Triply differential cross sections for single ionization of He by proton impact

Physics Letters A 353 (2006) 341–344 www.elsevier.com/locate/pla

Triply differential cross sections for single ionization of He by proton impact Ritu Dey ∗ , A.C. Roy Department of Physics, University of Kalyani, Kalyani 741235, India Received 28 October 2005; received in revised form 27 December 2005; accepted 2 January 2006 Available online 6 January 2006 Communicated by B. Fricke

Abstract We report triply differential cross sections for single ionization of helium by proton impact at the incident energy of 75 keV. The calculation is based on the Glauber approximation (GA) method. The effect of post collision interaction has been taken into account approximately. 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.  2006 Elsevier B.V. All rights reserved. PACS: 34.50.Fa

Collisions of ions with atoms represent one of the fundamental problems studied in atomic physics research. Of the many interesting phenomena which can occur in such collisions ionization is one where a single target electron or more are finally unbound. In the case of single ionization process triply differential cross section (TDCS) gives the most detailed information. Whereas in electron–atom collisions TDCS measurements are available in abundance [1], these data are very scarce in ion–atom collisions [2]. Kinematically complete experiment for positively charged ion impact has been first reported by Moshammer et al. [3]. Madison et al. [4] reported TDCS calculations along with the absolute measurements for C6+ impact ionization of helium at 100 MeV/amu for a variety of ejected electron energies and projectile scattering angles. Their calculation (3C-HF) was based on an asymptotically exact three-body 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 wavefunction for the active electron in the initial state, whereas for the ejected electron in the final state they employed a wave function calculated from the HF static potential for the residual helium ion. In the coplanar case, the 3C-HF results are in good agreement with

* Corresponding author.

E-mail address: [email protected] (R. Dey). 0375-9601/$ – see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.002

experiment at small and intermediate momentum transfers, but the agreement was not satisfactory at large momentum transfer. Dey et al. [5] applied the Glauber approximation (GA) method [6–8] to the above reaction process and found that at low and intermediate momentum transfers the GA method was in better agreement with experiment than the 3C-HF method in the binary region while the latter method showed some superiority over the GA in the recoil regime. This GA method has been extensively applied to calculate both the total and differential cross sections for single ionization of atoms by electron [9–13] and ion [14–18] impact. The method is not only successful for the evaluation of total cross sections [8], but it has also been found to yield differential cross sections in reasonable agreement with experiment in many cases. For example, Ray et al. [11] have applied the GA to calculate TDCS for electron impact ionization of helium for a coplanar asymmetric geometry at incident energies of 256 and 600 eV. The angular distributions of the ejected electrons predicted by the GA were in reasonably good agreement with the absolute experimental data. Again Ray et al. [12] have applied the GA to analyse the projectile angular distribution for electron–He collisions and pointed out that the doubly differential cross section predicted by the GA were in good agreement with the corresponding results obtained in the distorted wave Born calculation of McCarthy and Zhang [19] and experiment. The GA has also been found to be successful in predicting projectile angular distribution for proton–He

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collision [17]. Furthermore, Igarashi et al. [16] and Fang and Reading [15] have applied the GA to ion–atom collisions and reported SDCS for the single ionization of He by proton impact. The cross sections predicted by the GA in all the above cases were in reasonably good agreement with experiment. Very recently, Maydanyuk et al. [20] measured fully differential single ionization cross sections for 75 keV for a lighter projectile (proton), while the target was still He as before. They compared their data with the three-body distorted-wave-eikonal initial state (3DW-EIS) [21] and FBA calculations. They report that the experimental features are not well reproduced by their 3DW-EIS calculation at this low incident energy. The magnitudes of cross sections obtained in the 3DW-EIS calculations differed considerably from the experimental data, e.g., by a factor of 0.2, 0.25 and 0.6 at q = 0.67, 0.77 and 0.97 a.u., respectively. It has been pointed out that at this low velocity case (vp = 1.73 a.u.) an accurate treatment of higher order interactions is necessary in order to achieve a quantitative agreement with the measured data. The objective of this Letter is to apply the GA method to analyse the triply differential cross sections for the single ionization of helium by proton impact at the incident energy of 75 keV as a function of the ejected electron angle. 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. The Glauber amplitude for the ionization of He by proton impact is given by (atomic units are used throughout, unless otherwise indicated) [5,16]
ik db dr 1 dr 2 φf∗ (r 1 , r 2 )Γ (b; r 1 , r 2 ) F (q, k 2 ) = 2π × φi (r 1 , r 2 ) exp(iq . b), (1) where



