Structure determination of a rhombohedral Al–Ge phase by CBED and X-ray powder diffraction

Abstract

A new approach to the problem of crystal structure determination by electron diffraction is described based upon the intensities of higher-order Laue-zone (HOLZ) reflections in convergent-beam electron diffraction (CBED) patterns. Along a major zone axis, a subset of symmetry-related atoms may project into separate string potentials associated with localized Bloch states. In a quasi-kinematic limit, the amplitudes of HOLZ excess lines diffracted from the Bloch states are proportional to the partial structure factors for these atoms. If a structure is unknown, the Bloch states cannot be calculated but it is shown that the conditional Patterson transform of the HOLZ intensities around suitable axes may include narrow correlation peaks that identify interatomic vectors. This approach was tested by analysis of a metastable Al–Ge phase with space group R{\bar 3}c. A transform of HOLZ intensities measured around the c axis showed peaks displaced by only 0.21 Å from the screw axes, consistent with Ge atoms in the 18(e) positions. Although Al atoms were not visible in the Patterson maps, the crystal structure with nominal composition Al6Ge5 was identified as isostructural with the Zn4Sb3 phase by comparison of observed and calculated X-ray powder intensities.