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Comment on: “Photoionization of helium-like ions in asymptotic non-relativistic region” [Phys. Lett. A 358 (2006) 211]
Physics Letters A 367 (2007) 254–256 www.elsevier.com/locate/pla
Comment
Comment on: “Photoionization of helium-like ions in asymptotic non-relativistic region” [Phys. Lett. A 358 (2006) 211] T. Suri´c a,∗ , R.H. Pratt b a R. Boškovi´c Institute, Bijenicka 54, 10000 Zagreb, Croatia b Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
Received 28 February 2007; received in revised form 25 April 2007; accepted 27 April 2007 Available online 29 April 2007 Communicated by B. Fricke
Abstract The results of Mikhailov et al. [A.I. Mikhailov, A.V. Nefiodov, G. Plunien, Phys. Lett. A 358 (2006) 211] on single and double ionization of He at high energy can be obtained quite simply by combining two previous results long well known in the literature. The results of Mikhailov et al. can also be understood in a larger context, using an asymptotic Fourier transform approach, which also allows a justification of various statements assumed but not demonstrated by Mikhailov et al. © 2007 Elsevier B.V. All rights reserved. PACS: 32.80.Fb
It is interesting to note that the results of Mikhailov et al. [1] on single and double ionization of He at high energy can be obtained quite simply by combining two previous results long well known in the literature, (1) describing the high energy limits of photoionization in terms of the initial ground state wave function and (2) making a 1/Z expansion of that wave function. We want to point out that the results of Mikhailov et al. can also be understood in a larger context, using a more general approach based on an asymptotic Fourier transform formalism [2], which in addition allows a justification of various statements assumed but not demonstrated by Mikhailov et al. In their Letter, Mikhailov et al. first consider the process of single ionization of a He-like ion without excitation (the second electron is left in the ground state of the remaining ion), by absorption of a photon of high energy ω. The authors use a perturbative approach for the electron–electron interaction in calculating the matrix element for the process (and in addition
DOI of original article: 10.1016/j.physleta.2006.05.019. * Corresponding author.
E-mail address: [email protected] (T. Suri´c). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.04.096
the electron–photon interaction is approximated by its dipole term). They start with uncorrelated initial and final Coulombic states and, for the high energies which they consider, correctly take the e–e interaction only in the initial state, and only to first order. After a somewhat lengthy derivation, in which further simplifications are made which result in representing the final ejected electron by a plane wave, the authors arrive at the result σ1+ = σ0+ 1 + a1 Z −1 , (1) where σ0+
28 πα I 5/2 = 3mω ω
(2)
is the Born approximation total cross section for 1s ionization of a H-like ion of charge Z, I is the ionization energy, and the calculated correction for the He-like ion is the 1/Z term, with coefficient a1 −0.6676. Now it has been understood since the sixties [3–5] that the high energy limit for the total cross sections σn+ for single ionization with excitation to any state n (or without excitation, when n = 1, the case considered by Mikhailov at al.) can be described in terms of the He ground state wave function Ψ (r1 , r2 )
T. Suri´c, R.H. Pratt / Physics Letters A 367 (2007) 254–256
only, more precisely in terms of the portion of the ground state wave function Ψ (0, r2 ) when one electron is at the nucleus. The result obtained for the high energy limit of photoabsorption was [see e.g. Eq. (7) of [5], which we rewrite here in the form of Eq. (1)] σn+ = σ0+
2 π 3 Ψ (0, r )Ψ (r ) d r 2 n 2 2 , (mZα)3
(3)
where Ψn (r2 ) are H-like ns wave functions. The perturbative result of Mikhailov et al. corresponds to inserting the 1/Z expansion of the He-ground state described in Bethe and Salpeter [6], in Eq. (3) with n = 1, directly yielding Eq. (1) for σ1+ . If one would neglect initial state correlation the result for σ1+ would, of course, be just the Coulombic result, but the results for σn+ (n > 1), and for double ionization (see below), would be zero. The 1/Z expansion term gives the corrections to the leading order 1/Z result for σ1+ and the leading order 1/Z results for σn+ (n > 1) and for double ionization. These leading order 1/Z results for σn+ (n > 1) and for double ionization were obtained in [7,8]. To obtain the asymptotic double ionization total cross section σ ++ [4,5,7], where Ψn (r2 ) of Eq. (3) has been replaced by an ejected continuum electron Coulomb state, one must sum over all such states. Usually, this summation over continuum states is performed by using the completeness relation for Hydrogenic states. The double ionization total cross section is written (in terms of a sum over bound states) as (see e.g. Eqs. (3)–(6) of [4]) σ
++
2 π 3 ) d r Ψ (0, r 2 2 (mZα)3
∞ 2 3 − Ψ (0, r2 )Ψn (r2 ) d r2 .
