On the observability of field-assisted birefringent Delbr\"uck scattering

We consider the scattering of an x-ray free-electron laser (XFEL) beam on the superposition of a strong magnetic field $\bf{B}_{\rm ext}$ with the Coulomb field $\bf{E}_{\rm ext}$ of a nucleus with charge number $Z$. In contrast to pure Delbr\"uck scattering (Coulomb field only), the magnetic field $\bf{B}_{\rm ext}$ introduces an asymmetry (i.e., polarization dependence) and renders the effective interaction volume quite large, while the nuclear Coulomb field facilitates a significant momentum transfer $\Delta\bf k$. For a field strength of $B_{\rm ext}=10^6~\rm T$ (corresponding to an intensity of order $10^{22}~\rm W/cm^2$) and an XFEL frequency of 24 keV, we find a differential cross section $d\sigma/d\Omega\sim10^{-25}~Z^2/(\Delta{\bf k})^2$ in forward direction for one nucleus. Thus, this effect might be observable in the near future at facilities such as the Helmholtz International Beamline for Extreme Fields (HIBEF) at the European XFEL.