Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 389   | 1 | 2 |

Section Direct phase determination from electron micrographs

D. L. Dorsete* Direct phase determination from electron micrographs

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The `direct method' most familiar to the electron microscopist is the high-resolution electron micrograph of a crystalline lattice. Retrieval of an average structure from such a micrograph assumes that the experimental image conforms adequately to the `weak phase object' approximation, as discussed in Section 2.5.5.[link] If this is so, the use of image-averaging techniques, e.g. Fourier filtration or correlational alignment, will allow the unit-cell contents to be visualized after the electron-microscope phase contrast transfer function is deconvoluted from the average image, also discussed in Section 2.5.5.[link] Image analyses can also be extended to three dimensions, as discussed in Section 2.5.6[link], basically by employing tomographic reconstruction techniques to combine information from the several tilt projections taken from the crystalline object. The potential distribution of the unit cell to the resolution of the imaging experiment can then be used, via the Fourier transform, to obtain crystallographic phases for the electron-diffraction amplitudes recorded at the same resolution. This method for phase determination has been the mainstay of protein electron crystallography.

Once a set of phases is obtained from the Fourier transform of the deconvoluted image, they must, however, be referred to an allowed crystallographic origin. For many crystallographic space groups, this choice of origin may coincide with the location of a major symmetry element in the unit cell [see IT A (2005)[link]]. Hence, since the Fourier transform of translation is a phase term, if an image shift [[\delta({\bf r} + {\bf r}_{0})]] is required to translate the origin of the repeating mass unit [\varphi({\bf r})] from the arbitrary position in the image to a specific site allowed by the space group, [\hbox{g} ({\bf r}) = \varphi ({\bf r}) \otimes \delta ({\bf r} + {\bf r}_{0}) = \varphi ({\bf r} + {\bf r}_{0}),]where the operation `[\otimes]' denotes convolution. The Fourier transform of this shifted density function will be [G ({\bf s}) = F({\bf s}) \exp(2\pi i{\bf s} \cdot {\bf r}_{0}) = |F({\bf s})| \exp[i(\phi_{s} + 2\pi i{\bf s} \cdot {\bf r}_{0})].]In addition to the crystallographic phases [\phi_{s}], it will, therefore, be necessary to find the additional phase-shift term [2\pi i{\bf s} \cdot {\bf r}_{0}] that will access an allowed unit-cell origin. Such origin searches are carried out automatically by some commercial image-averaging computer-software packages.

In addition to applications to thin protein crystals (e.g. Henderson et al., 1990[link]; Jap et al., 1991[link]; Kühlbrandt et al., 1994[link]), there are numerous examples of molecular crystals that have been imaged to a resolution of 3–4 Å, many of which have been discussed by Fryer (1993)[link]. For π-delocalized compounds, which are the most stable in the electron beam against radiation damage, the best results (2 Å resolution) have been obtained at 500 kV from copper perchlorophthalocyanine epitaxically crystallized onto KCl. As shown by Uyeda et al. (1978–1979)[link], the averaged images clearly depict the positions of the heavy Cu and Cl atoms, while the positions of the light atoms in the organic residue are not resolved. (The utility of image-derived phases as a basis set for phase extension will be discussed below.) A number of aromatic polymer crystals have also been imaged to about 3 Å resolution, as reviewed recently (Tsuji, 1989[link]; Dorset, 1994b[link]).

Aliphatic molecular crystals are much more difficult to study because of their increased radiation sensitivity. Nevertheless, monolamellar crystals of the paraffin n-C44H90 have been imaged to 2.5 Å resolution with a liquid-helium cryomicroscope (Zemlin et al., 1985[link]). Similar images have been obtained at room temperature from polyethylene (Revol & Manley, 1986[link]) and also a number of other aliphatic polymer crystals (Revol, 1991[link]).

As noted by J. M. Cowley and J. C. H. Spence in Section 2.5.1[link], dynamical scattering can pose a significant barrier to the direct interpretation of high-resolution images from many inorganic materials. Nevertheless, with adequate control of experimental conditions (limiting crystal thickness, use of high-voltage electrons) some progress has been made. Pan & Crozier (1993)[link] have described 2.0 Å images from zeolites in terms of the phase-grating approximation. A three-dimensional structural study has been carried out on an aluminosilicate by Wenk et al. (1992)[link] with thin samples that conform to the weak-phase-object approximation at the 800 kV used for the imaging experiment. Heavy and light (e.g. oxygen) atoms were located in the micrographs in good agreement with an X-ray crystal structure. Heavy-atom positions from electron microscopic and X-ray structure analyses have also been favourably compared for two heavy-metal oxides (Hovmöller et al., 1984[link]; Li & Hovmöller, 1988[link]).


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