= σ0+
(4)
n=1
Once again, the e–e correlation effects on the asymptotic double ionization total cross section σ ++ are determined by the initial state wave function only, and therefore so also is the double to single ionization ratio R discussed by Mikhailov et al. As with σn+ (for n > 2), the 1/Z correction to the ground state wave function gives only the leading order 1/Z result for σ ++ and therefore also for R. This leading order result (which Mikhailov et al. denote by R0 ) for R had been obtained in [7,9] and it is R0 = 0.090/Z 2 . Mikhailov et al. use R0 , together with their result Eq. (1) and with the experimental value of R for He (Z = 2), in order to estimate the first 1/Z correction to R0 . We note in passing that Mikhailov et al. call σ1+ the single ionization cross section and use it in calculating the double to single ionization ratio R. In the literature, however, the term single ionization cross section is normally used for σ + = n σn+ , and σ + rather than σ1+ is usually used for calculating R. Now σ1+ is about 94% of σ + in the case of the He ground state (and even more for higher Z), but it is only about 60% of σ + for H− , and it is therefore inadequate for the estimate of R in this case.
255
Now we turn to the attempted extension of the result Eq. (1), by Mikhailov et al., to lower energies. In their Letter, Mikhailov et al. simply replace the Born result σ0+ (valid at high energy in this non-relativistic case) by the full Coulombic result σC for the total cross section for photoeffect from a H-like ion (which is the same as a He-like ion, except for doubling the number of electrons, once the e–e correlation is neglected), stating that it is “more preferable” and “still legitimate”, since the full Coulombic result only slowly converges to the Born result. However, they did not demonstrate that this was correct, i.e. that the 1/Z correction terms would have the same energy dependence. This issue has in fact been studied [2,10–12]. The full answer can be obtained, as in [11,13], by using the asymptotic Fourier transform approach [2,14], which makes it possible to describe high energy photoeffect processes in terms of the behavior of the wave functions in the vicinity of singularities (in the case of single ionization, with or without excitations, this simply means in the vicinity of the nucleus). The full wave functions are Coulombic in their shape (but not necessarily in their amplitude, which we call normalization) when one electron is in the vicinity of the nucleus, neglecting terms which result in (mα/p)2 corrections to the Coulombic shape of high energy single ionization, with (or without) excitation, and double ionization total cross sections. This means that the large and slowly converging correction πmZα/p of the full Coulombic result is indeed not affected by the correlations. Indeed the full, so-called, Stobbe factor S(p) = exp(−πmZα/p) can be factored out in the case of single ionization with excitation, or for the dominant double ionization cross sections, while the remainder will have fast convergence, as (mα/p)2 , toward the high energy limit. The Stobbe factor may in part be identified as coming from Coulomb normalization, but it also has further kinematic origins [13]. The full discussion and results (given in terms of fully correlated initial state wave functions and a perturbative treatment of final state correlations), including the corrections in (mα/p)2 for single ionization, are presented in [11]. Thus, the proposal of Mikhailov et al. is correct for σ1+ , neglecting terms in (mα/p)2 , to first order in 1/Z. Mikhailov et al. conclude their Letter by commenting on the convergence of the 1/Z expansion in high energy photoionization. They suggest that higher order corrections (1/Z 2 ) both for σ1+ and for double ionization (as well as the 1/Z correction for double ionization, which is not calculated in the Letter, but only estimated by comparison with experimental results) could be calculated within their approach. As we have discussed, these results can in fact be obtained simply from the perturbative (1/Z) expansion of the ground state of He. Such expansions, as had been discussed by Mikhailov et al., have been used in obtaining the ground state energy, and they are suc− cessful even for the ground state energy of H . The calculation of Ψ (0, r2 )Ψ1 (r2 ) d 3 r2 in Eq. (3) represents the calculation of another observable which is also given simply in terms of the ground state wave function. The 1/Z expansion can in fact be rather rapidly convergent for small Z even for more general quantities, as we saw, for example, in calculation of the double to single ionization ratio in Compton scattering [15].
