Determination of small crystal structures from a minimum set of diffraction intensities by homotopy continuation

No deterministic approach to obtaining a crystal structure from a set of diffraction intensities exists, despite significant progress in traditional probabilistic direct methods. One of the biggest hurdles in determining a crystal structure algebraically is solving a system of many polynomial equations of high power on intensities in terms of atomic coordinates. In this study, homotopy continuation is used for exhaustive investigation of such systems and an optimized homotopy continuation method is developed with random restarts to determine small (N < 5) crystal structures from a minimum set of error-free intensities.