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Multiply differential cross sections for single ionization of He by C6+ impact
Physics Letters A 332 (2004) 60–64 www.elsevier.com/locate/pla
Multiply differential cross sections for single ionization of He by C6+ impact Ritu Dey, A.C. Roy ∗ Department of Physics, University of Kalyani, Kalyani 741235, India Received 5 July 2004; received in revised form 10 September 2004; accepted 15 September 2004 Available online 29 September 2004 Communicated by B. Fricke
Abstract We report triply and doubly differential cross sections for single ionization of helium by C6+ impact at the incident energy of 100 MeV/amu. The calculation is based on the Glauber approximation (GA) method. A comparison is made of the present calculation with the results of other theoretical methods and experiment. At low and intermediate momentum transfers the present GA results are in reasonably good agreement with experiment. 2004 Elsevier B.V. All rights reserved. PACS: 34.50.Fa
Triply differential cross section (TDCS) gives the most detailed information of a single ionization process. Whereas in electron–atom collisions TDCS measurements are available in abundance [1], these data are very scarce in ion–atom collisions [2]. Kinematically complete experiments for positively charged ion impact has been reported by Moshammer et al. [3]. Recently Schulz et al. [4] performed relative measurements for TDCS for C6+ impact ionization of helium at the incident energy of 100 MeV/amu for a variety of ejected electron energies and projectile scattering angles. They also compared the results of
* Corresponding author.
E-mail address: [email protected] (A.C. Roy). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.09.039
the theoretical three Coulomb wave function (3C) calculation, second Born calculation, classical trajectory Monte Carlo (CTMC) calculation and first Born calculation with their measured data. Both the experiment and theory were found to have the same characteristic shape with binary and recoil peaks. More recently, Madison et al. [5] reported TDCS calculations along with the absolute measurements for C6+ impact ionization of helium at 100 MeV/amu. Their calculation (3C-HF) was based on an asymptotically exact threebody final state wave function that contained all active two-particle subsystem interactions to infinite order in perturbation theory. They used a Hartree–Fock (HF) bound state wave function for the active electron in the initial state, whereas for the ejected electron in the final state they employed a wave function calculated
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
from the HF static potential for the residual helium ion. In the coplanar case, the 3C-HF results are in good agreement with experiment at small and intermediate momentum transfers, but the agreement was not satisfactory at large momentum transfer. The objective of this Letter is to apply a theoretical method to analyse the multiply differential cross sections for the single ionization of helium by C6+ impact as a function of the ejected electron angle. We plan to apply the GA method [6,7] which has been extensively applied to electron–atom [8–11] and ion–atom collisions [12–15]. In contrast to the first Born approximation (FBA), this method includes contributions from projectile–core interaction and multiple scattering effects. In fact, the Glauber amplitude contains terms of all orders in V (i.e., the sum of the projectile–core and the projectile–electron interactions) in its phase in an approximate way. Ray et al. [10] has applied the GA to calculate doubly differential cross section (DDCS) for ionization of He by electron impact. They pointed out that the cross sections predicted by the GA were in good agreement with the corresponding results obtained in the distorted-wave Born calculation of McCarthy and Zhang and experiment. Recently, they have also applied this method with success