256
T. Suri´c, R.H. Pratt / Physics Letters A 367 (2007) 254–256
Acknowledgements This work has been supported in part by NSF Grant 0456499 and in part by MZOS Grant 098-0982931-2875. T.S. is grateful for the hospitality of the Department of Physics and Astronomy at the University of Pittsburgh during the visit when most of this work was done. References [1] A.I. Mikhailov, A.V. Nefiodov, G. Plunien, Phys. Lett. A 358 (2006) 211. [2] T. Suri´c, E.G. Drukarev, R.H. Pratt, Phys. Rev. A 67 (2003) 022709. [3] A. Dalgarno, A.L. Stewart, Proc. Phys. Soc. London 76 (1960) 49.
[4] F.W. Byron Jr., C.J. Joachain, Phys. Rev. 164 (1967) 1. [5] T. Åberg, Phys. Rev. A 2 (1970) 1726. [6] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One and Two-Electron Atoms, Springer-Verlag, Berlin, 1957, Section 27α, p. 128. [7] M.Ya. Amusia, E.G. Drukarev, V.G. Gorshkov, M.P. Kazachkov, J. Phys. B 8 (1975) 1248. [8] M.Ya. Amusia, A.I. Mikhailov, JETP 84 (1997) 474. [9] A.I. Mikhailov, I.A. Mikhailov, JETP 87 (1998) 833. [10] E.G. Drukarev, M.B. Trzhaskovskaya, J. Phys. B 31 (1998) 427. [11] T. Suri´c, R.H. Pratt, J. Phys. B 37 (2004) L93. [12] N.B. Avdonina, E.G. Drukarev, R.H. Pratt, Phys. Rev. A 65 (2002) 052705. [13] T. Suri´c, Radiat. Phys. Chem. 70 (2004) 253. [14] T. Suri´c, E.G. Drukarev, R.H. Pratt, JETP 97 (2003) 217. [15] T. Suri´c, K. Pisk, R.H. Pratt, Phys. Lett. A 211 (1996) 289.
Comment
Comment on: “Photoionization of helium-like ions in asymptotic non-relativistic region” [Phys. Lett. A 358 (2006) 211] T. Suri´c a,∗ , R.H. Pratt b a R. Boškovi´c Institute, Bijenicka 54, 10000 Zagreb, Croatia b Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
Received 28 February 2007; received in revised form 25 April 2007; accepted 27 April 2007 Available online 29 April 2007 Communicated by B. Fricke
Abstract The results of Mikhailov et al. [A.I. Mikhailov, A.V. Nefiodov, G. Plunien, Phys. Lett. A 358 (2006) 211] on single and double ionization of He at high energy can be obtained quite simply by combining two previous results long well known in the literature. The results of Mikhailov et al. can also be understood in a larger context, using an asymptotic Fourier transform approach, which also allows a justification of various statements assumed but not demonstrated by Mikhailov et al. © 2007 Elsevier B.V. All rights reserved. PACS: 32.80.Fb
It is interesting to note that the results of Mikhailov et al. [1] on single and double ionization of He at high energy can be obtained quite simply by combining two previous results long well known in the literature, (1) describing the high energy limits of photoionization in terms of the initial ground state wave function and (2) making a 1/Z expansion of that wave function. We want to point out that the results of Mikhailov et al. can also be understood in a larger context, using a more general approach based on an asymptotic Fourier transform formalism [2], which in addition allows a justification of various statements assumed but not demonstrated by Mikhailov et al. In their Letter, Mikhailov et al. first consider the process of single ionization of a He-like ion without excitation (the second electron is left in the ground state of the remaining ion), by absorption of a photon of high energy ω. The authors use a perturbative approach for the electron–electron interaction in calculating the matrix element for the process (and in addition
DOI of original article: 10.1016/j.physleta.2006.05.019. * Corresponding author.