Γ (b; r 1 , r 2 ) = 1 −

|b − s 1 | b

2iη 

|b − s 2 | b

2iη ,

z µ

q = k − k 1 and η = − pk . 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 proton, scattered projectile and ejected electron, respectively. In Eq. (1) q denotes the momentum transfer, b is the impact parameter vector. r 1 and r 2 are the position vectors of the electrons and s 1 and s 2 their projections onto the b plane. 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 [22]: φi (r 1 , r 2 ) = U (r 1 )U (r 2 ), where

 1 U (r) = (4π)− 2 Ae−αr + Be−βr , A = 2.60505, α = 1.41,

B = 2.08144, β = 2.61.

(2)

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 φk 2 orthogonalized to the ground state orbital  1 φf (r 1 , r 2 ) = 2− 2 φk 2 (r 1 )ν(r 2 ) + ν(r 1 )φk 2 (r 2 ) , (3) where 3

1



ν(r) = (λ ) 2 π − 2 e−λ r , 

φk 2 (r) = χk−2 (r) − U (r  ) χk−2 (r  ) U (r),  3 1 χk−2 (r) = (2π)− 2 exp γ π Γ (1 + iγ ) exp(ik 2 . r) 2   × 1 F1 −iγ , 1, −i(k2 r + 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 ) , k d kˆ 1 d kˆ 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 present TDCS calculation is performed using the technique of Roy, Das and Sil [10] that reduces the eight dimensional Glauber amplitude for the He(e, 2e)He+ process to a three-dimensional integral. Fig. 1 shows a comparison of present FBA and GA cross sections with the corresponding 3DW-EIS and FBA calculation of Maydanyuk et al. [20] and the absolute experimental TDCS data for 75 keV proton impact ionization of helium at the ejection energy of 5.5 eV. The present comparison is made for (i) q = 0.67 (θq = 18◦ ), (ii) q = 0.77 (θq = 34◦ ) and (iii) q = 0.97 (θq = 49◦ ) where θq is the direction of q. We notice that there is a significant difference between the present FBA and the Born calculation of Maydanyuk et al. [20]. It is worth mentioning that in the present FBA model, the initial target state is chosen to be the analytical fit to the Hartree–Fock wave function given by Byron and Joachain (Eq. (2)) whereas for the final state we have used a symmetrized product of the He+ ground state wave function for the bound electron times a Coulomb wave orthogonalized to the ground state orbital (Eq. (3)). The interaction potential is the sum of projectile–nucleus and projectile–electron interactions. In contrast to this, the FBA model of Maydanyuk et al. [20] considers a Hartree–Fock initial state and the projectile– target core interaction. Fig. 1 also reveals that the magnitudes of GA cross sections are in reasonably good agreement with the absolute data for the two smallest momentum transfers. However, for the largest momentum transfer (q = 0.97), the present GA results are smaller than the measured data by a factor of about 2 in the binary peak region. It is worth noting (see Fig. 1) that the 3DW-EIS cross sections overestimate the data in the same region for all the three momentum transfers. In the case of small and intermediate momentum transfers the GA results are in better agreement with experiment than the corresponding 3DW-EIS cross sections, especially in the binary region. One of the reasons for the success of present GA is due to the fact that it

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Fig. 1. Triply differential cross section for single ionization of helium by 75 keV proton impact as a function of the ejected electron angle for a fixed value of electron energy E2 = 5.5 eV. The momentum transfers are 0.67 a.u. (bottom), 0.77 a.u. (center), and 0.97 a.u. (top). The arrow and the vertical dotted line indicate the experimental binary peak maximum and the direction θq of q, respectively. The solid curve represents the present GA calculation. The dotted curve is the present FBA result. The dashed-dot and the dashed curves represent, respectively, the 3DW-EIS calculation and FBA calculations of Maydanyuk et al. [20]. The solid circles are the experimental data [20].

