Z. Physik A 283, 133 - 138 (1977) Zeitschrift for Physik A 9 by Springer-Verlag 1977 Glauber Amplitude for Ionization of Helium by Electron Impact Man Mohan and Thakur Vidhani Department of Physics and Astrophysics, University of Delhi, Delhi, India Received May 16, 1977 The high-energy Glauber approximation is used to derive the formula for scattering amplitude for ionization of helium atom by electron impact. The scattering amplitude is expressed as a one dimensional integral involving Meijer G-functions. This may further be expressed in a closed form as simple sums of Meijer functions by writing down the series expansion for Bessel and Meijer functions. Further, the asymptotic behaviour of these amplitudes is examined for both large and small momentum transfers. I. Introduction Glauber approximation [1] has been quite successful in high energy limit in studying various types of interactions. It is useful [2] in atomic scattering calculations where complete solutions are not known. Recently this theory has been successfully applied to various excitation and ionization processes in hy- drogen and hydrogen-like atoms [34]. Excitation of helium at various energies of incident electrons has been studied in greater detail [10-17]. Recently the Glauber cross-sections for ionization of helium has also been studied [18, 19]. For these collisions the Glauber approximation has been shown to be more useful than Born approximation for estimating differ- ential and integrated (over scattering angle) cross- sections. The experimental calculations for ionization of helium atom have been made by Schram et al. [20]. The motivation for using the Glauber approximation rather than Born approximation is that in this ap- proximation, the interaction of incident particle with the target nucleus is also taken into account. Further, the Glauber approximation also includes some mul- tiple scattering and is unitary in the high energy limit. However, at very high energies the Glauber ampli- tude should reduce to Born amplitude. In this paper we calculate the Glauber scattering amplitude for the ionization of helium by the impact of ordinary charged particles. The salient feature of our approach is that we reduce the e+He Glauber amplitude to a one dimensional integral involving Meijer G functions [21, 22]. This integral may be expressed in a closed form as simple sums of Meijer G-functions. Therefore, no numerical integration is required to calculate the scattering amplitude. In Sections 2 and 3 we give the theory for the ionization of helium atom and derive the expressions for the scattering amplitude in the form of a one dimensional integral involving Meijer functions. Finally, in Sec- tion 4 we discuss our results for both large and small momentum transfers. 2. Theory As the helium atom has two electrons and a nucleus therefore the interaction of incident electron with the helium is a four body problem. The Schr6dinger equation for such a four body process can be written (in the atomic units i.e., h = e = m = i) as: [_Iv12 g g l 1 1 1 1 , V~- V~ rl ra r3 1 1 1 -I + ~ + ~ + ~ ] 0 =e0 (1) where the mass of the nucleus is much greater than the electron mass. The above equation can be written as