- Email: [email protected]
Final-state interaction description of the autoionization of atoms by ion impact in the velocity matching regime
Nuclear Instruments and Methods in Physics Research B 205 (2003) 539–542 www.elsevier.com/locate/nimb
Final-state interaction description of the autoionization of atoms by ion impact in the velocity matching regime R. Della Picca, R.O. Barrachina
*
Centro At omico Bariloche and Instituto Balseiro, S.C. de Bariloche, 8400 Rıo Negro, Argentina
Abstract The line profile of the electron emitted in the autoionization of an atom by ion impact is strongly affected by the post-collision interaction (PCI) with the outgoing projectile. Here we analyze how these PCI effects are modified when the projectile velocity is in the close vicinity of the resonant electron velocity. In this energy range, the analysis of the resonance contribution is hindered by the characteristic Ôelectron capture to the continuumÕ divergence in the direct term. Here we present a theoretical study of the interplay between both contributions, based on a generalization of the final-state interaction model. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 34.50.Fa Keywords: Ionization; Autoionization; Final-state interaction
The electron spontaneously emitted from an atom in an autoionizing state has a sharply defined energy and a characteristic angular distribution. However, when this process is induced by the impact of a charged ion, its spectrum is broadened and shifted in energy due to the post-collision interaction (PCI) of the emitted electron and the outgoing projectile [1–3]. In 1989, Swenson et al. [4] experimentally observed that PCI can also produce a sharp enhancement of the line profile in the forward direction. This effect was theoretically described by means of a continuum distorted wave (CDW) approximation of the electron final state [5,6], and later ascribed to a forward Glory
*
Corresponding author. Tel.: +54-2944-44-5163; fax: +542944-44-5102. E-mail address: [email protected] (R.O. Barrachina).
mechanism [7]. Other effects, as the ‘‘state-tostate’’ interference between overlapping resonances [8] and the interference of the resonant and ‘‘direct’’ ionization amplitudes, can give rise to strongly modified profiles [9–12]. Recently, a careful experimental investigation of this latter effect has been carried out [13]. However, by keeping the electron observation angle away from the forward direction, the velocity matching condition, i.e. when the velocity of the emitted electron ve is near that of the outgoing projectile v, was carefully avoided. This matching velocity regime was experimentally explored by Vıkor et al. [14] for the Auger emission in a H þ He collision, but no detailed analysis of the line shapes was performed. The aim of the present work is to analyze a situation where the projectile velocity is in the close vicinity of the autoionization line. In this energy range, which increases linearly with the
0168-583X/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)02061-X
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R. Della Picca, R.O. Barrachina / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 539–542
projectile charge, the analysis of the resonance contribution is hindered by the characteristic electron capture to the continuum (ECC) divergence [15,16] in the direct term. Here we study the interplay between both contributions by means of a generalization of the final-state interaction (FSI) model [17]. Finally we propose a modified parameterization of the autoionization line shape in the vicinity of the ECC cusp. We consider the single ionization of a neutral atom by the impact of a projectile of mass MP and velocity v. The momentum distribution of the emitted electron reads [18] Z dr ^N ; ¼ ð2pÞ4 v1 jTj2 dk dk where the modulus of the momentum kN , which describes