Phase retrieval for 1D and 2D crystals

The extreme intensity and extremely short x-ray pulses produced by x-ray free electron lasers (XFELs), so-called 4th generation x-ray sources, are opening new horizons for x-ray crystallography (XRC). This new source of x-rays is not limited by the same rules that govern and plague synchrotron-based x-ray crystallography, which utilises 3rd generation x-ray sources. Small crystals, such as 1D and 2D crystals or even nano-crystals can now be used with X-ray free electron lasers allowing studies of essential proteins, such as membrane proteins, close to their natural environment. Studies on fibrous proteins can also potentially take advantage of what X-ray free electron lasers have to offer, allowing data to be collected using a single 1D crystal rather than a fiber. The crystallographic phase problem with such finite crystals is eased as the lack of periodicity allows for additional sampling of the Fourier amplitude in addition to the common Bragg sampling. Unfortunately, the phases are still not measurable and thus direct Fourier inversion is still not available. In this paper, we characterise uniqueness of the phase problem for a selection of crystalline specimens. Ab initio phase retrieval using the difference map algorithm for 1D and 2D crystals is also presented.