to evaluate singly differential cross sections for electronimpact ionization of He [11] and doubly differential cross sections for proton-impact ionization of helium [15]. In light of this, we extend the GA to evaluate TDCS and DDCS for C6+ –He collisions and compare our cross sections with other theoretical results and the experimental data [5]. The Glauber amplitude for the ionization of He by C6+ impact is given by (atomic units are used throughout, unless otherwise indicated) ik F (Q, k 2 ) = db dr 1 dr 2 φf∗ (r 1 , r 2 )Γ (b; r 1 , r 2 ) 2π × φi (r 1 , r 2 ) exp(iQ · b), (1) where Γ (b; r 1 , r 2 ) = 1 −
b − s1 b
2iη
b − s2 b
2iη ,
Q = k − k 1 and η = −zp µ/k. µ is the reduced mass of the projectile–atom system and zp denotes the charge of projectile. Here k, k 1 and k 2 are the momenta of the incoming C6+ , scattered projectile and
61
ejected electron, respectively, b, s 1 and s 2 are the respective projections of the position vectors of the incident particle and the two bound electrons onto the plane perpendicular to the direction of the Glauber path integration. In Eq. (1) Q, b, s 1 , and s 2 are all coplanar. φi (r 1 , r 2 ) and φf (r 1 , r 2 ) represent the wave functions of the initial and final states of the target, respectively. For the initial state of He, we have chosen the analytical fit to the Hartree–Fock wave function given by Byron and Joachain [16]: φi (r 1 , r 2 ) = U (r 1 )U (r 2 ),
(2)
where U (r) = (4π)−1/2 Ae−αr + Be−βr , A = 2.60505, B = 2.08144, α = 1.41, β = 2.61. For the final state wave function we have used a symmetrized product of the He+ ground state wave function for the bound electron times a Coulomb wave φk2 orthogonalized to the ground state orbital φf (r 1 , r 2 ) = 2−1/2 φk2 (r 1 )ν(r 2 ) + ν(r 1 )φk2 (r 2 ) , (3) where
ν(r) = (λ )3/2 π −1/2 e−λ r ,
φk2 (r) = χk−2 (r) − U (r ) χk−2 (χr ) U (r), 1 − −3/2 χk2 (r) = (2π) exp γ π (1 + iγ ) exp(ik 2 · r) 2 × 1 F 1 −iγ , 1, −i(k2r + k 2 · r) , γ = 1/k2
and λ = 2.
The triply differential cross section is given by 2 k1 k2 d 3σ = F (Q, k 2 ) , ˆ ˆ k d k 1 d k 2 dE2
(4)
where d kˆ 1 and d kˆ 2 denote, respectively, elements of solid angle for the scattered projectile and the ejected electron and dE2 represents the energy interval of the ejected electron. The doubly differential cross section as a function of the scattered projectile solid angle and the energy of the ejected electron is defined as 2 k1 k2 d 2σ = (5) d kˆ 2 d F (Q, k 2 ) . k d kˆ 1 dE2
62
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
Fig. 1. Triply differential cross section for single ionization of helium by 100 MeV/amu C6+ impact as a function of the ejected electron angle for a fixed value of electron energy E2 = 6.5 eV. The solid curve represents the present GA calculation. The short dashed curve is the present FBA result. The dashed-dot curve represents the 3C-HF calculation [5]. The solid circles are the experimental data [5].
The present calculation is performed using the technique of Roy, Das and Sil [9] that reduces the eightdimensional Glauber amplitude for the He(e, 2e)He+ process to a three-dimensional integral. Figs. 1, 2 and 3 show a comparison of present FBA and GA cross sections with the 3C-HF calculation of Madison et al. [5] and the absolute experimental data for 100 MeV/amu C6+ ionization of helium at ejection energies of 6.5, 17.5, and 37.5 eV, respectively. We see that the shapes of both the FBA and GA results are in good agreement with those of the available experimental data for momentum transfers |Q| = 0.88, 1.43 and 2.65 a.u. Furthermore, we see that the GA cross sections are in reasonably good agreement with the absolute data for the two smallest momentum transfers. However, for the largest momentum transfer and largest ejected electron energy, the present GA results
Fig. 2. Triply differential cross section for single ionization of helium for the same incident energy but E2 = 17.5 eV. The notation is the same as in Fig. 1.
