In some cosmic ray sources, since the magnetic field is so intense (see the right figure), the electron-magnetic field interaction will enter the QED regime. In this case the syncrotron radiation must be modified by QED, just like the Klein–Nishina effect. We studied the photon-pair cascade in intense magnetic field or photon field, and proivde simple formulae for the cascaded photon spectrum by fitting the Monte-Carlor results.
Ref: Wang et al. 2018, PRD, https://link.aps.org/doi/10.1103/PhysRevD.97.103016
A cascade means that: electrons produce high-energy gamma-ray photons via syncrotron or inverse Compton raidation, while the photons can again interact with the magnetic field or photon field to produce electron-positron pairs, and these secodaries can interact with the field again (see the bottom-left figure for a schematic view). We use Monte-Carlo (MC) simulation to obtain the specral energy distribution (SED) of the photons in such cascades. Surprisingly, we find the SEDs can be well fitted by simple formulae for the photon-pair cascade in magnetic field or thermal photon field (see the bottom-right figure for an example of the SED from synchrotron-pair cascade).