- Email: [email protected]
Extraction of interaction parameters from near threshold positron experiments
Nuclear Instruments and Methods in Physics Research B 267 (2009) 254–256
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Extraction of interaction parameters from near threshold positron experiments P.A. Macri a,*, R.O. Barrachina b a
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, FCEyN, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, 7600 Mar del Plata, Argentina b Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, Río Negro, Argentina
a r t i c l e
i n f o
Available online 17 October 2008 PACS: 34.70.+e Keywords: Effective range theory
a b s t r a c t We analyze different methods to extract, from experiments, positron–atom interaction parameters such as the scattering length and the effective range. The near-threshold distributions of positrons in positronium formation or elastic scattering have in common an s-wave evolution in an effective polarization potential. Different generalizations of the effective range theory are used in the literature to fit the experimental data. In this work, we propose a different approach that provides a significant improvement over other methods on a much broader range of energies. We discuss the applicability of this method for extracting information on low-energy electron–atom interactions in the case of positronium formation and elastic collisions involving positrons. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The energy dependence of cross sections near the threshold of opening reaction channels has been the focus of great interest for years. Much has been learned by studying bound state energies and closely associated threshold structures in atomic and molecular physics. The study of threshold effects in atomic and molecular physics has traditionally centered on the dynamics of electron impact and electronic structure since, until very recently, only electrons had wavelengths sufficiently long for a few parameters to characterize the scattering process over an appreciable energy range. New experimental techniques which exploit positron accumulators have markedly augmented the capabilities for investigating positron and positronium interactions with matter at very lowenergies [1]. These developments are opened now to the study of antimatter interactions owing to the steady advance of techniques for controlling and detecting slow positrons [2]. Almost 60 years ago, Wigner [3] showed that the longest part of the interaction determines the reaction cross section of two particles emerging with very low-energy. Moreover, this fact is independent of what the reaction mechanism is, as long as the long-range interaction of the product particles is the same. The celebrated Wigner threshold law (WTL), establishes that the ‘-wave cross section behaves as (atomic units are used throughout unless stated otherwise)
* Corresponding author. Fax: +54 223 4753150. E-mail address: [email protected] (P.A. Macri). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.015
r‘ / k4‘ ;
ð1Þ
for elastic scattering and as
r‘ / k2‘þ1 ;
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for inelastic scattering, where k ¼ 2lðE E0 Þ is the momentum among the emergent particles, l is the reduced mass and E0 is the threshold energy. However, WTL suffers several shortcomings. WTL is valid in an unknown and frequently very small energy domain. This is a very ordinary situation in atomic and molecular physics where rapid departures from WTL have been observed when resonant states are near, in energy ranges as small as a few leV [4]. WTL, as given by Eqs. (1) and (2), is valid only for short-range potential. And finally, what is more important for the purpose of the present work, WTL does not provide a link between the cross section and the interaction among the particles beyond the threshold energy E0 . The aim of this work is to avoid these shortcomings with the WTL formulating a generalized threshold law (GTL) as 4‘
r‘ /
k
jf‘ ðkÞj2
;
ð3Þ
for elastic scattering and as
r‘ /
k
2‘þ1
jf‘ ðkÞj2
;
ð4Þ
for inelastic scattering, where we have introduced the elastic ‘-wave Jost function in the final state f‘ ðkÞ which can be defined as the r ! 0 limit of the normalized radial wave function
P.A. Macri, R.O. Barrachina / Nuclear Instruments and Methods in Physics Research B 267 (2009) 254–256