the motion of the projectile P relative to the (recoil) target ion T of mass MT , is given by conservation of energy. For the case of a single autoionizing resonance, we separate the matrix element T in a direct and a resonant contribution, namely T ¼ Tdir þ Tres [12]. Both amplitudes can be affected by strong post-collisional interactions. In fact, the ejected electron travels in the twocenter field of the residual target and the receding projectile. The three-body final continuum state may be decomposed in the following way [7]: PT W ¼ DkT ;KT ðrT ; RT ÞWeT kT ðrT ÞWkN ðrN Þ K T R T k N rN exp i ; h eT
PT
are two-body continuum where W and W wavefunctions for the e þ T and P þ T systems, respectively, while the D factor incorporates the distortion due to the electron–projectile interaction. Jacobi coordinates and momenta, ðrT ; kT Þ and ðRT ; KT Þ, describe the motion of the electron relative to the particle T and the motion of P relative to the center of mass of T þ m, respectively. In the direct process, the FSI theory indicates that the corresponding transition matrix element can be written in the following way: Tdir ¼ Fdir f T dir [17], where Fdir is given by the distortion factor evaluated at the ‘‘coalescence region’’, namely Fdir ðkP Þ ¼ D ð0; 0Þ. This factor, which is related to the s-wave Jost function for the electron–projectile system, only depends on the modulus of the cor-
responding relative momentum kP . On the other hand, f T dir can be approximated by the transition matrix element for the direct process in the absence of that two-body interaction in the final state. Similarly, we separate the resonant amplitude in a reduced amplitude f T res , that accounts for the excitation to the autoionizing state by the projectile-target collision, and a multiplicative factor Fres describing the PCI between the projectile and the emitted electron. Let us assume that the projectile is sufficiently fast and heavy so that its velocity remains close to its initial value v all along the collision. In this case the distortion factor Fres can also be related to the D factor in a first order perturbation treatment [6]. We write Z 1 vx Fres i D kT 0; C=2 0 Z 2i x vy exp ð1 ieÞx VP dy dx; C 1 C=2 where VP ðrP Þ is the electron–projectile interaction and e ¼ ðkT2 =mT 2E0 Þ=C, with mT ¼ mMT = ðm þ MT Þ. E0 ¼ mT v20 =2 and 1=C are the resonant energy and characteristic lifetime of the autoionizing state of the target. With these two factorizations, the momentum distribution of the emitted electron reads 4 Z dr ð2pÞ 2 ^ ¼ jFdir ðkP Þ f T dir þ Fres ðkT ; vÞ f T res j dk N: dk v Now, it can be shown [18] that for any energetic collision, whenever the mass of the projectile is much larger than that of the emitted electron, it is possible to neglect the dependence on the momentum transfer of kT and kP in both distortion factors. As a consequence, the momentum k of the electron can be identified either with the Jacobi momentum kT , or with kP þ mv, and the previous cross section reads dr 2 dr 2 dr
jFdir ðkP Þj þ jFres ðk; vÞj dk dk dir dk res dr þ Real Fdir ðkP ÞFres ðk; vÞ ; ð1Þ dk int with kP ¼ jk mvj and obvious definitions for the direct (dir), resonant (res) and interference (int) reduced cross sections dr=dkj . The contribution
R. Della Picca, R.O. Barrachina / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 539–542 2
