REVIEW ARTICLES INSIGHT NATURE MATERIALS DOL:10.1038/NMAT2400 Box 1|Solving the phase problem by CXD. The phasing of the data is a critical step that uses a computer methods.A technique known as 'shrink wrap'is sometimes used algorithm that is illustrated schematically.The algorithm takes to allow the support to evolve on successive cycles as the algo- advantage of internal redundancies in the diffraction data when rithm progressesss.The oversampling condition is simply that the the measurement points are spaced close enough together to number of measurement points be at least twice the number of meet the 'oversampling'requirement.Fourier transforms ('F'; unknown density values within this support,as proposed in ref.56 fast Fourier transforms in the computer implementation)con- immediately after the publication of the Shannon theorem7.It was nect real-space arrays of data (left panels)with reciprocal space shown in ref.58 that this is a sufficient condition for a unique set arrays(right panels).Both sides are updated in every cycle of the of phases to be determined,in two or more dimensions.The best algorithm.Inputs to the algorithm are the measured intensity data known method for finding those phases and avoiding 'stagnation' (array amplitudes overwritten in the bottom-right panel)and a problems is the hybrid input-output method,which starts with postulated 3D 'support'volume (translucent box in the top-left a random phase'seed'and propagates a weighted combination of panel)in which all the complex sample density is constrained to the current and previous cycles on the real-space side of the algo- exist.The measurement of diffraction data is shown schematically rithm.This support-constrained phasing of the diffraction patterns on the right:a narrow,coherent X-ray beam illuminates one of of small objects is judged to be reliable because the same structure the Pb nanocrystals grown (in situ)on a substrate(viewed subse- emerges from different random starting phase sets40.One measure quently by SEM)to produce a diffraction pattern on an area detec- of its reliability is that only 50 iterations of hybrid input-output are tor.The dimensions of the support can be estimated directly from often enough to yield a consistent result with good data,whereas an the fringe spacings of the observed pattern or using autocorrelation earlier version of phasing algorithms required several thousand Coherent X-ray diffraction per fringe also ensures an 'oversampling'condition".Obtaining The sample-averaging limitations of the two methods described so enough coherent flux is only practicable with the brightness of far have been overcome by the development of coherent X-ray dif- undulator-based third-generation'synchrotron radiation sources, fraction(CXD)as a 3D structural analysis method for individual such as the ESRF and APS,used for the CXD results reported here. small crystals.As explained in Box 1,the inversion of the diffraction The crystal lattice introduces a powerful new constraint on the pattern yields a 3D image of the density distribution of the sample selection of a grain for imaging.A polycrystalline sample will have and a projection of the strain within it,so it is a genuine form of closely packed grains with numerous different orientations.Its phase-contrast X-ray microscopy.The basic principle of the CXD Bragg diffraction will resemble that of a powder but,with a small experiment is the illumination of the sample by a spatially coherent enough beam and a typical grain size of around a micrometre,the beam of X-rays,meaning that the transverse coherence length (of individual grains can still be separated.Even highly textured sam- up to a few micrometres)should exceed the dimensions of the sam- ples can have orientations distributed widely enough that the grains ple.Under these conditions,scattering from all parts of the sample can be distinguished.Once a Bragg peak is isolated and aligned,its can be expected to interfere in the far-field diffraction pattern. internal intensity distribution can be recorded by means of a CCD The coherence effects we exploit were first documented for hard at the end of a long detector arm.A'rocking'series of CCD images, X-rays in 1991 (ref.38).Coherent X-ray diffraction is usually meas- in which the crystal's angle to the beam is varied by a fraction of a ured in the same way as a normal diffraction experiment,but with a degree around its Bragg peak,yields a complete 3D data set. high-resolution charge-coupled device(CCD)or other X-ray detec- The 3D data are then inverted to form quantitative real-space tor positioned far enough away to resolve the finest fringes,called images using a computational method for solving the phase prob- 'speckles'by analogy with scattering of laser light.As discussed lem,described in Box 1.This computation effectively replaces the further below,the requirement of at least two detector pixel spacings objective lens in a traditional microscope.A typical CXD pattern is 294 NATURE MATERIALS VOL 8|APRIL 2009 www.nature.com/naturematerials 2009 Macmillan Publishers Limited.All rights reserved294 nature materials | VOL 8 | APRIL 2009 | www.nature.com/naturematerials