr + k · r)iα . (10)In order to simplify calculations, we now rewrite this ex-pression as follows:MVeik = − limk′→kε̂ ·(Z5/2√pi(2pi)3/2)kiαi×∇k′[∫dre−ik′k̂·r−Zr 1r(r + k̂·r)iα], (11)where we have considered k′ = k′k̂. Introducing theparabolic coordinates commonly used for the two body prob-lem [8]:ξ = r + k̂·r,η = r − k̂·r,φ = arctan(y/x),where z is defined along the k axis. Taking into account thatthe volume element in these coordinates is given bydr =(ξ + η)4dξdηdφ, (12)the integral between brackets in Eq. (11) turns separable:I = pi∫ ∞0∫ ∞0dξdηe−(ik′+Z) ξ2 ξiαe(ik′−Z) η2 . (13)The integrals can be easily performed and they giveI =4piΓ(1 + iα)(k′2 + Z2)(ik′ + Z2)−iα. (14)Rev. Mex. Fı´s. E 50 (1) (2004) 47–50AN EIKONAL APPROACH FOR THE ATOMIC PHOTOELECTRIC EFFECT ON H-LIKE ATOMS 49Finally, the transition amplitude in velocity gauge for theeikonal approximation is obtained:MVeik =(ε̂ · k̂)√ 2piZ5/2k−iZk Γ(1−iZk)(ik+Z2)iZk×(2k2(ik + Z) + Z(k2 + Z2)k(k2 + Z2)2(ik + Z)). (15)Following the same line of reasoning, it could be shownthat the length gauge transition amplitude is given by,MLeik = −(ε̂ · k̂)√ 2piZ3/2ik−iZk Γ(1− iZk)×(2−1−iZk Z(ik + Z)iZk (5k2 + Z2)k2(k − iZ)(k + iZ)2)(16)Since the angular factor that modulates the transition am-plitude is given by(ε̂ · k̂)= cos θ, we could expressdσGdΩ= BG cos2 θ. (17)The factor BG does not depend on angular coordinates.In other words, it contains all the information on the electronemission probability as a function of the photon energy. InFig. 1 we show B as a function of the emitted electron en-ergy for the different models considered to represent the finalwave function for the photoelectron.Both forms (velocity and length) of the interaction Hamil-tonian are presented. It could be seen that the plane wavemodel in velocity gauge overestimates the exact result givenby the Coulomb wave function of Eq. (3) for electrons emit-ted with more than about 5.5 eV. The length gauge predictionFIGURE 1. B as a function of the emitted electron energy for thedifferent models considered. Theories are: solid-line: the Coulombwave model; dot-dashed line: eikonal wave model in velocitygauge; dashed-line: eikonal wave model in length gauge; dotted-line: plane wave model in velocity gauge; short-dashed-line: planewave model in length gauge.for the same model, increases the overestimation by an exactfactor 4, as already mentioned. The alternative approximatedeikonal model, here suggested, shows a closer agreement be-tween gauges compared to the plane wave model in the in-termediate to high energy range. In this limit, the velocitygauge also tends to the exact result given by the Coulombwave function as the electron energy increases. In the thresh-old region, both approximated models seem to fail to describethe zero energy resonance obtained with the Coulomb wavefunction. This explicitly shows that these approximated mod-els could only be considered consistent in the intermediate tohigh energy limit.In Fig. 2 we present the differential cross section for anelectron emitted with E = 250eV . It could be seen that theeikonal model in both gauges is in better agreement with theCoulomb wave than the plane wave model in velocity gauge.The total cross section for the eikonal model could be easilyobtained in both gauges by an angular integration of Eq. (17),σV,L =16pi3k3c ωBV,L. (18)This total cross section represents the photoionizationemission probability as a function of the photon energy.As a conclusion, we have presented an alternative ap-proximation in order to introduce the atomic photoelectriceffect in quantum courses. We have represented the photo-electron with an asymptotically correct eikonal model. Wehave shown that this model reduces the gauges discrepanciestypical of the plane wave approximation used so far. We haveshown that in the intermediate to high energy limit this ap-proximation in velocity gauge tends to the exact value givenby the Coulomb wave. Furthermore, its implementation doesnot require further knowledge on special functions or integralrepresentations. This kind of analysis, could be helpful to es-timulate the student’s intuition on the spurius effects inherentto the use of approximated wave functions in the descriptionof physical processes.FIGURE 2. The angular distribution in atomic units for E =250eV . Results obtained with the Coulomb wave function, theplane wave and the eikonal wave are shown in both length and ve-locity gauges. Theories as in Fig. 1.Rev. Mex. Fı´s. E 50 (1) (2004) 47–5050 S. OTRANTO, G. GASANEO, AND E.D. ALBERDIAcknowledgmentsWe gratefully acknowledge Dr. Walter Cravero for fruitfuldiscussions and a careful reading of the manuscript. Thiswork is supported by PICT 99/03/06249 of the APCYT (Ar-gentina) and PGI 24/F027, Universidad Nacional del Sur (Ar-gentina).1. R. Do¨rner et al., Phys. Rev. A 57 (1998) 1074.2. J. Ullrich et al., J. Phys. B 30 (1997) 2917.3. C.R. Garibotti and J.E. Miraglia, Phys. Rev. A 21 (1980) 572.4. S. Otranto and C.R. Garibotti, Eur. Phys. J.D. 21 (2002) 285.5. G. Gasaneo, F.D. Colavecchia, C.R. Garibotti, J.E. Miraglia,and P. Macri, Phys. Rev. A 55 (1997) 2809.6. C. Cohen-Tannoudji, B. Diu, and F. Laloe¨, Quantum MechanicsVol. II (John Wiley & Sons, New York, 1977).7. B.H. Bransden and C.J. Joachain, Physics of Atoms andMolecules (Longman, London, 1983).8. L. Schiff, Quantum Mechanics (McGraw-Hill, New York,1968).9. C.K. Au, Am. J. Phys 50 (1982) 260.10. R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).11. L.D. Landau and E.M. Lifshitz, Quantum Mechanics(Butterworth-Heinemann, Oxford, 1977).12. M. Abramowitz and I.A. Stegun, Handbook of MathematicalFunctions (Dover, New York, 1970).13. A. Nordsieck, Phys. Rev. 93 (1954) 785.14. M.S. Gravielle and J.E. Miraglia, Com. Phys. Comm. 69 (1992)53.15. F.D. Colavecchia, G. Gasaneo, and C.R. Garibotti, J. Math.Phys 38 (1997) 6603.Rev. Mex. Fı´s. E 50 (1) (2004) 47–50

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An eikonal approach for the atomic photoelectric effect on H-like atoms

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