of the distortion factor jFdir ðkP Þj in Eq. (1) is relatively smooth over the range where the autoionization line shape is significant, whenever the projectile velocity remains far from this region [13,19]. Otherwise, its dominant structure obscures any analysis of the interplay between the direct and resonant contributions. Furthermore, since the extent of the autoionization line is proportional to the strength of the PCI, this effect is more significant for highly charged projectiles. It seems clear that this important shortcoming has to be solved before the details of the line shape can be theoretically analyzed or extracted from experimental results in this particular region of the velocity space. The key element to a solution relies in the fact that the PCI distortion of the direct term Fdir ðkP Þ can be extracted from Fres ðk; vÞ, namely, Fres ðk; vÞ ¼ Fdir ðjk mvjÞ Kðk; vÞ, where Kðk; vÞ does not diverge or vanish in the matching regime [6]. This result allows us to factorize jFdir ðkP Þj2 from the cross section Eq. (1), dr dr 2 dr 2
jFdir ðkP Þj jKðk; vÞj þ dk dk dir dk res dr þ Real Kðk; vÞ ; dk int 2
where ðdr=dkÞ=jFdir ðkP Þj is free of the ECC structure. This does not only provide an effective
541
method for the experimental analysis of the autoionization line shape in the vicinity of the ECC cusp, but also for the theoretical study of the interplay between the direct and resonant contributions through the PCI reduced distortion factor KðkÞ. In comparison, a factorization which assumes that F dir ðkP Þ dr=dkjint is independent of the electron energy [13] is adequate for electron observation angles away from the forward direction, but clearly fails in the vicinity of the ECC structure. Furthermore, note that this ECC structure is not an exclusive attribute of the direct term, but also appears in the resonant and interference contributions. Therefore, the simple substraction from the experimental data of a direct ionization background does not represent a sensible procedure to single out the autoionization lines in the forward direction. Up to this point, no assumption has been made regarding the interactions between the intervening particles in the final state. Now, we assume that the projectile is a positively charged ion of charge ZP . Different approximations of the factor D lead to different expressions for Fdir and K. For instance, let us consider a CDW approximation [20,21] of the final state of the electron, D Cð1 þ imP Þ expðpmP =2Þ1 F1 ðimP ; 1; ixÞ with x ¼ kP rP þ kP rP and mP ¼ ZP =kP . 1 F1 ða; b; zÞ is the regular confluent hypergeometric functions, conditionally
Fig. 1. CDW approximation of the distortion factor jKðk; vÞj2 , Eq. (2), in the vicinity of k ¼ mv for the autoionization of helium excited to a ð2s2 Þ1 S state by the impact of ions of velocity v ¼ 1 a.u. and charges ZP ¼ 1 and 6.
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R. Della Picca, R.O. Barrachina / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 539–542
convergent on the unit circle [22]. Within this ap2 proximation, we get jFdir ðkP Þj ¼ ð2pmP Þ= ð1 expð2pmP ÞÞ, and [5,6,23] Cð1 þ imÞ in Kðk; vÞ ¼ i F imP ; 1 þ im; 1; ; 1þim 2 1 1 ie ð1 ieÞ ð2Þ where we have defined m ¼ ZP =v and n ¼ ðkP v kP vÞ=ðC=2Þ. For a small absolute value of the argument in=ð1 ieÞ (and fixed mP ) the amplitude Kðk; vÞ reduces to K0 ðm; eÞ iCð1 þ imÞ ð1 ieÞ1im which is similar to the amplitude obtained by Devdariani et al. [2]. On the other hand, for large jn=ð1 ieÞj (and fixed mP ) van der Straten and MorgensternÕs amplitude Fdir K
nimP K0 ðm mP ; eÞ is obtained [3]. However, none of these approximations is valid in the vicinity of the ECC cusp. In this region the argument of the hypergeometric vanishes while mP diverges simultaneously. As a result, the distortion factor displays a strong anisotropy given by K
K0 ðm; eÞ1 F1 ð1 þ im; 1; mP n=ð1 ieÞÞ. This factor is shown in Fig. 1 for the autoionization of helium 1 excited to a ð2s2 Þ S state by the impact of ions of velocity v ¼ 1 a.u. and charges ZP ¼ 1 and 6. It adds up to a similar but opposite asymmetry observed in the direct term dr=dkjdir [24]. The effect of the nearby autoionization line is clear for increasing projectile charges. Eventually, this asymmetry switches towards the backward direction for v0 v < ZP C=2v0 , reinforcing the effect of the direct term.
