Simulation of X-ray topographs: a new method to calculate the diffracted field

Abstract

Abstract The precision of the numerical algorithms used to integrate the Takagi-Taupin equations has been in the past a severe limitation for the simulation of accurate topographs. The intensity, especially in the direct image of the defect, is underestimated. This has forbidden the use of the reciprocity theorem for the simulation of traverse and white-beam syn- chrotron topographs. A new algorithm is described, based on two different methods of expressing the partial-derivative equations, which permits a faster and more accurate calculation. I. Introduction X-ray topography is a widespread method for single- crystal characterization. Computer simulation of topographs is useful for image interpretation because it allows quantitative analysis of the perfection of crystals. The comparison between the computed and the experimental images makes it possible to test the validity of a deformation model for the defects seen in the image and to determine quantitatively param- eters that are not accessible through the experiment such as the sign and magnitude of the Burgers vector of a dislocation or the nature of a stacking fault. Simulation of section topographs is now well estab- lished (Epelboin, 1985). As for traverse topographs, 0108 -7673 / 93/030460-08506.00 Petrashen, Chukovskii & Shulpina (1980) have attempted to calculate the intensity along a line of the image and Epelboin & Soyer (1985) have simu- lated whole images. The latter have shown that the precision of the algorithms was not sufficient for the reciprocity theorem of optics to be used as suggested by Petrashen (1976). Three aspects must be considered when computing X-ray topographs: (i) the kind of wave incident on the surface of the crystal; (ii) the numerical method to solve the propagation equations inside the crystal; (iii) the network of integration used to integrate these equations. Let us briefly review each of them. X-ray topogra- phy may be classified into two groups: plane-wave and spherical-wave topography. Laboratory and syn- chrotron-radiation sources produce spherical waves (Aristov, Kohn, Polovinkina & Snigirev, 1982; Carvalho & Epelboin, 1990), so that to obtain a plane wave it is necessary to put a specially designed mono- chromator in front of the specimen. Petrashen