are surprisingly smaller than the measured data by a factor of about 2. It is worth noting (see Fig. 3) that the 3C-HF calculation yields cross sections which are also a factor of 2 smaller than the data. Improvements in the quality of helium wave functions are not expected to resolve the big discrepancy. Insight into this problem may be gained from a straightforward numerical solution of the fundamental equations with the correct asymptotic behaviour for all regions of the configuration space for a three body Coulomb system [17]. In the case of small and intermediate momentum transfers (see Figs. 1 and 2), whereas the GA method is in better agreement with experiment than the 3C-HF method in the binary region, the latter method shows some superiority over the GA in the recoil regime. The FBA, on the other hand, is found to give an overestimate of the binary peak in all the cases studied here and is inferior to the GA. In Fig. 4 we display the present FBA and GA cross sections along with the measured DDCS data
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
Fig. 3. Triply differential cross section for single ionization of helium for the same incident energy but E2 = 37.5 eV. The notation is the same as in Fig. 1.
of Madison et al. [5] for the electron energy of 6.5 eV. Also included in this figure is the FBA-HF curve of Madison et al. which represents the limit of the 3C-HF calculation when the projectile wave function in the final state is a plane wave and the final state projectile–electron interaction is ignored. As the measured DDCS were reported as a function of the perpendicular component of momentum transfer Q⊥ , we have multiplied, for comparison, our DDCS cross sec2 tions ˆd σ given by Eq. (5) by the appropriate factor. d k 1 dE2
In the present case, since very small projectile scattering angle (∼ µ rad) is involved, and energy loss Ep is very small compared to the velocity of the incident particle vp , Q⊥ and Q are same up to two decimal places. So we have rewritten Eq. (5) (as in Ref. [5]) as follows: d 2 σ tan θ1 d 2σ = . dQ⊥ dE2 d kˆ 1 dE2 k1
(6)
63
Fig. 4. Doubly differential cross section as a function of the perpendicular component of momentum transfer |Q⊥ | for E2 = 6.5 eV. The solid curve represents the present GA calculation. The short dashed curve is the present FBA result. The dashed-dot curve represents the FBA-HF calculation [5]. The solid circles are the experimental data [5].
We find that the GA cross sections agree closely with experiment and are significantly better than the present FBA as well as the FBA-HF especially for |Q⊥ | > 3.0. The important difference between the FBA and the GA for large |Q⊥ | values has been elaborated by a number of authors [18,19]. The origin of this difference may be attributed to the projectile– core interaction term which is missing in the FBA, but present in the GA. It is worth noting that Fukuda et al. [18] observed significant improvement in their calculation of singly differential cross sections for p–He collisions on the introduction of the eikonal phase factor which contained this internuclear term. A similar improvement has also been observed by Rodriguez and Barrachina [19] who incorporated the internuclear interaction by means of an eikonal phase in their continuum-distorted-wave-eikonal-initial-state calculation of DDCS for projectile angular distribution for the single ionization of helium by proton impact.
64
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
In summary, we have applied the GA method to calculate TDCS and DDCS for C6+ impact ionization of helium at the incident energy of 100 MeV/amu. We find that the GA results are in good agreement with the absolute experimental data [5] at small and intermediate momentum transfers than the FBA results. Detailed experimental studies of the present reaction process, especially for lower incident energies would be extremely valuable for assessing the effectiveness of different theoretical models.
Acknowledgements The authors would like to thank Dr. Robert Moshammer for giving the experimental data. They are indebted to Prof. D. Madison for providing the tabular values of their 3C-HF cross sections.
References [1] H. Ehrhardt, K. Jung, G. Knoth, P. Schlemmer, Z. Phys. D 1 (1986) 3. [2] J. Ullrich, R. Moshammer, A. Dorn, R. Dorner, L.Ph.H. Schmidt, H. Schmidt-Bocking, Rep. Prog. Phys. 66 (2003) 1463.