References [1] R.B. Barker, H.W. Berry, Phys. Rev. 151 (1966) 14. [2] A.Z. Devdariani, V.N. Ostrovskii, Yu.N. Sebayakin, Sov. Phys. JETP 46 (1977) 215. [3] P. van der Straten, R. Morgenstern, J. Phys. B 19 (1986) 1361. [4] J.K. Swenson, C.C. Havener, N. Stolterfoht, K. Sommer, F.W. Meyer, Phys. Rev. Lett. 63 (1989) 35. [5] M. Yu Kuchiev, S.A. Sheinerman, J. Phys. B 21 (1988) 2027. [6] R.O. Barrachina, J.H. Macek, J. Phys. B 22 (1989) 2151. [7] I. Samengo, R.O. Barrachina, J. Phys. B 29 (1996) 4179. [8] I.L. Cordrey, J.H. Macek, Phys. Rev. A 48 (1993) 1264. [9] U. Fano, Phys. Rev. 124 (1961) 1866. [10] B.W. Shore, Rev. Mod. Phys. 39 (1967) 439. [11] V.V. Balashov, S.S. Lipovetskii, V.S. Senashenko, Sov. Phys. JETP 36 (1973) 858. [12] A.L. Godunov, Sh.D. Kunikeev, N.V. Novikov, V.S. Senashenko, Sov. Phys. JETP 69 (1989) 927. [13] P. Moretto-Capelle, M. Benhenni, D. Bordenave-Montesquieu, A. Bordenave-Montesquieu, J. Phys. B 29 (1996) 2007. . K€ [14] Gy. Vıkor, S. Ricz, B. Sulik, L. T oth, A over, J. Vegh, J. Phys. B 29 (1996) L787. [15] G.B. Crooks, M.E. Rudd, Phys. Rev. Lett. 25 (1970) 1599. [16] J. Macek, Phys. Rev. A 1 (1970) 235. [17] R.O. Barrachina, Nucl. Instr. and Meth. B 124 (1997) 198. [18] J. Fiol, R.O. Barrachina, V.D. Rodrıguez, J. Phys. B 35 (2002) 149. [19] S.A. Sheinerman, Nucl. Instr. and Meth. B 86 (1994) 105. [20] I.M. Cheshire, Proc. Phys. Soc. 84 (1964) 89. [21] Dz. Belkic, R. Gayet, A. Salin, Phys. Rep. 56 (1979) 279. [22] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970, p. 504. [23] Sh.D. Kunikeev, V.S. Senashenko, Sov. Phys. JETP 82 (1996) 839. [24] R.G. Pregliasco, C.R. Garibotti, R.O. Barrachina, J. Phys. B 27 (1994) 1151.
Final-state interaction description of the autoionization of atoms by ion impact in the velocity matching regime R. Della Picca, R.O. Barrachina
*
Centro At omico Bariloche and Instituto Balseiro, S.C. de Bariloche, 8400 Rıo Negro, Argentina
Abstract The line profile of the electron emitted in the autoionization of an atom by ion impact is strongly affected by the post-collision interaction (PCI) with the outgoing projectile. Here we analyze how these PCI effects are modified when the projectile velocity is in the close vicinity of the resonant electron velocity. In this energy range, the analysis of the resonance contribution is hindered by the characteristic Ôelectron capture to the continuumÕ divergence in the direct term. Here we present a theoretical study of the interplay between both contributions, based on a generalization of the final-state interaction model. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 34.50.Fa Keywords: Ionization; Autoionization; Final-state interaction
The electron spontaneously emitted from an atom in an autoionizing state has a sharply defined energy and a characteristic angular distribution. However, when this process is induced by the impact of a charged ion, its spectrum is broadened and shifted in energy due to the post-collision interaction (PCI) of the emitted electron and the outgoing projectile [1–3]. In 1989, Swenson et al. [4] experimentally observed that PCI can also produce a sharp enhancement of the line profile in the forward direction. This effect was theoretically described by means of a continuum distorted wave (CDW) approximation of the electron final state [5,6], and later ascribed to a forward Glory
*
Corresponding author. Tel.: +54-2944-44-5163; fax: +542944-44-5102. E-mail address: [email protected] (R.O. Barrachina).