E-mail address: [email protected] (T. Suri´c). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.04.096
the electron–photon interaction is approximated by its dipole term). They start with uncorrelated initial and final Coulombic states and, for the high energies which they consider, correctly take the e–e interaction only in the initial state, and only to first order. After a somewhat lengthy derivation, in which further simplifications are made which result in representing the final ejected electron by a plane wave, the authors arrive at the result σ1+ = σ0+ 1 + a1 Z −1 , (1) where σ0+
28 πα I 5/2 = 3mω ω
(2)
is the Born approximation total cross section for 1s ionization of a H-like ion of charge Z, I is the ionization energy, and the calculated correction for the He-like ion is the 1/Z term, with coefficient a1 −0.6676. Now it has been understood since the sixties [3–5] that the high energy limit for the total cross sections σn+ for single ionization with excitation to any state n (or without excitation, when n = 1, the case considered by Mikhailov at al.) can be described in terms of the He ground state wave function Ψ (r1 , r2 )
T. Suri´c, R.H. Pratt / Physics Letters A 367 (2007) 254–256
only, more precisely in terms of the portion of the ground state wave function Ψ (0, r2 ) when one electron is at the nucleus. The result obtained for the high energy limit of photoabsorption was [see e.g. Eq. (7) of [5], which we rewrite here in the form of Eq. (1)] σn+ = σ0+
2 π 3 Ψ (0, r )Ψ (r ) d r 2 n 2 2 , (mZα)3
(3)
where Ψn (r2 ) are H-like ns wave functions. The perturbative result of Mikhailov et al. corresponds to inserting the 1/Z expansion of the He-ground state described in Bethe and Salpeter [6], in Eq. (3) with n = 1, directly yielding Eq. (1) for σ1+ . If one would neglect initial state correlation the result for σ1+ would, of course, be just the Coulombic result, but the results for σn+ (n > 1), and for double ionization (see below), would be zero. The 1/Z expansion term gives the corrections to the leading order 1/Z result for σ1+ and the leading order 1/Z results for σn+ (n > 1) and for double ionization. These leading order 1/Z results for σn+ (n > 1) and for double ionization were obtained in [7,8]. To obtain the asymptotic double ionization total cross section σ ++ [4,5,7], where Ψn (r2 ) of Eq. (3) has been replaced by an ejected continuum electron Coulomb state, one must sum over all such states. Usually, this summation over continuum states is performed by using the completeness relation for Hydrogenic states. The double ionization total cross section is written (in terms of a sum over bound states) as (see e.g. Eqs. (3)–(6) of [4]) σ
++
2 π 3 ) d r Ψ (0, r 2 2 (mZα)3
∞ 2 3 − Ψ (0, r2 )Ψn (r2 ) d r2 .
= σ0+
(4)
n=1
Once again, the e–e correlation effects on the asymptotic double ionization total cross section σ ++ are determined by the initial state wave function only, and therefore so also is the double to single ionization ratio R discussed by Mikhailov et al. As with σn+ (for n > 2), the 1/Z correction to the ground state wave function gives only the leading order 1/Z result for σ ++ and therefore also for R. This leading order result (which Mikhailov et al. denote by R0 ) for R had been obtained in [7,9] and it is R0 = 0.090/Z 2 . Mikhailov et al. use R0 , together with their result Eq. (1) and with the experimental value of R for He (Z = 2), in order to estimate the first 1/Z correction to R0 . We note in passing that Mikhailov et al. call σ1+ the single ionization cross section and use it in calculating the double to single ionization ratio R. In the literature, however, the term single ionization cross section is normally used for σ + = n σn+ , and σ + rather than σ1+ is usually used for calculating R. Now σ1+ is about 94% of σ + in the case of the He ground state (and even more for higher Z), but it is only about 60% of σ + for H− , and it is therefore inadequate for the estimate of R in this case.
