Symmetry enhancement in convergent-beam electron diffraction

Abstract

Convergent-beam electron diffraction (eBED) patterns do not always show the true symmetries of a specimen crystal, but can show higher s~~etries than those of the specimen (s~~etry enhancement). If the phase of the structure factor can not be detected, a symmetry enhancement may take place. Lacking of an inversion center in a crystal can not be detected by X-ray diffraction, but can be unarnbiguously knmffi by CBED.~Vhen the structure factors at two reciprocal. lattice points which are related by a twofold axis have-the same amplitude but have different phases, we say that a pseudo twofold a~is 2* is present. If a mirror plane is present, a pseudo twofold axis is accomp~~ied perpendicularly to the mirror plane. In this case, two structure factors related by the pseudo tHO fold axis are complex conjugate each other. A pseudo mirror plane m* is defined in a similar manner.A pseudo twofold axis and a pseudo mirror plane can cause a symmetry-enhancement in the bright field pattern of CBED. However, the enhanc~~ent can not be observed, since the true mirror plane and the true twofold axis which give the same sy~metries caused by the 2* and m* coexist in usual cases. In a general case, the syLmnetry enhancement c~T), not be expected in the \-lhole pattern and dark field pattern.In a zincblende type crystal, Hhose space group is F43m, the approximate sy~metry enhancement takes place in the 220 dark field pattern by 2* at [Ill] electron incidence, that is, HOLZ lines in the reflection shm-ls the symmetry m2. The reflections belonging to the O-th Laue zone produce 2mm symmetry in the 220 dark-field pattern, since the phases of the reflections are ze~o. The imaginary part of the 220 reflection caused by two successive HOLZ reflections (Umweganregung) is proportional to fA2fBl -fAlfB2, where fxi is the st~cture factor of X atom for the i-th reflection. wnen the scattering angle dependence of the structure factor for A atom is similar to that of B atom, the imaginary part and the resultant phase of the reflection becomes small, resulting a symmetry enhancement of m2' The contribution of the UmweganregQ~g via more th~~ three reflections may be small. A 34 bea~ dynamical computation really shows m2 symmetry in the 220 reflection.The point group of 10H-SiC is 3m. However, it is known that the point group determined by X-ray diffraction is 6/rnrnm, which is Ra~sdell type symmetry enhancement [L.S. Rfu"sdell and J.A. Kohn, Acta Cryst. i (1951) Ill]. When the crystal is exa~ined by CBED, the diffraction group and point group obtained at 0001 incidence were 3mlR and 6m2, and those obtained at 1100 incidence were mlR and 6rnrn. The symmetry enhanc~~ent occurs in the bright field pattern and the approXL~ate symmetry enhancement occurs in the dark field patterns by m* in the (0001) and the (1120) planes and by 2* in the c axis and in the [1100] and the [1120] directions.