Indexing of Electron Diffraction Patterns of Icosahedral and Decagonal Phases

Abstract

The atomic arrangements of solids fall into two broad categories. First refers to long range translational order giving rise to sharp diffraction patterns. Second relates to an atomic order that displays diffuse halos. Prior to the discovery of Quasiperiodic translational order in rapidly solidified Al-Mn alloys [1], sharp diffraction peaks were considered to be synonymous to possession of periodic translational orders in solids. The electron diffraction patterns reported in reference [1] were found to be invariant under icosahedral point group (mぬ博の博) symmetry. Any elementary text on crystallography begins by showing that 5-fold symmetry is incompatible with periodic lattice translations. Thus, it was proved beyond doubt that Shetchman et al.[1] have discovered a new state of order in solids. Readers may go through the notes given in annexure A for the excitement and importance of this discovery. The underlying atomic arrangement [1] was believed to possess “Quasiperiodic” translational order and due to invariance of diffraction patterns under icosahedral point group, such a class of solids was later termed as icosahedral quasicrystals (IQC). If one observes the location of diffracted spots in this class of solids then it is not periodic but all of them are as sharp as any crystalline diffraction patterns. These observations clearly established the fact that IQC displays new type of long range translational order known as Quasiperiodic translational order. We refer the readers to the annotations and reprints of papers in reference [2] for getting familiar with all the terminologies in this area. Having recorded diffraction patterns, the first step is to index them. For indexing, we need a set of basis vectors that are integrally independent. All three dimensionally periodic solids need three integrally independent basis vectors to index their diffraction patterns. The minimum number of integrally independent basis vectors is known as rank of any solids possessing long range translational orders [3-4]. Thus, all 3d periodic solids have rank equals to three. Remember, one uses four basis vectors for hexagonal crystals in Miller-Bravais (MB) scheme but all of them are not independent. As a consequence of this, assignment of indices to a diffracted spot in MB scheme is made unique by invoking condition that sum of indices along the three planar basis vectors is zero. We shall deliberate on this aspect further while discussing indexing of decagonal quasicrystals (DQC).