255
Now we turn to the attempted extension of the result Eq. (1), by Mikhailov et al., to lower energies. In their Letter, Mikhailov et al. simply replace the Born result σ0+ (valid at high energy in this non-relativistic case) by the full Coulombic result σC for the total cross section for photoeffect from a H-like ion (which is the same as a He-like ion, except for doubling the number of electrons, once the e–e correlation is neglected), stating that it is “more preferable” and “still legitimate”, since the full Coulombic result only slowly converges to the Born result. However, they did not demonstrate that this was correct, i.e. that the 1/Z correction terms would have the same energy dependence. This issue has in fact been studied [2,10–12]. The full answer can be obtained, as in [11,13], by using the asymptotic Fourier transform approach [2,14], which makes it possible to describe high energy photoeffect processes in terms of the behavior of the wave functions in the vicinity of singularities (in the case of single ionization, with or without excitations, this simply means in the vicinity of the nucleus). The full wave functions are Coulombic in their shape (but not necessarily in their amplitude, which we call normalization) when one electron is in the vicinity of the nucleus, neglecting terms which result in (mα/p)2 corrections to the Coulombic shape of high energy single ionization, with (or without) excitation, and double ionization total cross sections. This means that the large and slowly converging correction πmZα/p of the full Coulombic result is indeed not affected by the correlations. Indeed the full, so-called, Stobbe factor S(p) = exp(−πmZα/p) can be factored out in the case of single ionization with excitation, or for the dominant double ionization cross sections, while the remainder will have fast convergence, as (mα/p)2 , toward the high energy limit. The Stobbe factor may in part be identified as coming from Coulomb normalization, but it also has further kinematic origins [13]. The full discussion and results (given in terms of fully correlated initial state wave functions and a perturbative treatment of final state correlations), including the corrections in (mα/p)2 for single ionization, are presented in [11]. Thus, the proposal of Mikhailov et al. is correct for σ1+ , neglecting terms in (mα/p)2 , to first order in 1/Z. Mikhailov et al. conclude their Letter by commenting on the convergence of the 1/Z expansion in high energy photoionization. They suggest that higher order corrections (1/Z 2 ) both for σ1+ and for double ionization (as well as the 1/Z correction for double ionization, which is not calculated in the Letter, but only estimated by comparison with experimental results) could be calculated within their approach. As we have discussed, these results can in fact be obtained simply from the perturbative (1/Z) expansion of the ground state of He. Such expansions, as had been discussed by Mikhailov et al., have been used in obtaining the ground state energy, and they are suc− cessful even for the ground state energy of H . The calculation of Ψ (0, r2 )Ψ1 (r2 ) d 3 r2 in Eq. (3) represents the calculation of another observable which is also given simply in terms of the ground state wave function. The 1/Z expansion can in fact be rather rapidly convergent for small Z even for more general quantities, as we saw, for example, in calculation of the double to single ionization ratio in Compton scattering [15].
256
T. Suri´c, R.H. Pratt / Physics Letters A 367 (2007) 254–256
Acknowledgements This work has been supported in part by NSF Grant 0456499 and in part by MZOS Grant 098-0982931-2875. T.S. is grateful for the hospitality of the Department of Physics and Astronomy at the University of Pittsburgh during the visit when most of this work was done. References [1] A.I. Mikhailov, A.V. Nefiodov, G. Plunien, Phys. Lett. A 358 (2006) 211. [2] T. Suri´c, E.G. Drukarev, R.H. Pratt, Phys. Rev. A 67 (2003) 022709. [3] A. Dalgarno, A.L. Stewart, Proc. Phys. Soc. London 76 (1960) 49.
[4] F.W. Byron Jr., C.J. Joachain, Phys. Rev. 164 (1967) 1. [5] T. Åberg, Phys. Rev. A 2 (1970) 1726. [6] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One and Two-Electron Atoms, Springer-Verlag, Berlin, 1957, Section 27α, p. 128. [7] M.Ya. Amusia, E.G. Drukarev, V.G. Gorshkov, M.P. Kazachkov, J. Phys. B 8 (1975) 1248. [8] M.Ya. Amusia, A.I. Mikhailov, JETP 84 (1997) 474. [9] A.I. Mikhailov, I.A. Mikhailov, JETP 87 (1998) 833. [10] E.G. Drukarev, M.B. Trzhaskovskaya, J. Phys. B 31 (1998) 427. [11] T. Suri´c, R.H. Pratt, J. Phys. B 37 (2004) L93. [12] N.B. Avdonina, E.G. Drukarev, R.H. Pratt, Phys. Rev. A 65 (2002) 052705. [13] T. Suri´c, Radiat. Phys. Chem. 70 (2004) 253. [14] T. Suri´c, E.G. Drukarev, R.H. Pratt, JETP 97 (2003) 217. [15] T. Suri´c, K. Pisk, R.H. Pratt, Phys. Lett. A 211 (1996) 289.