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

¹ University of Crete, Department of Physics & Institute of Theoretical & Computational Physics, 70013 Heraklion, Greece
e-mail: kylafis@physics.uoc.gr
² Institute of Astrophysics, Foundation for Research and Technology-Hellas, 71110 Heraklion, Crete, Greece
³ Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, 85741 Garching, Germany

Received: 26 August 2020
Accepted: 27 July 2021

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/m_ec² and B ≡ ℬ/ℬ_cr, where E is the photon energy and ℬ_cr 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.