The phase problem for one-dimensional crystals

The phase problem for diffraction amplitudes measured from a one-dimensional crystal is examined. In the absence of any a priori information, the solution to this problem is shown to be unique up to a parameterized, low-dimensional set of solutions. Minimal additional a priori information is expected to render the solution unique. The effects of additional information such as positivity, molecular envelope and helical symmetry on uniqueness are characterized. The results are pertinent to structural studies of polymeric and rod-like biomolecular assemblies that form one-dimensional, rather than three-dimensional, crystals. This shows the potential for ab initio phasing of diffraction data from single such assemblies measured using new X-ray free-electron laser sources. Such an approach would circumvent the complicated inversion of cylindrically averaged diffraction that is necessary in traditional X-ray fibre diffraction analysis.