  We examine the process of Compton scattering, in the presence of a Lorentz-
and CPT-violating modification to the structure of the electron. We calculate
the complete tree-level contribution to the cross section; our result is valid
to all orders in the Lorentz-violating parameter. We find a cross section that
differs qualitatively from the Klein-Nishina result at small frequencies, and
we also encounter a previously undescribed complication that will arise in the
calculation of many Lorentz-violation cross sections: The Lorentz violation
breaks the spin degeneracy of the external states, so we cannot use a closure
relation to calculate the unpolarized cross section.
