On possible extensions of X-ray crystallography through diffraction-pattern oversampling

It is known that sampling the diffraction pattern of a finite specimen, at a spacing somewhat finer than the Nyquist spacing (the inverse of the size of the diffracting specimen), corresponds to generating a no-density region surrounding the electron density of the specimen. This no-density region c an then be used to retrieve the phase information. In earlier papers [Miao, Sayre & Chapman (1998). J. Opt. Soc. Am. A15, 1662-1669; Sayre, Chapman & Miao (1998). Acta Cryst. A54, 232-239], it was demonstrated, in the case of noncrystalline specimens, that this no-density region could be used to ret rieve the phase information; here the same is performed for crystalline and near-crystalline specimens. By employment of an iterative algorithm, the p hase information could be recovered from computer-generated oversampled dif fraction patterns of small specimens that are (a) perfect or imperfect crys tals, or (b) have a repeated motif without orientational regularity, or (c) are an unrepeated motif, such as an amorphous glass, a single molecule or a single biological cell. Cases (a) and (b) represent an extension over wor k recently published [Miao, Charalambous, Kirz & Sayre (1999). Nature (Lond on), 400, 342-344]. Our algorithm requires an approximate envelope for the specimen. It does not require any structural knowledge concerning the speci men and does not require data to atomic resolution (although it can use suc h data if present). After a few hundred to a few thousand iterations, the c orrect phase set and image are recovered. The oversampling technique thus g reatly extends the specimen range of X-ray crystallography but it imposes a high radiation dose on the specimens compared with the situation in crysta llography, in which it is usual for the pattern to be sampled at the (much less fine) Bragg spacing (the inverse of the size of the unit cell). In cas es where the specimen is a crystal, there are also possibilities for oversa mpling relative to Bragg (instead of Nyquist) sampling, thus providing a le sser degree of oversampling and the possibility of lower dosage. Damage of the specimen in consequence of the dose will in many cases seriously affect the quality and resolution of the imaging, but in at least one case [the b iological cell in (c) above] the imaging obtainable with the aid of a cryog enic protective technique should surpass any other present method of whole- cell imaging. In addition, with the possible appearance in the future of fr ee electron lasers (>10(12) photons and <200 fs per pulse), it is possible to circumvent the radiation-damage problem by recording diffraction pattern s before damage manifests itself.