ð2‘ þ 1Þ!!w‘;k ðrÞ ðkrÞ‘þ1 =f‘ ðkÞ [5]. This GTL, establishes a clear link between a resonant structure in a multichannel collision and the low-energy elastic scattering process by the same final-state interaction. Moreover, effective range expansions for the Jost function allows to extract low-energy interaction parameters as the scattering length and the effective range. In the case of electron or positron interacting with atoms and molecules, however, the active interactions are of a long-range type and the standard effective range expansions are not valid [5]. Low-energy expansions for the phase shift d‘ ðkÞ are known for potentials with inverse power-law tails, i.e. VðrÞ / 1=rn as r ! 1, with 1 6 n 6 3 [6]. Furthermore, the important polarization potential (n ¼ 4) case was analyzed in 1961 by O’Malley et al. [7], but restricted to the leading term of 2 the K-matrix elements K ‘ ðkÞ ¼ tan g‘ ðkÞ for ‘ > 0 and up to the k order for ‘ ¼ 0. However, beyond the analysis of the K matrix elements, the knowledge of the full Jost functions is essential to reach the link between elastic and multichannel collisions. To this end, in a recent publication we have evaluated the ‘-wave Jost function for the polarization potential and calculated its effective range expan4 sion for ‘ ¼ 0 up to the order k [8]. Unfortunately, even though this expansion might be appropriate for many different applications, we showed that its validity is limited to a rather small energy range, not providing a reliable starting point for the parametrization of resonant structures in multichannel collisions. It is important to point out that Fabrikant’s approach [9] and the multichannel theory of Watanabe and Greene [10], which are commonly used in the literature to fit the experimental data [11], also represent effective 4 2 range theories of orders k and k , respectively. Thus, here we propose a different approach that provides a large improvement over other methods on a much broader range of energies. First, we cast the intricate polarization effects of the positron–atom interaction in terms of a very simple model potential
255
and
tan g‘ ¼
ðtan2 ðpn=2Þ c2 Þ þ B‘ ðkÞð1 c2 Þ tanðnp=2Þ : ð1 c2 Þ tanðpn=2Þ þ B‘ ðkÞð1 c2 tan2 ðnp=2ÞÞ
ð7Þ
In the Eqs. (6) and (7) above, A‘ ; B‘ are even functions of k which depend on the polarizability b2 and the range R. On the other hand, n and c are the characteristic exponent and quotient of the Mathieu’s equation. Thus, instead of dealing with a very complicated generalized effective range expansion with at least four parameters (to account for the first two orders of the real and the imaginary parts of the Jost function), the analytical expressions given by Eqs. (6) and (7) with only one free parameter R, can be used to fit the experimental data. Once the range R is adjusted, the other coefficients in the expansion can be easily found. For instance, the scattering length can be obtained as
ao ¼
b ð2b=RÞ cosð2b=RÞ sinð2b=RÞ : 2 sinðb=RÞ ð2b=RÞ cosðb=RÞ sinðb=RÞ
ð8Þ
2. Results
ð6Þ
We employ our method to determine the scattering length of several systems involving positrons. In the first place we consider the ab-initio calculations of the s-wave elastic scattering of positrons by hydrogen atoms [12] as the fitting data set. In Fig. 1, we show how the data are smoothly described by our Jost function calculation. The cross section rapidly deviates from the WTL and exhibits a Ramsauer–Townsend minimum around 0:2 eV. The scattering length extracted from the data is a0 ¼ 2:083 au in full agreement with a0 ¼ 2:096 au obtained by the ab-initio computations [12]. Next, we analyze s-wave ab-initio calculations of elastic scattering of protons by positronium atoms [12,13]. In Fig. 2 we can see that again the cross section departs from the WTL for very low energies and presents a Ramsauer–Townsend minimum near 0:26 eV. Our fitting method allows us to reproduce all the features depicted in the figure and extract a scattering length a0 ¼ 15:94 denouncing the presence of a low lying virtual state in the system. This is in complete agreement with the value a0 ¼ 15:89 au [12]. In Fig. 3 we show experimental data for elastic scattering of positron with argon atoms [14]. In order to fit this broad energy range data we include the full s-wave and the leading order of
Fig. 1. ‘ ¼ 0 partial contribution to the elastic cross section for positron–hydrogen scattering as a function of the incident positron energy. Triangles, ab-initio calculations of Gien [12].
Fig. 2. ‘ ¼ 0 partial contribution to the elastic cross section for proton-positronium scattering as a function of the incident positron energy. Triangles, ab-initio calculations of Gien[12]. Circles, ab-initio calculations of Humberston et al. [13].
