We perform a study on Rayleigh scattering of highly linearly polarized hard x rays on a thin Au foil target. In the study the angular distribution of the scattered radiation is analyzed in a relativistic regime both within and out of the plane of polarization of the incident beam. Within this experiment we scatter a synchrotron beam with a photon energy of 175 keV on a high-$Z$ target foil, namely, gold. Our findings correlate well with state-of-the-art calculations of the scattering process performed in the framework of quantum electrodynamics and may have considerable impact on future experiments regarding a polarization-resolved analysis of Delbr\"uck scattering. Furthermore, we show that the angular distribution of Rayleigh scattering can be used for a highly sensitive determination of the degree and orientation of the linear polarization of the incident hard-x-ray beam, if we rely on the theoretical framework.
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