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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 can 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 non-crystalline specimens, that this no-density region could be used to retrieve the phase information; here the same is performed for crystalline and near-crystalline specimens. By employment of an iterative algorithm, the phase information could be recovered from computer-generated oversampled diffraction patterns of small specimens that are (a) perfect or imperfect crystals, 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 work recently published [Miao, Charalambous, Kirz & Sayre (1999). Nature (London), 400, 342-344]. Our algorithm requires an approximate envelope for the specimen. It does not require any structural knowledge concerning the specimen and does not require data to atomic resolution (although it can use such data if present). After a few hundred to a few thousand iterations, the correct phase set and image are recovered. The oversampling technique thus greatly extends the specimen range of X-ray crystallography but it imposes a high radiation dose on the specimens compared with the situation in crystallography, 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 cases where the specimen is a crystal, there are also possibilities for oversampling relative to Bragg (instead of Nyquist) sampling, thus providing a lesser 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 biological cell in (c) above] the imaging obtainable with the aid of a cryogenic protective technique should surpass any other present method of whole-cell imaging. In addition, with the possible appearance in the future of free electron lasers (>1012 photons and <200 fs per pulse), it is possible to circumvent the radiation-damage problem by recording diffraction patterns before damage manifests itself.
Keywords: oversampling.

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