mechanism [7]. Other effects, as the ‘‘state-tostate’’ interference between overlapping resonances [8] and the interference of the resonant and ‘‘direct’’ ionization amplitudes, can give rise to strongly modified profiles [9–12]. Recently, a careful experimental investigation of this latter effect has been carried out [13]. However, by keeping the electron observation angle away from the forward direction, the velocity matching condition, i.e. when the velocity of the emitted electron ve is near that of the outgoing projectile v, was carefully avoided. This matching velocity regime was experimentally explored by Vıkor et al. [14] for the Auger emission in a H þ He collision, but no detailed analysis of the line shapes was performed. The aim of the present work is to analyze a situation where the projectile velocity is in the close vicinity of the autoionization line. In this energy range, which increases linearly with the
0168-583X/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)02061-X
540
R. Della Picca, R.O. Barrachina / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 539–542
projectile charge, the analysis of the resonance contribution is hindered by the characteristic electron capture to the continuum (ECC) divergence [15,16] in the direct term. Here we study the interplay between both contributions by means of a generalization of the final-state interaction (FSI) model [17]. Finally we propose a modified parameterization of the autoionization line shape in the vicinity of the ECC cusp. We consider the single ionization of a neutral atom by the impact of a projectile of mass MP and velocity v. The momentum distribution of the emitted electron reads [18] Z dr ^N ; ¼ ð2pÞ4 v1 jTj2 dk dk where the modulus of the momentum kN , which describes the motion of the projectile P relative to the (recoil) target ion T of mass MT , is given by conservation of energy. For the case of a single autoionizing resonance, we separate the matrix element T in a direct and a resonant contribution, namely T ¼ Tdir þ Tres [12]. Both amplitudes can be affected by strong post-collisional interactions. In fact, the ejected electron travels in the twocenter field of the residual target and the receding projectile. The three-body final continuum state may be decomposed in the following way [7]: PT W ¼ DkT ;KT ðrT ; RT ÞWeT kT ðrT ÞWkN ðrN Þ K T R T k N rN exp i ; h eT
PT
are two-body continuum where W and W wavefunctions for the e þ T and P þ T systems, respectively, while the D factor incorporates the distortion due to the electron–projectile interaction. Jacobi coordinates and momenta, ðrT ; kT Þ and ðRT ; KT Þ, describe the motion of the electron relative to the particle T and the motion of P relative to the center of mass of T þ m, respectively. In the direct process, the FSI theory indicates that the corresponding transition matrix element can be written in the following way: Tdir ¼ Fdir f T dir [17], where Fdir is given by the distortion factor evaluated at the ‘‘coalescence region’’, namely Fdir ðkP Þ ¼ D ð0; 0Þ. This factor, which is related to the s-wave Jost function for the electron–projectile system, only depends on the modulus of the cor-
responding relative momentum kP . On the other hand, f T dir can be approximated by the transition matrix element for the direct process in the absence of that two-body interaction in the final state. Similarly, we separate the resonant amplitude in a reduced amplitude f T res , that accounts for the excitation to the autoionizing state by the projectile-target collision, and a multiplicative factor Fres describing the PCI between the projectile and the emitted electron. Let us assume that the projectile is sufficiently fast and heavy so that its velocity remains close to its initial value v all along the collision. In this case the distortion factor Fres can also be related to the D factor in a first order perturbation treatment [6]. We write Z 1 vx Fres i D kT 0; C=2 0 Z 2i x vy exp ð1 ieÞx VP dy dx; C 1 C=2 