[3] R. Moshammer, J. Ullrich, M. Unverzagt, W. Schmidt, P. Jardin, R.E. Olson, R. Mann, R. Dörner, V. Mergel, U. Buck, H. Schmidt-Böcking, Phys. Rev. Lett. 73 (1994) 3371. [4] M. Schulz, R. Moshammer, D.H. Madison, R.E. Olson, P. Marchalant, C.T. Whelan, H.R.J. Walters, S. Jones, M. Foster, H. Kollmus, A. Cassimi, J. Ullrich, J. Phys. B 34 (2001) L305. [5] D. Madison, M. Schulz, S. Jones, M. Foster, R. Moshammer, J. Ullrich, J. Phys. B 35 (2002) 3297. [6] R.J. Glauber, in: W.E. Brittin, et al. (Eds.), in: Lectures in Theoretical Physics, vol. 1, Interscience, New York, 1959, p. 315. [7] E. Gerjuoy, B.K. Thomas, Rep. Prog. Phys. 37 (1974) 1345. [8] V. Franco, Phys. Rev. A 8 (1973) 2927, and references therein. [9] A.C. Roy, A.K. Das, N.C. Sil, Phys. Rev. A 28 (1983) 181. [10] H. Ray, U. Werner, A.C. Roy, Phys. Rev. A 44 (1991) 7834. [11] A.C. Roy, R. Dey, B. Najjari, Nucl. Instrum. Methods B 197 (2002) 175. [12] V. Franco, B.K. Thomas, Phys. Rev. A 4 (1971) 945. [13] J.H. McGuire, Phys. Rev. A 26 (1982) 143, and references therein. [14] X. Fang, J.F. Reading, Nucl. Instrum. Methods B 53 (1991) 453. [15] R. Dey, A.C. Roy, Nucl. Instrum. Methods B 225 (2004) 207. [16] F.W. Byron Jr., C.J. Joachain, Phys. Rev. 146 (1966) 1. [17] A.L. Godunov, V.A. Schipakov, M. Schulz, J. Phys. B 31 (1998) 4943. [18] H. Fukuda, I. Shimamura, L. Vegh, T. Watanabe, Phys. Rev. A 44 (1991) 1565. [19] V.D. Rodriguez, R.Q. Barrachina, Phys. Rev. A 57 (1998) 215.
Multiply differential cross sections for single ionization of He by C6+ impact Ritu Dey, A.C. Roy ∗ Department of Physics, University of Kalyani, Kalyani 741235, India Received 5 July 2004; received in revised form 10 September 2004; accepted 15 September 2004 Available online 29 September 2004 Communicated by B. Fricke
Abstract We report triply and doubly differential cross sections for single ionization of helium by C6+ impact at the incident energy of 100 MeV/amu. The calculation is based on the Glauber approximation (GA) method. A comparison is made of the present calculation with the results of other theoretical methods and experiment. At low and intermediate momentum transfers the present GA results are in reasonably good agreement with experiment. 2004 Elsevier B.V. All rights reserved. PACS: 34.50.Fa
Triply differential cross section (TDCS) gives the most detailed information of a single ionization process. Whereas in electron–atom collisions TDCS measurements are available in abundance [1], these data are very scarce in ion–atom collisions [2]. Kinematically complete experiments for positively charged ion impact has been reported by Moshammer et al. [3]. Recently Schulz et al. [4] performed relative measurements for TDCS for C6+ impact ionization of helium at the incident energy of 100 MeV/amu for a variety of ejected electron energies and projectile scattering angles. They also compared the results of
* Corresponding author.