( VðrÞ ¼
b2 =2R4 ; r < R b2 =2r4 ;
rPR
ð5Þ
where b2 is the dipole polarizability. For this potential the Jost functions f‘ ðkÞ ¼ v‘ ðkÞ ½1 i tan g‘ ðkÞ, can be analytically obtained by means of the solutions of the Mathieu’s equation as [8]
n bk A‘ ðkÞ 4 ð1 c2 Þ tanðpn=2Þ þ B‘ ðkÞð1 c2 tan2 ðpn=2ÞÞ ;
σ
σ
π
π
v‘ ðkÞ ¼
P.A. Macri, R.O. Barrachina / Nuclear Instruments and Methods in Physics Research B 267 (2009) 254–256
σ
256
the data set composed by the ab-initio calculations of Gien [12] and Humberston et al. [13]. The range where the WTL applies is almost not visible here. This can be easily interpreted in terms of Eq. (4). There must be a zero of the elastic s-wave Jost function for the elastic scattering of positronium by protons. This is in fact confirmed by our previous findings related to Fig. 2. From the positronium formation cross section we obtain a scattering length of a0 ¼ 15:94 au in perfect match with our fit for the direct elastic collisions between protons and positronium. We want to note that the present method allows to fit the computed cross sections in the whole energy range, in contrast with the many channel effective range theory fits [11] which miss most of the data with the exception of those very close to the threshold. 3. Summary
Fig. 3. Total cross section for positron–argon scattering versus the incident positron energy. Circles, experiments from Karwasz et al. [14].
In conclusion, in this work we have generalized the Wigner threshold law in such a way to connect the observed cross section with low-energy parameters which characterize the interaction. We also developed a method capable to fit scattering data accurately and to extract information of positron–atom interaction. In particular, the obtained scattering length are in excellent agreement with theoretical and experimental estimates. We also showed that is possible to fit ab-initio calculations and experimental data in large energy domains beyond the capability of standard effective range theories.
σ
π
Acknowledgements This work was partially supported by the Agencia Nacional de Promoción Cientı´fica y Tecnológica (Grants 03-12567 and 0320548), Consejo Nacional de Investigaciones Cientı´ficas y Técnicas (Grant PIP 5595) and Universidad Nacional de Cuyo (Grant 06/ C229), Argentina. References
Fig. 4. ‘ ¼ 0 partial contribution to the Ps formation cross section for positron– hydrogen scattering as a function of the incident positron energy. Triangles, abinitio calculations of Gien [12]. Circles, ab-initio calculations of Humberston et al. [13].
the p; d; f -wave Jost functions. A scattering length of a0 ¼ 5:27 au was obtained in this case, in concordance with a0 ¼ 5:30 from a polarized orbital method calculation [15]. Finally, in Fig. 4 we consider the s-wave positronium formation cross section of hydrogen by impact of positrons. We want to fit
[1] C.M. Surko, F.A. Gianturco (Eds.), New Directions in Antimatter Chemistry and Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. [2] C.M. Surko, G.F. Gribakin, S.J. Buckman, J. Phys. B 38 (2005) R57. [3] E.P. Wigner, Phys. Rev. 73 (1948) 1002. [4] Y.K. Bae, J.R. Peterson, Phys. Rev. A 32 (1985) 1917. [5] J.R. Taylor, Scattering Theory, Wiley, New York, 1972. [6] R.O. Barrachina, Nucl. Instr. and Meth. B 124 (1997) 198. [7] T.F. O’Malley, L. Spruch, L. Rosenberg, J. Math. Phys. 2 (1961) 491. [8] P.A. Macri, R.O. Barrachina, Phys. Rev. A 65 (2002) 062718. [9] I.I. Fabrikant, Opt. Spectrosc. (USSR) 53 (1982) 131. [10] S. Watanabe, C.H. Greene, Phys. Rev. A 22 (1980) 158. [11] S.J. Ward, J.C. Hunnell, J.H. Macek, Nucl. Instr. and Meth. B 192 (2002) 54. [12] T.T. Gien, Phys. Rev. A 56 (1997) 1332. [13] J.W. Humberston, P. Van Reeth, M.S.T. Watts, W.E. Meyerhof, J. Phys. B 30 (1997) 2477. [14] G.P. Karwasz, D. Pliszka, R.S. Brusa, Nucl. Instr. and Meth. B 247 (2006) 68. [15] R.P. McEachran, A.D. Stauffer, L.E.M. Campbel, J. Phys. B 12 (1979) 1031.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Extraction of interaction parameters from near threshold positron experiments P.A. Macri a,*, R.O. Barrachina b a
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, FCEyN, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, 7600 Mar del Plata, Argentina b Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, Río Negro, Argentina