review articles | insight NaTure maTerials doi: 10.1038/nmat2400 coherent X-ray diffraction The sample-averaging limitations of the two methods described so far have been overcome by the development of coherent X-ray dif￾fraction (CXD) as a 3D structural analysis method for individual small crystals. As explained in Box 1, the inversion of the diffraction pattern yields a 3D image of the density distribution of the sample and a projection of the strain within it, so it is a genuine form of phase-contrast X-ray microscopy. The basic principle of the CXD experiment is the illumination of the sample by a spatially coherent beam of X-rays, meaning that the transverse coherence length (of up to a few micrometres) should exceed the dimensions of the sam￾ple. Under these conditions, scattering from all parts of the sample can be expected to interfere in the far-field diffraction pattern. The coherence effects we exploit were first documented for hard X-rays in 1991 (ref. 38). Coherent X-ray diffraction is usually meas￾ured in the same way as a normal diffraction experiment, but with a high-resolution charge-coupled device (CCD) or other X-ray detec￾tor positioned far enough away to resolve the finest fringes, called ‘speckles’ by analogy with scattering of laser light. As discussed further below, the requirement of at least two detector pixel spacings per fringe also ensures an ‘oversampling’ condition39. Obtaining enough coherent flux is only practicable with the brightness of undulator-based ‘third-generation’ synchrotron radiation sources, such as the ESRF and APS, used for the CXD results reported here. The crystal lattice introduces a powerful new constraint on the selection of a grain for imaging. A polycrystalline sample will have closely packed grains with numerous different orientations. Its Bragg diffraction will resemble that of a powder but, with a small enough beam and a typical grain size of around a micrometre, the individual grains can still be separated. Even highly textured sam￾ples can have orientations distributed widely enough that the grains can be distinguished. Once a Bragg peak is isolated and aligned, its internal intensity distribution can be recorded by means of a CCD at the end of a long detector arm. A ‘rocking’ series of CCD images, in which the crystal’s angle to the beam is varied by a fraction of a degree around its Bragg peak, yields a complete 3D data set. The 3D data are then inverted to form quantitative real-space images using a computational method for solving the phase prob￾lem, described in Box 1. This computation effectively replaces the objective lens in a traditional microscope. A typical CXD pattern is Box 1 | solving the phase problem by CXD. F F–1 500 nm The phasing of the data is a critical step that uses a computer algorithm that is illustrated schematically. The algorithm takes advantage of internal redundancies in the diffraction data when the measurement points are spaced close enough together to meet the ‘oversampling’ requirement39. Fourier transforms (‘F’; fast Fourier transforms in the computer implementation) con￾nect real-space arrays of data (left panels) with reciprocal space arrays (right panels). Both sides are updated in every cycle of the algorithm. Inputs to the algorithm are the measured intensity data (array amplitudes overwritten in the bottom-right panel) and a postulated 3D ‘support’ volume (translucent box in the top-left panel) in which all the complex sample density is constrained to exist. The measurement of diffraction data is shown schematically on the right: a narrow, coherent X-ray beam illuminates one of the Pb nanocrystals grown (in situ) on a substrate (viewed subse￾quently by SEM) to produce a diffraction pattern on an area detec￾tor. The dimensions of the support can be estimated directly from the fringe spacings of the observed pattern or using autocorrelation methods. A technique known as ‘shrink wrap’ is sometimes used to allow the support to evolve on successive cycles as the algo￾rithm progresses55. The oversampling condition is simply that the number of measurement points be at least twice the number of unknown density values within this support, as proposed in ref. 56 immediately after the publication of the Shannon theorem57. It was shown in ref. 58 that this is a sufficient condition for a unique set of phases to be determined, in two or more dimensions. The best known method for finding those phases and avoiding ‘stagnation’ problems is the hybrid input–output method59, which starts with a random phase ‘seed’ and propagates a weighted combination of the current and previous cycles on the real-space side of the algo￾rithm. This support-constrained phasing of the diffraction patterns of small objects is judged to be reliable because the same structure emerges from different random starting phase sets60. One measure of its reliability is that only 50 iterations of hybrid input–output are often enough to yield a consistent result with good data, whereas an earlier version of phasing algorithms required several thousand61. nmat_2400_APR09.indd 294 13/3/09 12:04:30 © 2009 Macmillan Publishers Limited. All rights reserved