  Accurate modelling of the interaction between fast electrons and partially
ionized atoms is important for evaluating tokamak disruption mitigation schemes
based on material injection. This requires accounting for the effect of
screening of the impurity nuclei by the cloud of bound electrons. In this
paper, we detail the derivation of a generalized collision operator including
the effect of partial screening. We calculate the effective ion length-scales,
which are needed in the components of the collision operator, for a number of
ion species commonly appearing in fusion experiments. We show that for high
electric fields, the secondary runaway growth rate can be substantially larger
than in a fully ionized plasma with the same effective charge, although the
growth rate is significantly reduced at near-critical electric fields.
Furthermore, by comparison with the Boltzmann collision operator, we show that
the Fokker--Planck formalism is valid even for large impurity content.