where VP ðrP Þ is the electron–projectile interaction and e ¼ ðkT2 =mT 2E0 Þ=C, with mT ¼ mMT = ðm þ MT Þ. E0 ¼ mT v20 =2 and 1=C are the resonant energy and characteristic lifetime of the autoionizing state of the target. With these two factorizations, the momentum distribution of the emitted electron reads 4 Z dr ð2pÞ 2 ^ ¼ jFdir ðkP Þ f T dir þ Fres ðkT ; vÞ f T res j dk N: dk v Now, it can be shown [18] that for any energetic collision, whenever the mass of the projectile is much larger than that of the emitted electron, it is possible to neglect the dependence on the momentum transfer of kT and kP in both distortion factors. As a consequence, the momentum k of the electron can be identified either with the Jacobi momentum kT , or with kP þ mv, and the previous cross section reads dr 2 dr 2 dr
jFdir ðkP Þj þ jFres ðk; vÞj dk dk dir dk res dr þ Real Fdir ðkP ÞFres ðk; vÞ ; ð1Þ dk int with kP ¼ jk mvj and obvious definitions for the direct (dir), resonant (res) and interference (int) reduced cross sections dr=dkj . The contribution
R. Della Picca, R.O. Barrachina / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 539–542 2
of the distortion factor jFdir ðkP Þj in Eq. (1) is relatively smooth over the range where the autoionization line shape is significant, whenever the projectile velocity remains far from this region [13,19]. Otherwise, its dominant structure obscures any analysis of the interplay between the direct and resonant contributions. Furthermore, since the extent of the autoionization line is proportional to the strength of the PCI, this effect is more significant for highly charged projectiles. It seems clear that this important shortcoming has to be solved before the details of the line shape can be theoretically analyzed or extracted from experimental results in this particular region of the velocity space. The key element to a solution relies in the fact that the PCI distortion of the direct term Fdir ðkP Þ can be extracted from Fres ðk; vÞ, namely, Fres ðk; vÞ ¼ Fdir ðjk mvjÞ Kðk; vÞ, where Kðk; vÞ does not diverge or vanish in the matching regime [6]. This result allows us to factorize jFdir ðkP Þj2 from the cross section Eq. (1), dr dr 2 dr 2
jFdir ðkP Þj jKðk; vÞj þ dk dk dir dk res dr þ Real Kðk; vÞ ; dk int 2
where ðdr=dkÞ=jFdir ðkP Þj is free of the ECC structure. This does not only provide an effective
541
method for the experimental analysis of the autoionization line shape in the vicinity of the ECC cusp, but also for the theoretical study of the interplay between the direct and resonant contributions through the PCI reduced distortion factor KðkÞ. In comparison, a factorization which assumes that F dir ðkP Þ dr=dkjint is independent of the electron energy [13] is adequate for electron observation angles away from the forward direction, but clearly fails in the vicinity of the ECC structure. Furthermore, note that this ECC structure is not an exclusive attribute of the direct term, but also appears in the resonant and interference contributions. Therefore, the simple substraction from the experimental data of a direct ionization background does not represent a sensible procedure to single out the autoionization lines in the forward direction. Up to this point, no assumption has been made regarding the interactions between the intervening particles in the final state. Now, we assume that the projectile is a positively charged ion of charge ZP . Different approximations of the factor D lead to different expressions for Fdir and K. For instance, let us consider a CDW approximation [20,21] of the final state of the electron, D Cð1 þ imP Þ expðpmP =2Þ1 F1 ðimP ; 1; ixÞ with x ¼ kP rP þ kP rP and mP ¼ ZP =kP . 1 F1 ða; b; zÞ is the regular confluent hypergeometric functions, conditionally
Fig. 1. CDW approximation of the distortion factor jKðk; vÞj2 , Eq. (2), in the vicinity of k ¼ mv for the autoionization of helium excited to a ð2s2 Þ1 S state by the impact of ions of velocity v ¼ 1 a.u. and charges ZP ¼ 1 and 6.
