A large fraction of accreting black hole and neutron stars systems present clear evidence of the reprocessing of X-rays in the atmosphere of an optically-thick accretion disk. The main hallmarks of X-ray reflection include fluorescent K-shell emission lines from iron ($sim 6.4-6.9$ keV), the absorption iron K-edge ($sim 7-9$ keV), and a broad featureless component known as the Compton hump ($sim 20-40$ keV). This Compton hump is produced as the result of the scattering of high-energy photons ($E gtrsim 10$ keV) of the relatively colder electrons ($T_e sim 10^5-10^7$ K) in the accretion disk, in combination with photoelectric absorption from iron. The treatment of this process in most current models of ionized X-ray reflection has been done using an approximated Gaussian redistribution kernel. This approach works sufficiently well up to $sim100$ keV, but it becomes largely inaccurate at higher energies and at relativistic temperatures ($T_esim10^9$ K). We present new calculations of X-ray reflection using a modified version of our code XILLVER, including an accurate solution for Compton scattering of the reflected unpolarized photons in the disk atmosphere. This solution takes into account quantum electrodynamic and relativistic effects allowing the correct treatment of high photon energies and electron temperatures. We show new reflection spectra computed with this model, and discuss the improvements achieved in the reproducing the correct shape of the Compton hump, the discrepancies with previous calculations, and the expected impact of these new models in the interpretation of observational data.
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