Differential and Total Cross Sections for Antiproton Impact Ionization of Atomic Hydrogen and Helium M. McGovern, 1 D. Assafr˜ ao, 2 J.R. Mohallem, 2 Colm T. Whelan, 3 and H.R.J. Walters 1 1 Department of Applied Mathematics and Theoretical Physics, Queen’s University, Belfast BT7 1NN, United Kingdom 2 Laborat´oriode ´ Atomos e Mol´ eculas Especiais, Departamento de F´ ısica, ICEx, Universidade Federal de Minas Gerais, P.O Box 702, 30123-970 Belo Horizonte, MG, Brazil. 3 Department of Physics, Old Dominion University, Norfolk, VA 23529-0116 , USA We develop a method for extracting fully differential cross sections for ionization from an impact parameter treatment of the collision. The approach uses pseudostates. The method is applied to antiproton impact ionization of H and He. It is not restricted to antiprotons but can be used with other projectiles. The method automatically includes the interaction between the projectile and the target nucleus. This interaction is shown to be very important in low energy antiproton ionization. Integrated cross sections for elastic scattering, discrete excitation, ionization and total scattering are also calculated. For ionization, these are compared with experimental data on H and He. At impact energies greater than 30 keV there is agreement with these data. At lower energies the calculated cross sections for He are in disagreement with the trend of the older experimental data of Andersen et al and Hvelplund et al but are in qualitative accord with the recent measurement of Knudsen et al. However, there is not overall quantitative agreement with the new measurements. First Born calculations are also presented as a benchmark for checking the pseudostate approximation. PACS numbers: 34.10.+x, 34.50.Fa I. INTRODUCTION The main thrust of this paper is directed towards dif- ferential cross sections for single ionization of atomic hy- drogen and helium and with particular reference to the most detailed cross section, the triple differential cross section (TDCS). We show how, within a coupled pseu- dostate formalism, the TDCS can be extracted from an impact parameter treatment. Byproducts of this work are cross sections for total ionization and discrete exci- tation of the atom. While there is not so much work on differential ionization [1–8], there is a large body of lit- erature on the total ionization cross section [4–6, 8–44]. Here we largely confirm what has gone before but, we believe, with some significant new insights. In section II we derive an approximation which is fully differential in the motion of the “heavy” projectile and the ionized electron. Here, using pertubation theory, we establish a connection between the wave treatment of projectile motion and the straight line impact parameter method (IPM). This enables us to describe the deflec- tion of the heavy projectile even though the straight line IPM seemingly does not allow such. Next we show how differential electron ejection can be obtained from the pseudostates. Combining this with the results for pro- jectile deflection we are able to construct an amplitude that is fully differential in both particles. As a useful benchmark to judge the coupled pseu- dostate results we also calculate first Born cross sections in the wave treatment (section III). By setting all cou- plings to zero except those that connect to the inital atomic state, and by decoupling the initial state from it- self, the coupled IPM equations can be run in first Born mode. Comparision with the corresponding first Born wave treatment then enables us to judge how well the pseudostates are performing in representing the ionized electron continuum as well as the validity of the IPM, at least at the first Born level. The first Born calculations also clearly define the asymptotic limit to which the cou- pled pseudostate results should converge with increasing impact energy. Our results are presented in section IV where compari- sion is made with the available experimental data on total ionization. This gives us the opportunity to assess some very recent new measurements on helium [45] which show a completely different trend to the earlier data [46, 47] at the lower impact energies. Conclusions are summarised in section V. Throughout we use atomic units (a.u.) in which  = m e = e = 1. II. THEORY We consider a bare charged particle of mass m P and charge Z P incident with velocity v 0 upon an N-electron neutral atom of nuclear mass m T which is at rest in the laboratory. It is convenient to work in the relative coordi- nate system in which the target remains at rest through- out the collision and the projectile has an effective (re- duced) mass of μ = mP (mT +N) (mP +mT +N) . We denote by R(r i ) the position vector of the projectile (ith electron) relative to the target nucleus and by r and X the collective co- ordinates r ≡ (r 1 , r 2 ,..., r N ) and X ≡ (x 1 , x 2 ,..., x N ) where x i ≡ (r i ,s i ) and s i is the spin coordinate of the ith electron. Let H A be the atomic Hamiltonian and V (r, R) the Coulombic interaction between the projectile and the target: