The synchrotron and cyclo-synchrotron absorption cross-section

We emphasize the usefulness of the concept of a cross-section for the synchrotron and the cyclo-synchrotron absorption process. We show that the previously known relation between the cross-section and the single particle emissivity is most easily derived by applying the Einstein coefficients and their relations to a system with three energy levels. Using this relation and known emissivities we derive useful analytical expressions for the synchrotron and cyclotron absorption cross-section. As a side result we show that, although the angle averaged synchrotron cross-section is well known to be positive, the non-averaged cross-section is negative for photons whose direction of motion is outside the characteristic emission cone of angular width 1/γ.