The effect of anharmonic atomic vibrations on the elastic scattering of X-rays by a crystal is treated by time-independent perturbation theory. The complete expression for the anharmonic temperature factor (TF) in the high-temperature limit is presented for atoms in general positions. The phonon-phonon interactions in an anharmonic crystal may be identified with X-ray diffraction data by using various temperature and scattering vector q dependencies of different anharmonic contri-butions to the TF. The expression for the intensity of elastic scattering by an anharmonic crystal contains an additional factor -the anharmonic cross-term. This term has to be taken into account for obtaining the unbiased thermal parameters of fourth order in | q | from X-ray diffraction data.