Diffractive Imaging of Single Particles

Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

Radiation damage is the main problem which prevents the determination of the structure of a single biological macromolecule at atomic resolution using any kind of microscopy. This is true whether neutrons, electrons or X-rays are used as the illumination. For neutrons, the cross-section for nuclear capture and the associated energy deposition and radiation damage could be reduced by using samples that are fully deuterated and 15N-labelled and by using fast neutrons, but single molecule biological microscopy is still not feasible. For naturally occurring biological material, electrons at present provide the most information for a given amount of radiation damage. Using phase contrast electron microscopy on biological molecules and macromolecular assemblies of approximately 10(5) molecular weight and above, there is in theory enough information present in the image to allow determination of the position and orientation of individual particles: the application of averaging methods can then be used to provide an atomic resolution structure. The images of approximately 10,000 particles are required. Below 10(5) molecular weight, some kind of crystal or other geometrically ordered aggregate is necessary to provide a sufficiently high combined molecular weight to allow for the alignment. In practice, the present quality of the best images still falls short of that attainable in theory and this means that a greater number of particles must be averaged and that the molecular weight limitation is somewhat larger than the predicted limit. For X-rays, the amount of damage per useful elastic scattering event is several hundred times greater than for electrons at all wavelengths and energies and therefore the requirements on specimen size and number of particles are correspondingly larger. Because of the lack of sufficiently bright neutron sources in the foreseeable future, electron microscopy in practice provides the greatest potential for immediate progress.