Dynamical theory of electron diffraction for the electron microscopic image of crystal lattices I. Images of single crystals

Abstract

The dynamical theory of electron diffraction is applied to the interpretation of electron micro­ scopic images of lattice planes of plate- and wedge-shaped crystals. The wave functions and corresponding intensities predicting interference fringes on the exit surface of a crystal are derived. It is shown in both cases that the fringes are composed of parallel lines and the spacing of the fringes at the exact Bragg angle coincides with that of the original lattice but the positions of the lines do not coincide with those of potential maxima in the crystal, i.e. intensity profiles of the fringes do not represent the variation of mass-thickness in the crystal. The intensity profiles and the spacings of the fringes vary with the thickness of crystal and the deviation from the Bragg angle. The fringes from a bent plate-shaped crystal, which are formed on the extinction contour bands, show the same spacing as that of the crystal lattice along the centre of the contour but they have an increased or decreased spacing near the edge of the contour. The fringes which are formed on the subsidiary extinction contour also show spacing anomaly; they are shifted by half the corresponding amount for the principal contour. The spacing of the fringes of a wedge-shaped crystal coincides with that of the original lattice at the exact Bragg angle, but the contrast of the lines reverses wherever the thickness of the crystal increases by an amount of XE/2V _g (A, wave length; E , accelerating potential; V _g , Fourier coefficient of inner potential of the crystal). For deviation from the Bragg angle, the spacing of the fringes, in general, does not coincide with that of the original lattice and, moreover, the contrast of the lines reverses wherever the thickness of the crystal increases by an amount of The anomalies of spacing and reversal of contrast which are expected from the present theory were observed in the electron microscopic images of metal-phthalocyanine and sodium faujasite crystals respectively. The effects of absorption by the crystal and divergence of illumination on the contrast of the image are discussed and the possibility of obtaining two-dimensional projections of the atomic arrangement in a crystal by using electron microscopic images is also discussed.