Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 362-363   | 1 | 2 |

Section An account of absorption

B. K. Vainshteinc An account of absorption

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Elastic interaction of an incident wave with a weak phase object is defined on its exit surface by the distribution of potential projection [\varphi (xy)]; however, in the general case, the electron scattering amplitude is a complex one (Glauber & Schomaker, 1953[link]). In such a way, the image itself has the phase and amplitude contrast. This may be taken into account if one considers not only the potential projection [\varphi (xy)], but also the `imaginary potential' [\mu (xy)] which describes phenomenologically the absorption in thin specimens. Then, instead of ([link], the wave on the exit surface of a specimen can be written as [q (xy) = 1 - i\sigma \varphi (xy) - \mu (xy) \eqno(]and in the back focal plane if [ \Phi = {\scr F}\varphi] and [ M = {\scr F}\mu] [Q (uv) = \delta (uv) - i\sigma \Phi (uv) - M (uv). \eqno(]Usually, μ is small, but it can, nevertheless, make a certain contribution to an image. In a sufficiently good linear approximation, it may be assumed that the real part cos χ of the phase function ([link] affects [M(uv)], while [\Phi (xy)], as we know, is under the action of the imaginary part sin χ.

Thus, instead of ([link], one can write [Q (\exp i\chi) = \delta ({\bf u}) - i\sigma \Phi ({\bf u}) \sin \chi - M ({\bf u}) \cos \chi, \eqno(]and as the result, instead of ([link], [ \eqalignno{ I (xy) &= 1 + 2 \sigma \varphi (xy) * {\scr F}^{-1} (\sin \chi) * a (U) &\cr &\quad - 2 \mu (xy) * {\scr F}^{-1} (\cos \chi) * a (U). &(\cr}]

The functions [\varphi (xy)] and [\mu (xy)] can be separated by object imaging using the through-focus series method. In this case, using the Fourier transformation, one passes from the intensity distribution ([link] in real space to reciprocal space. Now, at two different defocus values [\Delta f_{1}] and [\Delta f_{2}] [([link]), ([link],b[link])] the values [\Phi ({\bf u})] and [M({\bf u})] can be found from the two linear equations ([link]. Using the inverse Fourier transformation, one can pass on again to real space which gives [\varphi ({\bf x})] and [\mu ({\bf x})] (Schiske, 1968[link]). In practice, it is possible to use several through-focus series and to solve a set of equations by the least-squares method.

Another method for processing takes into account the simultaneous presence of noise [N({\bf x})] and transfer function zeros (Kirkland et al., 1980[link]). In this method the space frequencies corresponding to small values of the transfer function modulus are suppressed, while the regions where such a modulus is large are found to be reinforced.


Glauber, R. & Schomaker, V. (1953). The theory of electron diffraction. Phys. Rev. 89, 667–670.
Kirkland, E. J., Siegel, B. M., Uyeda, N. & Fujiyoshi, Y. (1980). Digital reconstruction of bright field phase contrast images from high resolution electron micrographs. Ultramicroscopy, 5, 479–503.
Schiske, P. (1968). Zur Frage der Bildrekonstruktion durch Fokusreihen. 1 Y Eur. Reg. Conf. Electron Microsc. Rome, 1, 145–146.

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