Signal, noise, and resolution in correlated fluctuations from snapshot small-angle x-ray scattering.

It has been suggested that the three-dimensional structure of one particle may be reconstructed using the scattering from many identical, randomly oriented copies ab initio, without modeling or a priori information. This may be possible if these particles are frozen in either space or time, so that the conventional two-dimensional small-angle x-ray scattering (SAXS) distribution contains fluctuations and is no longer isotropic. We consider the magnitude of the correlated fluctuation SAXS (CFSAXS) signal for typical x-ray free-electron laser (XFEL) beam conditions and compare this against the errors derived with the inclusion of Poisson photon counting statistics. The resulting signal-to-noise ratio (SNR) is found to rapidly approach a limit independent of the number of particles contributing to each diffraction pattern, so that the addition of more particles to a "single-particle-per-shot" experiment may be of little value, apart from reducing solvent background. When the scattering power is significantly less than one photon per particle per Shannon pixel, the SNR grows in proportion to incident flux. We provide simulations for protein molecules in support of these analytical results, and discuss the effects of solvent background scatter. We consider the SNR dependence on resolution and particle size, and discuss the application of the method to glasses and liquids, and the implications of more powerful XFELs, smaller focused beams, and higher pulse repetition rates for this approach. We find that an accurate CFSAXS measurement may be acquired to subnanometer resolution for protein molecules if a 9-keV beam containing 10(13) photons is focused to a ~100-nm spot diameter, provided that the effects of solvent background can be reduced sufficiently.