Cross-sections of relativistic quantum-mechanical versus those of classical magnetic resonant scattering

Abstract

Context. Radiative transfer calculations in strong (few ×1012 G) magnetic fields, which are observed in X-ray pulsars, require accurate differential cross-sections of resonant scattering. While such cross-sections exist, their application is cumbersome. Aims. Here, we compare the classical (non-relativistic) with the quantum-mechanical (relativistic) resonant differential scattering cross-sections and offer a prescription for the use of the much simpler classical expressions with impressively accurate results. Methods. We expanded the quantum-mechanical differential cross-sections and kept the terms up to the first order in ϵ ≡ E/mec2 and B ≡ ℬ/ℬcr, where E is the photon energy and ℬcr is the critical magnetic field. We recovered the classical differential cross-sections along with the terms that are due to spin flip, which is a pure quantum-mechanical phenomenon. Results. When adding the spin-flip terms to the polarization-dependent classical differential cross-sections by hand, we find that they are in excellent agreement with the quantum mechanical ones for all energies near resonance and all angles. We plotted both of them and the agreement is impressive. Conclusions. We give a prescription for the use of the classical differential cross-sections for radiative transfer calculations that guarantees accurate results.