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The experimental determination of the lifetime of pionium provides a veryimportant test on chiral perturbation theory. This quantity is determined inthe DIRAC experiment at CERN. In the analysis of this experiment, the breakupprobabilities of of pionium in matter are needed to high accuracy as atheoretical input. We study in detail the influence of the target electrons.They contribute through screening and incoherent effects. We use Dirac-Hartree-Fock-Slater wavefunctions in order to determine the corresponding form factors.We find that the inner-shell electrons contribute less than the weakly boundouter electrons. Furthermore, we establish a more rigorous estimate for themagnitude of the contributions form the transverse current (magnetic terms thusfar neglected in the calculations).Comment: Journal of Physics B: Atomic, Molecular and Optical Physics;  (accepted; 22 pages, 6 figures, 26 references) Revised version: more detailed  description of DIRAC experiment; failure of simplest models for incoherent  scattering demonstrated by example

Coherent and incoherent atomic scattering: formalism and application to pionium interacting with matter