Recover the phases from intensity data of x-ray diffraction

For many years, people believed that in conventional x-ray crystallography one can only record the scattering intensities but not the phases. In order to study structures at the atomic scale, one has to use multibeam measurement or to rely on additional knowledge such as bonding length and to fit the intensity data by trial structure models. In this letter, however, we show that the phases are in fact hidden in the intensity data and can be recovered by studying the peak shoulders. To demonstrate, the x-ray diffraction data of aluminum powder were used to recover the phases and to reconstruct the electron density map.