542
R. Della Picca, R.O. Barrachina / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 539–542
convergent on the unit circle [22]. Within this ap2 proximation, we get jFdir ðkP Þj ¼ ð2pmP Þ= ð1 expð2pmP ÞÞ, and [5,6,23] Cð1 þ imÞ in Kðk; vÞ ¼ i F imP ; 1 þ im; 1; ; 1þim 2 1 1 ie ð1 ieÞ ð2Þ where we have defined m ¼ ZP =v and n ¼ ðkP v kP vÞ=ðC=2Þ. For a small absolute value of the argument in=ð1 ieÞ (and fixed mP ) the amplitude Kðk; vÞ reduces to K0 ðm; eÞ iCð1 þ imÞ ð1 ieÞ1im which is similar to the amplitude obtained by Devdariani et al. [2]. On the other hand, for large jn=ð1 ieÞj (and fixed mP ) van der Straten and MorgensternÕs amplitude Fdir K
nimP K0 ðm mP ; eÞ is obtained [3]. However, none of these approximations is valid in the vicinity of the ECC cusp. In this region the argument of the hypergeometric vanishes while mP diverges simultaneously. As a result, the distortion factor displays a strong anisotropy given by K
K0 ðm; eÞ1 F1 ð1 þ im; 1; mP n=ð1 ieÞÞ. This factor is shown in Fig. 1 for the autoionization of helium 1 excited to a ð2s2 Þ S state by the impact of ions of velocity v ¼ 1 a.u. and charges ZP ¼ 1 and 6. It adds up to a similar but opposite asymmetry observed in the direct term dr=dkjdir [24]. The effect of the nearby autoionization line is clear for increasing projectile charges. Eventually, this asymmetry switches towards the backward direction for v0 v < ZP C=2v0 , reinforcing the effect of the direct term.
References [1] R.B. Barker, H.W. Berry, Phys. Rev. 151 (1966) 14. [2] A.Z. Devdariani, V.N. Ostrovskii, Yu.N. Sebayakin, Sov. Phys. JETP 46 (1977) 215. [3] P. van der Straten, R. Morgenstern, J. Phys. B 19 (1986) 1361. [4] J.K. Swenson, C.C. Havener, N. Stolterfoht, K. Sommer, F.W. Meyer, Phys. Rev. Lett. 63 (1989) 35. [5] M. Yu Kuchiev, S.A. Sheinerman, J. Phys. B 21 (1988) 2027. [6] R.O. Barrachina, J.H. Macek, J. Phys. B 22 (1989) 2151. [7] I. Samengo, R.O. Barrachina, J. Phys. B 29 (1996) 4179. [8] I.L. Cordrey, J.H. Macek, Phys. Rev. A 48 (1993) 1264. [9] U. Fano, Phys. Rev. 124 (1961) 1866. [10] B.W. Shore, Rev. Mod. Phys. 39 (1967) 439. [11] V.V. Balashov, S.S. Lipovetskii, V.S. Senashenko, Sov. Phys. JETP 36 (1973) 858. [12] A.L. Godunov, Sh.D. Kunikeev, N.V. Novikov, V.S. Senashenko, Sov. Phys. JETP 69 (1989) 927. [13] P. Moretto-Capelle, M. Benhenni, D. Bordenave-Montesquieu, A. Bordenave-Montesquieu, J. Phys. B 29 (1996) 2007. . K€ [14] Gy. Vıkor, S. Ricz, B. Sulik, L. T oth, A over, J. Vegh, J. Phys. B 29 (1996) L787. [15] G.B. Crooks, M.E. Rudd, Phys. Rev. Lett. 25 (1970) 1599. [16] J. Macek, Phys. Rev. A 1 (1970) 235. [17] R.O. Barrachina, Nucl. Instr. and Meth. B 124 (1997) 198. [18] J. Fiol, R.O. Barrachina, V.D. Rodrıguez, J. Phys. B 35 (2002) 149. [19] S.A. Sheinerman, Nucl. Instr. and Meth. B 86 (1994) 105. [20] I.M. Cheshire, Proc. Phys. Soc. 84 (1964) 89. [21] Dz. Belkic, R. Gayet, A. Salin, Phys. Rep. 56 (1979) 279. [22] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970, p. 504. [23] Sh.D. Kunikeev, V.S. Senashenko, Sov. Phys. JETP 82 (1996) 839. [24] R.G. Pregliasco, C.R. Garibotti, R.O. Barrachina, J. Phys. B 27 (1994) 1151.