a r t i c l e
i n f o
Available online 17 October 2008 PACS: 34.70.+e Keywords: Effective range theory
a b s t r a c t We analyze different methods to extract, from experiments, positron–atom interaction parameters such as the scattering length and the effective range. The near-threshold distributions of positrons in positronium formation or elastic scattering have in common an s-wave evolution in an effective polarization potential. Different generalizations of the effective range theory are used in the literature to fit the experimental data. In this work, we propose a different approach that provides a significant improvement over other methods on a much broader range of energies. We discuss the applicability of this method for extracting information on low-energy electron–atom interactions in the case of positronium formation and elastic collisions involving positrons. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The energy dependence of cross sections near the threshold of opening reaction channels has been the focus of great interest for years. Much has been learned by studying bound state energies and closely associated threshold structures in atomic and molecular physics. The study of threshold effects in atomic and molecular physics has traditionally centered on the dynamics of electron impact and electronic structure since, until very recently, only electrons had wavelengths sufficiently long for a few parameters to characterize the scattering process over an appreciable energy range. New experimental techniques which exploit positron accumulators have markedly augmented the capabilities for investigating positron and positronium interactions with matter at very lowenergies [1]. These developments are opened now to the study of antimatter interactions owing to the steady advance of techniques for controlling and detecting slow positrons [2]. Almost 60 years ago, Wigner [3] showed that the longest part of the interaction determines the reaction cross section of two particles emerging with very low-energy. Moreover, this fact is independent of what the reaction mechanism is, as long as the long-range interaction of the product particles is the same. The celebrated Wigner threshold law (WTL), establishes that the ‘-wave cross section behaves as (atomic units are used throughout unless stated otherwise)
* Corresponding author. Fax: +54 223 4753150. E-mail address: [email protected] (P.A. Macri). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.015
r‘ / k4‘ ;
ð1Þ
for elastic scattering and as
r‘ / k2‘þ1 ;
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for inelastic scattering, where k ¼ 2lðE E0 Þ is the momentum among the emergent particles, l is the reduced mass and E0 is the threshold energy. However, WTL suffers several shortcomings. WTL is valid in an unknown and frequently very small energy domain. This is a very ordinary situation in atomic and molecular physics where rapid departures from WTL have been observed when resonant states are near, in energy ranges as small as a few leV [4]. WTL, as given by Eqs. (1) and (2), is valid only for short-range potential. And finally, what is more important for the purpose of the present work, WTL does not provide a link between the cross section and the interaction among the particles beyond the threshold energy E0 . The aim of this work is to avoid these shortcomings with the WTL formulating a generalized threshold law (GTL) as 4‘
r‘ /
k
jf‘ ðkÞj2
;
ð3Þ
for elastic scattering and as
r‘ /
k
2‘þ1
jf‘ ðkÞj2
;
ð4Þ
for inelastic scattering, where we have introduced the elastic ‘-wave Jost function in the final state f‘ ðkÞ which can be defined as the r ! 0 limit of the normalized radial wave function
P.A. Macri, R.O. Barrachina / Nuclear Instruments and Methods in Physics Research B 267 (2009) 254–256
