the ground to the arbitrary exited state was calculated by Daugherty & Harding [20] and by Meszaros [21]. However, these QED calculations assume infinitely long-lived intermediate state and, therefore, are more relevant to photon energies far from the resonances. In order to calculate the resonant cross section one has to introduce a finite lifetime or decay width to the virtual electrons for cyclotronic transitions to lower Landau levels [22]. For the specific case of ground-state to ground-state transition in the electron rest frame, when incident photons are parallel to the B-field, Gonthier et al. [23] showed that the commonly used spin-average width of Landau levels does not correctly account for the spin dependence of the temporal decay and results in a wrong value of the cross section at the resonance as well as at very low photon energies, where the level width becomes comparable to the energy of the initial photon.Scattering from the ground Landau level is commonly used as a basic approach in case of a strong field:heB/(m e c) > k B T , where k B is the Boltzmann constant and T is the electron temperature, when the majority of electrons occupy the ground energy level [2,4,6,10,[24][25][26][27][28]. For the case of initial electron on the ground Landau level and the initial photon with momentum parallel to the magnetic field direction, the cross section has only one resonance and takes the simplest form. A simple approximation for the scattering cross section in this case was found by Gonthier et al. [29]. Their approximation represents the exact cross section quite well below the resonance and above it even for extremely strong fields (B < 10 15 G).Moving electrons scatter the photons differently because of relativistic effects. As a result, the electron distribution over momentum affects the exact cross section and broadens the resonance features. This effect could be important for formation of spectral features in X-ray pulsars [30,31] and for the estimations of radiation pressure [8-10], because the resonant scattering increases the effective interaction cross section dramatically. It is also important to use correct Landau level width and calculate correctly the exact resonant cross section here. The influence of electron distribution varies much with the photon momentum direction because electrons take part mostly in a motion along the B-field lines and the corresponding Doppler broadening varies a lot [32,33]. The scattering cross section for the case of thermal electrons was calculated and compared with cyclotron absorption by Harding & Daugherty [32]. However, only polarization-averaged cross section for the case of initial electron at rest in the ground state was explored and an incorrect width of Landau levels based on Johnson-Lippmann wave-functions [34] was taken into account (see [35] and [23] for detailed discussion).Description of additional effects such as vacuum polarization, two-photon scattering [36], pair creation [37] demands the use of high order perturbation theory. They are beyon...
