  Radiative transfer calculations in strong (few $\times 10^{12}$ G) magnetic
fields, observed in X-ray pulsars, require accurate resonant differential
scattering cross sections. Such cross sections exist, but they are quite
cumbersome. 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. We have expanded the
quantum-mechanical differential cross sections and kept terms up to first order
in $\epsilon \equiv E/m_ec^2$ and $B \equiv {\cal B}/{\cal B}_{cr}$, where $E$
is the photon energy and ${\cal B}_{cr}$ is the critical magnetic field, and
recovered the classical differential cross sections plus terms that are due to
spin flip, which is a pure quantum-mechanical phenomenon. Adding by hand the
spin-flip terms to the polarization-dependent classical differential cross
sections, we find that they are in excellent agreement with the quantum
mechanical ones for all energies near resonance and all angles. We have plotted
both of them and the agreement is impressive. We give a prescription for the
use of the classical differential cross sections that guarantees very accurate
results.
