International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 311312

The intensity distribution of an electron wave in the image plane depends not only on the coherent and inelastic scattering, but also on the instrumental functions. The electron wave transmitted through an object interacts with the electrostatic potential which is produced by the nuclei charges and the electronic shells of the atoms. The scattering and absorption of electrons depend on the structure and thickness of a specimen, and the atomic numbers of the atoms of which it is composed. If an object with the threedimensional distribution of potential is sufficiently thin, then the interaction of a plane electron wave with it can be described as the interaction with a twodimensional distribution of potential projection , where b is the specimen thickness. It should be noted that, unlike the threedimensional function of potential with dimension , the twodimensional function of potential projection has the potentiallength dimension which, formally, coincides with the charge dimension. The transmission function, in the general case, has the form (2.5.2.42), and for weak phase objects the approximation is valid.
In the back focal plane of the objective lens the wave has the form where is the Scherzer phase function (Scherzer, 1949) of an objective lens (Fig. 2.5.5.1), is the aperture function, the spherical aberration coefficient, and Δf the defocus value [(2.5.2.32) –(2.5.2.35)].
The brightfield image intensity (in object coordinates) is where . The phase function (2.5.5.7) depends on defocus, and for a weak phase object (Cowley, 1981) where , which includes only an imaginary part of function (2.5.5.6). While selecting defocus in such a way that under the Scherzer defocus conditions [(2.5.2.44), (2.5.2.45)] , one could obtain In this very simple case the image reflects directly the structure of the object – the twodimensional distribution of the projection of the potential convoluted with the spread function . In this case, no image restoration is necessary. Contrast reversal may be achieved by a change of defocus.
At high resolution, this method enables one to obtain an image of projections of the atomic structure of crystals and defects in the atomic arrangement – vacancies, replacements by foreign atoms, amorphous structures and so on; at resolution worse than atomic one obtains images of dislocations as continuous lines, inserted phases, inclusions etc. (Cowley, 1981). It is also possible to obtain images of thin biological crystals, individual molecules, biological macromolecules and their associations.
Image restoration. In the case just considered (2.5.5.10), the projection of potential , convoluted with the spread function, can be directly observed. In the general case (2.5.5.9), when the aperture becomes larger, the contribution to image formation is made by large values of spatial frequencies U, in which the function sin χ oscillates, changing its sign. Naturally, this distorts the image just in the region of appropriate high resolution. However, if one knows the form of the function sin χ (2.5.5.7), the true function can be restored.
This could be carried out experimentally if one were to place in the back focal plane of an objective lens a zone plate transmitting only onesign regions of sin χ (Hoppe, 1971). In this case, the information on is partly lost, but not distorted. To perform such a filtration in an electron microscope is a rather complicated task.
Another method is used (Erickson & Klug, 1971). It consists of a Fourier transformation of the measured intensity distribution TQ (2.5.5.6) and division of this transform, according to (2.5.5.7a,b), by the phase function sin χ. This gives Then, the new Fourier transformation yields (in the weakphaseobject approximation) the true distribution The function sin χ depending on defocus Δf should be known to perform this procedure. The transfer function can also be found from an electron micrograph (Thon, 1966). It manifests itself in a circular image intensity modulation of an amorphous substrate or, if the specimen is crystalline, in the `noise' component of the image. The analogue method (optical Fourier transformation for obtaining the image ) can be used (optical diffraction, see below); digitization and Fourier transformation can also be applied (Hoppe et al., 1973).
The thin crystalline specimen implies that in the back focal objective lens plane the discrete kinematic amplitudes are arranged and, by the above method, they are corrected and released from phase distortions introduced by the function sin χ (see below) (Unwin & Henderson, 1975).
For the threedimensional reconstruction (see Section 2.5.6) it is necessary to have the projections of potential of the specimen tilted at different angles α to the beam direction (normal beam incidence corresponds to ). In this case, the defocus Δf changes linearly with increase of the distance l of specimen points from the rotation axis . Following the above procedure for passing on to reciprocal space and correction of sin χ, one can find (Henderson & Unwin, 1975).
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