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Incommensurate superstructures are interesting problems for high resolution electron microscopy. If modulations are formed within a plane of a host lattice parallel to the incident beam direction, their structures can be known directly from images (1,2,3). In this paper, an incommensurate superstructure of hexagonal potassium tungsten bronze, K0.3WO3, is observed by a 100B high-resolution electron microscope and its structure model is proposed. The hexagonal tungsten bronze was determined by Magneli (4) and its structure is shown in Fig. 1. The structure consists of WO6 octahedra and alkali metal ions. The alkali metal ions situated in hexagonal tunnels do not lie at the same level of the WO6 octahedra but at ¼ c above or below them. It was assumed that the alkali metal ions were distributed in a random fashion. Fig. 2 shows a structure image of K0.3WO3, taken along the [100] direction. The white spots correspond to hexagonal tunnels in Fig. 1. Fig. 3 shows an electron diffraction pattern taken along the [010] direction. Some additional weak superstructure spots are observed. The superstructure spot (indicated by A) is situated at a non-integral multiple position of the subcell spots along the c*-axis, indicating that the incommensurate superstructure has a multiplicity of 2.2 x c of the subcell. The non-integral periodicity can be seen in a high-resolution image taken along the [010] direction, as shown in Fig. 4. At the thick crystal regions, some weak dark bands running parallel to the c axis are observed, in which they have two different widths (2 x c or 2.5 x c) along the c axis. An average distance between the adjacent dark bands becomes about 2.2 x c, which is consistent with an optical diffraction pattern (inset in Fig. 4). One of the possible models for the incommensurate superstructure of the hexagonal tungsten bronze is proposed in Fig. 5. The superstructure arises from the local ordering of K ion vacancies located in the tunnels along the c axis. The vacancies are formed at every fourth or fifth site of K ions. We call them structures with n = 4 and n = 5. The incommensurate superstructure results from a mixture of the structure elements with n = 4 and n = 5, causing the formation of a non-integral periodicity observed presently. It should be noted in Fig. 4 that the image from a thin region does not show the superstructure but the image from a thick region does. It seems that this phenomenon arises from dynamical diffraction effects. This will be discussed in detail on the basis of the image calculation.

A technique has been developed for the interpretation of composition depth profiles from angleresolved x-ray data using a Monte Carlo electron scattering simulation. Conventional methods for the interpretation of angle-resolved depth profiles used in the fields of x-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES) assume that the outgoing signal is exponentially attenuated along its path. This assumption if not valid for angle-resolved x-ray techniques, as the x-ray signal is dependent on both the paths of the incident electrons and the path of the emitted x-rays. In this case, while the latter can be treated using an exponential attenuation, the path of the incident beam is more complex and corresponds to the well known “pear-shaped” interaction volume. In order to reliably interpret angle-resolved depth profiles in which the angle of the incident beam is varied, it is necessary to be able to obtain the distribution of x-ray emission within the sample for any angle of incidence.