contains the interaction between the projectile and residual target ion in the transition operator while the 3DW-EIS includes this interaction asymptotically via a Coulomb correction factor. The FBA, on the other hand, is found to give an overestimate of the binary and recoil peak in the cases of intermediate momentum transfers (q = 0.67 and 0.77). The good agreement of FBA with experiment at q = 0.97 seems to be fortuitous. All the theoretical calculations, however, fail to reproduce the special structure observed in the measured data at q = 0.67 at in the neighbourhood of binary peak. The reasons for the discrepancy between present theory and experiment at q = 0.67 at the ejection angle θe < 0 still remains unexplained. Attempts may be made to apply a non-perturbative method to explain the existence of the above-mentioned feature. Both the present GA and FBA methods predict binary peaks which are located along the direction of the momentum transfer. The experimental peaks are, however, seen to be slightly shifted. One of the reasons for this deviation is attributed to the post collision interaction (PCI) between the three Coulomb particles in the final state. In order to incorporate this effect in the present GA, we have followed the method of Salin [23] where the PCI is included by multiplying the transition amplitude by a Coulomb factor proportional to |ve − vp |−1 where ve and vp are the electron and proton velocities respectively. Fig. 2 shows a shape comparison of the present PCI modified Glauber approximation (GA-PCI), GA and 3DW-EIS calculations along with the measured data. As we are interested in a comparative study

Table 1 Comparison of the angular position of the binary peak maximum, as given by different theories q (a.u.)

GA (deg)

GA-PCI (deg)

3DW-EIS (deg)

Expt. (deg)

0.67 0.77 0.97

18 34 49

14 29 44

15 30 45

25 35 45

of angular distributions (Fig. 2), we have multiplied all the theoretical results by appropriate factors to give the experimental binary peak height. We find that at q = 0.97 the binary peak position predicted by the GA and 3DW-EIS methods are in good agreement with experiment, whereas at lower q where the PCI effect is expected to be more pronounced the agreement especially for the lower momentum transfer is far from satisfactory. It may be noted that despite the entirely different methods used for the description of PCI in the GA and 3DW-EIS calculations, both the theoretical methods predict nearly the same angular position of the binary peak maximum (see Table 1). It seems that a more sophisticated description of the post collision interaction is needed for a quantitative agreement of theories and experiment. In summary, we have applied the GA method to calculate TDCS for proton impact ionization of helium at the incident energy of 75 keV for a fixed ejection electron energy 5.5 eV. We find that the GA cross sections are in better agreement with the

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Fig. 2. Triply differential cross section for single ionization of helium by 75 keV proton impact as a function of the ejected electron angle for a fixed value of electron energy E2 = 5.5 eV. The momentum transfers are 0.67 a.u. (bottom), 0.77 a.u. (center), and 0.97 a.u. (top). The arrow and the vertical dotted line indicate the experimental binary peak maximum and the direction θq of q, respectively. The solid curve represents the present GA calculation with PCI. The dashed curve is the present GA result. The dashed-dot curve represents the 3DW-EIS calculation [20] and the solid circles are the experimental data [20]. All the theoretical cross sections have been multiplied by appropriate factors to give the binary peak height.

absolute experimental data [20] especially at small and intermediate momentum transfers (q = 0.67 and 0.77) than the FBA and 3DW-EIS results. However, at large momentum transfer (q = 0.97), whereas the GA underestimates the experimental data in the vicinity of binary peak the 3DW-EIS method yield cross sections which overestimate the data in the above region. Acknowledgements The authors would like to thank Prof. Michael Schulz for providing their experimental and theoretical data. References [1] H. Ehrhardt, K. Jung, G. Knoth, P. Schlemmer, Z. Phys. D 1 (1986) 3. [2] J. Ullrich, R. Moshammer, A. Dorn, R. Dörner, L.Ph.H. Schmidt, H. Schmidt-Böcking, 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] D. Madison, M. Schulz, S. Jones, M. Foster, R. Moshammer, J. Ullrich, J. Phys. B 35 (2002) 3297.

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