ð2‘ þ 1Þ!!w‘;k ðrÞ ðkrÞ‘þ1 =f‘ ðkÞ [5]. This GTL, establishes a clear link between a resonant structure in a multichannel collision and the low-energy elastic scattering process by the same final-state interaction. Moreover, effective range expansions for the Jost function allows to extract low-energy interaction parameters as the scattering length and the effective range. In the case of electron or positron interacting with atoms and molecules, however, the active interactions are of a long-range type and the standard effective range expansions are not valid [5]. Low-energy expansions for the phase shift d‘ ðkÞ are known for potentials with inverse power-law tails, i.e. VðrÞ / 1=rn as r ! 1, with 1 6 n 6 3 [6]. Furthermore, the important polarization potential (n ¼ 4) case was analyzed in 1961 by O’Malley et al. [7], but restricted to the leading term of 2 the K-matrix elements K ‘ ðkÞ ¼ tan g‘ ðkÞ for ‘ > 0 and up to the k order for ‘ ¼ 0. However, beyond the analysis of the K matrix elements, the knowledge of the full Jost functions is essential to reach the link between elastic and multichannel collisions. To this end, in a recent publication we have evaluated the ‘-wave Jost function for the polarization potential and calculated its effective range expan4 sion for ‘ ¼ 0 up to the order k [8]. Unfortunately, even though this expansion might be appropriate for many different applications, we showed that its validity is limited to a rather small energy range, not providing a reliable starting point for the parametrization of resonant structures in multichannel collisions. It is important to point out that Fabrikant’s approach [9] and the multichannel theory of Watanabe and Greene [10], which are commonly used in the literature to fit the experimental data [11], also represent effective 4 2 range theories of orders k and k , respectively. Thus, here we propose a different approach that provides a large improvement over other methods on a much broader range of energies. First, we cast the intricate polarization effects of the positron–atom interaction in terms of a very simple model potential
255
and
tan g‘ ¼
ðtan2 ðpn=2Þ c2 Þ þ B‘ ðkÞð1 c2 Þ tanðnp=2Þ : ð1 c2 Þ tanðpn=2Þ þ B‘ ðkÞð1 c2 tan2 ðnp=2ÞÞ
ð7Þ
In the Eqs. (6) and (7) above, A‘ ; B‘ are even functions of k which depend on the polarizability b2 and the range R. On the other hand, n and c are the characteristic exponent and quotient of the Mathieu’s equation. Thus, instead of dealing with a very complicated generalized effective range expansion with at least four parameters (to account for the first two orders of the real and the imaginary parts of the Jost function), the analytical expressions given by Eqs. (6) and (7) with only one free parameter R, can be used to fit the experimental data. Once the range R is adjusted, the other coefficients in the expansion can be easily found. For instance, the scattering length can be obtained as
ao ¼
b ð2b=RÞ cosð2b=RÞ sinð2b=RÞ : 2 sinðb=RÞ ð2b=RÞ cosðb=RÞ sinðb=RÞ
ð8Þ
2. Results
ð6Þ
We employ our method to determine the scattering length of several systems involving positrons. In the first place we consider the ab-initio calculations of the s-wave elastic scattering of positrons by hydrogen atoms [12] as the fitting data set. In Fig. 1, we show how the data are smoothly described by our Jost function calculation. The cross section rapidly deviates from the WTL and exhibits a Ramsauer–Townsend minimum around 0:2 eV. The scattering length extracted from the data is a0 ¼ 2:083 au in full agreement with a0 ¼ 2:096 au obtained by the ab-initio computations [12]. Next, we analyze s-wave ab-initio calculations of elastic scattering of protons by positronium atoms [12,13]. In Fig. 2 we can see that again the cross section departs from the WTL for very low energies and presents a Ramsauer–Townsend minimum near 0:26 eV. Our fitting method allows us to reproduce all the features depicted in the figure and extract a scattering length a0 ¼ 15:94 denouncing the presence of a low lying virtual state in the system. This is in complete agreement with the value a0 ¼ 15:89 au [12]. In Fig. 3 we show experimental data for elastic scattering of positron with argon atoms [14]. In order to fit this broad energy range data we include the full s-wave and the leading order of
Fig. 1. ‘ ¼ 0 partial contribution to the elastic cross section for positron–hydrogen scattering as a function of the incident positron energy. Triangles, ab-initio calculations of Gien [12].
Fig. 2. ‘ ¼ 0 partial contribution to the elastic cross section for proton-positronium scattering as a function of the incident positron energy. Triangles, ab-initio calculations of Gien[12]. Circles, ab-initio calculations of Humberston et al. [13].
