A Review of Transmission Electron Microscopy of Quasicrystals—How Are Atoms Arranged?
Quasicrystals (QCs) possess rotational symmetries forbidden in the conventional crystallography and lack translational symmetries. Their atoms are arranged in an ordered but non-periodic way. Transmission electron microscopy (TEM) was the right tool to discover such exotic materials and has always been a main technique in their studies since then. It provides the morphological and crystallographic information and images of real atomic arrangements of QCs. In this review, we summarized the
... ummarized the achievements of the study of QCs using TEM, providing intriguing structural details of QCs unveiled by TEM analyses. The main findings on the symmetry, local atomic arrangement and chemical order of QCs are illustrated. The lattice of icosahedral QCs can be obtained by the projection of a 6D hypercubic into the 3D space. Theoretically, there are three types of 6D hypercubic lattice-primitive, body-centered and face-centered-corresponding to icosahedral lattice in the 3D space  . The lattices generated by the projection of the three different 6D hypercubic lattices into the 3D space are called primitive, body-centered and face-centered icosahedral QCs, respectively. All the icosahedral QCs discovered so far are primitive or face-centered. They can be easily distinguished by their 2-fold patterns: along their fivefold direction, the distance between the diffraction spots is inflated by τ 3 and τ (τ = (1 + 5 1/2 )/2) Crystals 2016, 6, 105 4 of 16 for primitive and face-centered icosahedral QCs, respectively . The first QC discovered is primitive icosahedral phase . Later, this type of icosahedral phase was found in Ag-In-Yb , Cd-Mg-Rare Earth . The face-centered icosahedral QCs were first discovered by Tsai et al.  in Al-Cu-Fe and later in Al-Pd-Mn  and Al-Pd-Re , and Zn-Mg-Rare Earth  etc. 2D QCs Most of the 2D QCs found so far are decagonal phases, which have one 10-fold axis and two types of characteristic 2-fold axes (each type having 10 rotational axes, schematically shown in Figure 2a) . The twofold axes are perpendicular to the 10-fold axis and the angle between the neighboring 2-fold axes is 18 • . Along the tenfold axis, quasiperiodic atomic layers are stacked periodically. The diffraction spots are thus arranged periodically in this direction and quasiperiodically perpendicular to this direction. The periodicity along the 10-fold axis can be determined from the 2-fold symmetry patterns. Different periodicities along the 10-fold axis have been reported in different alloys: 0.4 nm [44-46], 0.8 nm [47,48], 1.2 nm [32,49] and 1.6 nm [50,51], which correspond to a stacking of 2, 4, 6 and 8 atomic layers, respectively [52,53]. Figure 2b-d are the SAED patterns taken along the periodic axis c* and two types of 2-fold symmetry axes (A and B) of decagonal Al 70 Ni 20 Rh 10 , respectively. The periodicity along c* is determined to be 0.4 nm from Figure 2c.