tokens: is more interesting. Even when the initial photon energy lies well below the first Landau resonance, the scattering cross section spikes (but does not diverge) at a value of ω f (and therefore ω i ) somewhat larger than 2 m . This spike arises from the pole in the matrix element associated with annihilation of the initial electron with a virtual positron into the final-state photon (Herold 1979 ; Daugherty Harding 1986 ). It appears at a lower photon frequency than the first Landau resonance (energy ) if B > 4 B Q . Figure 3. Differential nonresonant electron scattering cross section vs. initial photon frequency ω i (top panels) and final photon direction cosine μ f (bottom panels), in the regime B ≫ B Q . Curves in the top panels represent . Black curves: μ f > 0; blue curve: μ f = 0; red curves: μ f 0. The cross section rises as the scattered photon approaches the threshold for single-photon pair creation. In the bottom panels, the blue curve marks the classical limit, and black and red curves are used for clarity. Dotted curve: the final-state photon rapidly converts to an electron–positron pair. The partial cross section for scattering into non-pair-creating states is shown in Figure 4 . The high-energy behavior of the total cross section is opposite to the Klein–Nishina result for scattering in vacuum: the cross section grows as ω rises above m . Nonetheless, scattering is still suppressed for photons that have a large energy owing to relativistic motion of the scattering charge along B (Lorentz factor γ ≫ 1), because the factor decreases as ∼1/ γ 2 . Figure 4. Integral nonresonant electron scattering cross section plotted vs. initial photon frequency ω i for a range of initial direction cosine μ i , in the regime B ≫ B Q . The high-frequency peak marks the opening up of pair conversion in the final state (see Figure 6 ). The high-frequency dip in the curves shown in Figure 4 represents the opening up of the final-state phase space to pair creation over some range of scattering angles. A scattered photon can convert directly to an electron–positron pair if , meaning that the cross section for scattering-assisted pair creation, e ± + γ → e ± + e + + e − , can substantially exceed the vacuum value. This phenomenon is examined in more detail in Section 4 . More generally, a high-frequency Klein–Nishina scaling for the scattering cross section—which is approached in the case of scattering at high-order Landau resonances in sub-QED magnetic fields (Gonthier et al. 2000 )—loses meaning as B rises above 4 B Q , because the scattered photon has a high probability of converting to a pair (Beloborodov Thompson 2007 ). 3.2. Finite-<italic>B</italic> Correction to the Cross Section A simple correction to the electron–photon scattering cross section representing a finite magnetic field is available. The overlap of the photon wave function (wavevector k μ ) with a pair of harmonic oscillator wave functions, such as appears in the scattering matrix,
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