**Figure 2.** (a) TDCS for the ionization of atomic hydrogen by 250 eV electron impact for θ_{s} = 5° as a function of the ejected electron angle θ_{e} relative to the incident electron direction. The ejected electron energy is *E _{e}* = 5 eV. The results of the first Born approximation are represented by a dotted line, those of the second Born approximation by a full curve, those of the BBK model by a dashed line and experiments by squares. (b) Same as (a), but the results of the first Born approximation are represented by a dotted line, those of the second Born approximation calculated by including only the contribution of excited states by a dashed line, those of the second Born approximation calculated by including only the continuum states by a dash–dotted line and those of the second Born approximation calculated by including all the contributions by a full curve.

**Abstract**

The second Born approximation is often used, particularly when we study double processes such as the ionization–excitation and the double ionization of atoms and molecules by charged particles. But when we apply this approximation, it needs the knowledge of all excited states of the target. In this study, we apply the second Born approximation by using 294 excited and pseudo-states for the ionization of atomic hydrogen by electrons. We compare the results of our model with those given by other models and to all experiments performed with an incident energy of 250 eV. We show that our new version of the second Born approximation gives better agreement than previous versions even for high values of the energy of the ejected electrons (50 eV).