Convergent-beam electron diffraction study of incommensurately modulated crystals. I. (3+1)-dimensional point groups

Abstract

The interrelation between the symmetries of convergent-beam electron diffraction (CBED) patterns and the (3+l)-dimensional point groups of one-dimensional incommensurately modulated crystals is discussed. The symmetry subsymbol {\overline 1} for the modulated structure implies that the incommensurate reflections show no symmetry in the CBED pattern. The symmetry subsymbols other than {\overline 1} imply that the incommensurate reflections exhibit the same symmetries as those of the average structure. These results are demonstrated using the CBED patterns obtained from the one-dimensional incommensurately modulated crystals of Sr2Nb2O7 and Bi2Sr2CaCu2O8+δ. It is shown that the CBED method can identify the (3+1)-dimensional point groups of one-dimensional incommensurately modulated structures.