E-mail address: [email protected] (A.C. Roy). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.09.039
the theoretical three Coulomb wave function (3C) calculation, second Born calculation, classical trajectory Monte Carlo (CTMC) calculation and first Born calculation with their measured data. Both the experiment and theory were found to have the same characteristic shape with binary and recoil peaks. More recently, Madison et al. [5] reported TDCS calculations along with the absolute measurements for C6+ impact ionization of helium at 100 MeV/amu. Their calculation (3C-HF) was based on an asymptotically exact threebody final state wave function that contained all active two-particle subsystem interactions to infinite order in perturbation theory. They used a Hartree–Fock (HF) bound state wave function for the active electron in the initial state, whereas for the ejected electron in the final state they employed a wave function calculated
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
from the HF static potential for the residual helium ion. In the coplanar case, the 3C-HF results are in good agreement with experiment at small and intermediate momentum transfers, but the agreement was not satisfactory at large momentum transfer. The objective of this Letter is to apply a theoretical method to analyse the multiply differential cross sections for the single ionization of helium by C6+ impact as a function of the ejected electron angle. We plan to apply the GA method [6,7] which has been extensively applied to electron–atom [8–11] and ion–atom collisions [12–15]. In contrast to the first Born approximation (FBA), this method includes contributions from projectile–core interaction and multiple scattering effects. In fact, the Glauber amplitude contains terms of all orders in V (i.e., the sum of the projectile–core and the projectile–electron interactions) in its phase in an approximate way. Ray et al. [10] has applied the GA to calculate doubly differential cross section (DDCS) for ionization of He by electron impact. They pointed out that the cross sections predicted by the GA were in good agreement with the corresponding results obtained in the distorted-wave Born calculation of McCarthy and Zhang and experiment. Recently, they have also applied this method with success to evaluate singly differential cross sections for electronimpact ionization of He [11] and doubly differential cross sections for proton-impact ionization of helium [15]. In light of this, we extend the GA to evaluate TDCS and DDCS for C6+ –He collisions and compare our cross sections with other theoretical results and the experimental data [5]. The Glauber amplitude for the ionization of He by C6+ impact is given by (atomic units are used throughout, unless otherwise indicated) ik F (Q, k 2 ) = db dr 1 dr 2 φf∗ (r 1 , r 2 )Γ (b; r 1 , r 2 ) 2π × φi (r 1 , r 2 ) exp(iQ · b), (1) where Γ (b; r 1 , r 2 ) = 1 −
b − s1 b
2iη
b − s2 b
2iη ,
Q = k − k 1 and η = −zp µ/k. µ is the reduced mass of the projectile–atom system and zp denotes the charge of projectile. Here k, k 1 and k 2 are the momenta of the incoming C6+ , scattered projectile and
61
ejected electron, respectively, b, s 1 and s 2 are the respective projections of the position vectors of the incident particle and the two bound electrons onto the plane perpendicular to the direction of the Glauber path integration. In Eq. (1) Q, b, s 1 , and s 2 are all coplanar. φi (r 1 , r 2 ) and φf (r 1 , r 2 ) represent the wave functions of the initial and final states of the target, respectively. For the initial state of He, we have chosen the analytical fit to the Hartree–Fock wave function given by Byron and Joachain [16]: φi (r 1 , r 2 ) = U (r 1 )U (r 2 ),
(2)
where U (r) = (4π)−1/2 Ae−αr + Be−βr , A = 2.60505, B = 2.08144, α = 1.41, β = 2.61. For the final state wave function we have used a symmetrized product of the He+ ground state wave function for the bound electron times a Coulomb wave φk2 orthogonalized to the ground state orbital φf (r 1 , r 2 ) = 2−1/2 φk2 (r 1 )ν(r 2 ) + ν(r 1 )φk2 (r 2 ) , (3) where
ν(r) = (λ )3/2 π −1/2 e−λ r ,
φk2 (r) = χk−2 (r) − U (r ) χk−2 (χr ) U (r), 1 − −3/2 χk2 (r) = (2π) exp γ π (1 + iγ ) exp(ik 2 · r) 2 × 1 F 1 −iγ , 1, −i(k2r + k 2 · r) , γ = 1/k2
and λ = 2.