( VðrÞ ¼
b2 =2R4 ; r < R b2 =2r4 ;
rPR
ð5Þ
where b2 is the dipole polarizability. For this potential the Jost functions f‘ ðkÞ ¼ v‘ ðkÞ ½1 i tan g‘ ðkÞ, can be analytically obtained by means of the solutions of the Mathieu’s equation as [8]
n bk A‘ ðkÞ 4 ð1 c2 Þ tanðpn=2Þ þ B‘ ðkÞð1 c2 tan2 ðpn=2ÞÞ ;
σ
σ
π
π
v‘ ðkÞ ¼
P.A. Macri, R.O. Barrachina / Nuclear Instruments and Methods in Physics Research B 267 (2009) 254–256
σ
256
the data set composed by the ab-initio calculations of Gien [12] and Humberston et al. [13]. The range where the WTL applies is almost not visible here. This can be easily interpreted in terms of Eq. (4). There must be a zero of the elastic s-wave Jost function for the elastic scattering of positronium by protons. This is in fact confirmed by our previous findings related to Fig. 2. From the positronium formation cross section we obtain a scattering length of a0 ¼ 15:94 au in perfect match with our fit for the direct elastic collisions between protons and positronium. We want to note that the present method allows to fit the computed cross sections in the whole energy range, in contrast with the many channel effective range theory fits [11] which miss most of the data with the exception of those very close to the threshold. 3. Summary
Fig. 3. Total cross section for positron–argon scattering versus the incident positron energy. Circles, experiments from Karwasz et al. [14].
In conclusion, in this work we have generalized the Wigner threshold law in such a way to connect the observed cross section with low-energy parameters which characterize the interaction. We also developed a method capable to fit scattering data accurately and to extract information of positron–atom interaction. In particular, the obtained scattering length are in excellent agreement with theoretical and experimental estimates. We also showed that is possible to fit ab-initio calculations and experimental data in large energy domains beyond the capability of standard effective range theories.
σ
π
Acknowledgements This work was partially supported by the Agencia Nacional de Promoción Cientı´fica y Tecnológica (Grants 03-12567 and 0320548), Consejo Nacional de Investigaciones Cientı´ficas y Técnicas (Grant PIP 5595) and Universidad Nacional de Cuyo (Grant 06/ C229), Argentina. References
Fig. 4. ‘ ¼ 0 partial contribution to the Ps formation cross section for positron– hydrogen scattering as a function of the incident positron energy. Triangles, abinitio calculations of Gien [12]. Circles, ab-initio calculations of Humberston et al. [13].
the p; d; f -wave Jost functions. A scattering length of a0 ¼ 5:27 au was obtained in this case, in concordance with a0 ¼ 5:30 from a polarized orbital method calculation [15]. Finally, in Fig. 4 we consider the s-wave positronium formation cross section of hydrogen by impact of positrons. We want to fit
[1] C.M. Surko, F.A. Gianturco (Eds.), New Directions in Antimatter Chemistry and Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. [2] C.M. Surko, G.F. Gribakin, S.J. Buckman, J. Phys. B 38 (2005) R57. [3] E.P. Wigner, Phys. Rev. 73 (1948) 1002. [4] Y.K. Bae, J.R. Peterson, Phys. Rev. A 32 (1985) 1917. [5] J.R. Taylor, Scattering Theory, Wiley, New York, 1972. [6] R.O. Barrachina, Nucl. Instr. and Meth. B 124 (1997) 198. [7] T.F. O’Malley, L. Spruch, L. Rosenberg, J. Math. Phys. 2 (1961) 491. [8] P.A. Macri, R.O. Barrachina, Phys. Rev. A 65 (2002) 062718. [9] I.I. Fabrikant, Opt. Spectrosc. (USSR) 53 (1982) 131. [10] S. Watanabe, C.H. Greene, Phys. Rev. A 22 (1980) 158. [11] S.J. Ward, J.C. Hunnell, J.H. Macek, Nucl. Instr. and Meth. B 192 (2002) 54. [12] T.T. Gien, Phys. Rev. A 56 (1997) 1332. [13] J.W. Humberston, P. Van Reeth, M.S.T. Watts, W.E. Meyerhof, J. Phys. B 30 (1997) 2477. [14] G.P. Karwasz, D. Pliszka, R.S. Brusa, Nucl. Instr. and Meth. B 247 (2006) 68. [15] R.P. McEachran, A.D. Stauffer, L.E.M. Campbel, J. Phys. B 12 (1979) 1031.