The triply differential cross section is given by 2 k1 k2 d 3σ = F (Q, k 2 ) , ˆ ˆ k d k 1 d k 2 dE2
(4)
where d kˆ 1 and d kˆ 2 denote, respectively, elements of solid angle for the scattered projectile and the ejected electron and dE2 represents the energy interval of the ejected electron. The doubly differential cross section as a function of the scattered projectile solid angle and the energy of the ejected electron is defined as 2 k1 k2 d 2σ = (5) d kˆ 2 d F (Q, k 2 ) . k d kˆ 1 dE2
62
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
Fig. 1. Triply differential cross section for single ionization of helium by 100 MeV/amu C6+ impact as a function of the ejected electron angle for a fixed value of electron energy E2 = 6.5 eV. The solid curve represents the present GA calculation. The short dashed curve is the present FBA result. The dashed-dot curve represents the 3C-HF calculation [5]. The solid circles are the experimental data [5].
The present calculation is performed using the technique of Roy, Das and Sil [9] that reduces the eightdimensional Glauber amplitude for the He(e, 2e)He+ process to a three-dimensional integral. Figs. 1, 2 and 3 show a comparison of present FBA and GA cross sections with the 3C-HF calculation of Madison et al. [5] and the absolute experimental data for 100 MeV/amu C6+ ionization of helium at ejection energies of 6.5, 17.5, and 37.5 eV, respectively. We see that the shapes of both the FBA and GA results are in good agreement with those of the available experimental data for momentum transfers |Q| = 0.88, 1.43 and 2.65 a.u. Furthermore, we see that the GA cross sections are in reasonably good agreement with the absolute data for the two smallest momentum transfers. However, for the largest momentum transfer and largest ejected electron energy, the present GA results
Fig. 2. Triply differential cross section for single ionization of helium for the same incident energy but E2 = 17.5 eV. The notation is the same as in Fig. 1.
are surprisingly smaller than the measured data by a factor of about 2. It is worth noting (see Fig. 3) that the 3C-HF calculation yields cross sections which are also a factor of 2 smaller than the data. Improvements in the quality of helium wave functions are not expected to resolve the big discrepancy. Insight into this problem may be gained from a straightforward numerical solution of the fundamental equations with the correct asymptotic behaviour for all regions of the configuration space for a three body Coulomb system [17]. In the case of small and intermediate momentum transfers (see Figs. 1 and 2), whereas the GA method is in better agreement with experiment than the 3C-HF method in the binary region, the latter method shows some superiority over the GA in the recoil regime. The FBA, on the other hand, is found to give an overestimate of the binary peak in all the cases studied here and is inferior to the GA. In Fig. 4 we display the present FBA and GA cross sections along with the measured DDCS data
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
Fig. 3. Triply differential cross section for single ionization of helium for the same incident energy but E2 = 37.5 eV. The notation is the same as in Fig. 1.
of Madison et al. [5] for the electron energy of 6.5 eV. Also included in this figure is the FBA-HF curve of Madison et al. which represents the limit of the 3C-HF calculation when the projectile wave function in the final state is a plane wave and the final state projectile–electron interaction is ignored. As the measured DDCS were reported as a function of the perpendicular component of momentum transfer Q⊥ , we have multiplied, for comparison, our DDCS cross sec2 tions ˆd σ given by Eq. (5) by the appropriate factor. d k 1 dE2
In the present case, since very small projectile scattering angle (∼ µ rad) is involved, and energy loss Ep is very small compared to the velocity of the incident particle vp , Q⊥ and Q are same up to two decimal places. So we have rewritten Eq. (5) (as in Ref. [5]) as follows: d 2 σ tan θ1 d 2σ = . dQ⊥ dE2 d kˆ 1 dE2 k1
(6)
63
Fig. 4. Doubly differential cross section as a function of the perpendicular component of momentum transfer |Q⊥ | for E2 = 6.5 eV. The solid curve represents the present GA calculation. The short dashed curve is the present FBA result. The dashed-dot curve represents the FBA-HF calculation [5]. The solid circles are the experimental data [5].
We find that the GA cross sections agree closely with experiment and are significantly better than the present FBA as well as the FBA-HF especially for |Q⊥ | > 3.0. The important difference between the FBA and the GA for large |Q⊥ | values has been elaborated by a number of authors [18,19]. The origin of this difference may be attributed to the projectile– core interaction term which is missing in the FBA, but present in the GA. It is worth noting that Fukuda et al. [18] observed significant improvement in their calculation of singly differential cross sections for p–He collisions on the introduction of the eikonal phase factor which contained this internuclear term. A similar improvement has also been observed by Rodriguez and Barrachina [19] who incorporated the internuclear interaction by means of an eikonal phase in their continuum-distorted-wave-eikonal-initial-state calculation of DDCS for projectile angular distribution for the single ionization of helium by proton impact.
64
R. Dey, A.C. Roy / Physics Letters A 332 (2004) 60–64
In summary, we have applied the GA method to calculate TDCS and DDCS for C6+ impact ionization of helium at the incident energy of 100 MeV/amu. We find that the GA results are in good agreement with the absolute experimental data [5] at small and intermediate momentum transfers than the FBA results. Detailed experimental studies of the present reaction process, especially for lower incident energies would be extremely valuable for assessing the effectiveness of different theoretical models.
Acknowledgements The authors would like to thank Dr. Robert Moshammer for giving the experimental data. They are indebted to Prof. D. Madison for providing the tabular values of their 3C-HF cross sections.
References [1] H. Ehrhardt, K. Jung, G. Knoth, P. Schlemmer, Z. Phys. D 1 (1986) 3. [2] J. Ullrich, R. Moshammer, A. Dorn, R. Dorner, L.Ph.H. Schmidt, H. Schmidt-Bocking, Rep. Prog. Phys. 66 (2003) 1463.
[3] R. Moshammer, J. Ullrich, M. Unverzagt, W. Schmidt, P. Jardin, R.E. Olson, R. Mann, R. Dörner, V. Mergel, U. Buck, H. Schmidt-Böcking, Phys. Rev. Lett. 73 (1994) 3371. [4] M. Schulz, R. Moshammer, D.H. Madison, R.E. Olson, P. Marchalant, C.T. Whelan, H.R.J. Walters, S. Jones, M. Foster, H. Kollmus, A. Cassimi, J. Ullrich, J. Phys. B 34 (2001) L305. [5] D. Madison, M. Schulz, S. Jones, M. Foster, R. Moshammer, J. Ullrich, J. Phys. B 35 (2002) 3297. [6] R.J. Glauber, in: W.E. Brittin, et al. (Eds.), in: Lectures in Theoretical Physics, vol. 1, Interscience, New York, 1959, p. 315. [7] E. Gerjuoy, B.K. Thomas, Rep. Prog. Phys. 37 (1974) 1345. [8] V. Franco, Phys. Rev. A 8 (1973) 2927, and references therein. [9] A.C. Roy, A.K. Das, N.C. Sil, Phys. Rev. A 28 (1983) 181. [10] H. Ray, U. Werner, A.C. Roy, Phys. Rev. A 44 (1991) 7834. [11] A.C. Roy, R. Dey, B. Najjari, Nucl. Instrum. Methods B 197 (2002) 175. [12] V. Franco, B.K. Thomas, Phys. Rev. A 4 (1971) 945. [13] J.H. McGuire, Phys. Rev. A 26 (1982) 143, and references therein. [14] X. Fang, J.F. Reading, Nucl. Instrum. Methods B 53 (1991) 453. [15] R. Dey, A.C. Roy, Nucl. Instrum. Methods B 225 (2004) 207. [16] F.W. Byron Jr., C.J. Joachain, Phys. Rev. 146 (1966) 1. [17] A.L. Godunov, V.A. Schipakov, M. Schulz, J. Phys. B 31 (1998) 4943. [18] H. Fukuda, I. Shimamura, L. Vegh, T. Watanabe, Phys. Rev. A 44 (1991) 1565. [19] V.D. Rodriguez, R.Q. Barrachina, Phys. Rev. A 57 (1998) 215.