and rotational symmetry. In real space, its structural elements (atoms or molecules) are arranged periodically in three-dimensional space according to certain rules. Therefore, periodicity is considered as the most essential characteristic of a crystal structure. Its morphology is mostly manifested as a highly symmetrical polyhedron (Figure 1.3a). In reciprocal space, the periodically arranged structural units of a single crystal material would produce diffraction spots with translation and rotation repeatability. The diffused diffraction spots form diamond patterns centered on the transmission spot (Figure 1.3b). The diamond angle and edge length are the direct transformation of crystal lattice parameters. For polycrystalline materials, the diffraction patterns are sharp diffraction rings centered on the transmission spot.
Figure 1.3 The morphologies of three different kinds of solid materials, as well as their corresponding electron diffraction patterns. (a, b) Crystal, (c, d) quasicrystal, and (e, f) amorphous materials. Source: Panels (a, b) Reproduced with permission from Zhang et al. [4]. Copyright 2009, Royal Society of Chemistry. Panel. (c) Reproduced with permission from Fisher et al. [5]. Copyright 2000, Elsevier Inc. Panel. (d) Reproduced with permission from Zhang et al. [6]. Copyright 1985, Taylor and Francis Group. Panel. (f) Reproduced with permission from Yue et al. [7]. Copyright 2015, Science China Press.
For quasicrystal materials (Figure 1.3c), the atomic arrangement has rotational symmetry but does not have translational symmetry. Its biggest feature is the symmetry that is incompatible with the traditional crystal space lattice (e.g. fifth symmetric axis). In reciprocal space, it also exhibits similar diffraction patterns as the crystals with regular and diffused diffraction spots (Figure 1.3d). The difference is that there is only rotation regularity and no translation regularity.
Because crystals and quasicrystals have great consistency in structure, modern solid-state physics is also accustomed to classifying quasicrystals together into crystals, i.e. materials with sharp diffraction spots (i.e. periodic arrangement of atoms in real space) as crystals, which have the following characteristics:
1 (1) The atomic arrangement of crystal units has long-range symmetry and regularity.
2 (2) Crystals show self-limitation, which means natural-grown crystals without external interference will eventually grow into regular morphologies with high symmetry. It is the geometric basis for the determination of crystals.
3 (3) Crystals obey the law of constancy of interfacial angles, which is the first law of geometric crystallography, and is also the basis for judging crystals in morphology. It states that the angles between two corresponding faces on the crystals of any solid chemical or mineral species are constant and are characteristic of the species. The law holds for any crystals, regardless of size, locality of occurrence, or whether they are natural or man-made.
4 (4) Single crystals are anisotropic.
5 (5) Crystal material has a fixed melting point, and its temperature remains unchanged during the phase transition process.
6 (6) Crystals can produce X-ray diffraction with specific regularity: It is the basis for modern crystallography to judge whether a substance is a crystal or not.
1.2.2 Amorphous Materials
For amorphous materials, the arrangement of atoms does not have long-range symmetry, neither rotational symmetry nor translational symmetry. In real space, it is generally believed that amorphous materials have SRO only in a few angstroms but do not have LRO. It cannot spontaneously embody regular morphology (except spherical), so amorphous materials are generally known as formless (Figure 1.3e). In reciprocal space, the diffraction pattern does not have any diffraction spots or sharp rings but a circular diffraction halo (Figure 1.3f).
Compared with crystalline materials, the features of amorphous materials can be summarized as follows:
1 (1) Atoms in amorphous materials only have fixed atomic arrangement rules in the nearest and the next neighbor (<1 nm). The order of the longer range is still unclear.
2 (2) Because of the long-range disorder of the atomic arrangement, the regular morphology of amorphous materials cannot be obtained by natural growth under non-limited conditions. Therefore, amorphous materials always embody as formless or spherical, driven by surface energy.
3 (3) Amorphous material is physically and chemically isotropic: the homogeneity of the atomic environment determines that they are not as anisotropic as crystals;
4 (4) Compared with crystal, the amorphous material is metastable. The amorphous structure will relax to crystalline state at a high-temperature/high-pressure processing.
5 (5) There is no fixed melting point for amorphous material. It only showed a glass transition temperature. There is no unchanged temperature platform during the phase transition process.
6 (6) Amorphous material does not produce regular X-ray diffraction. The typical X-ray diffraction pattern of amorphous material is a hump at a specific location, rather than a series of peaks in crystal. Its typical electronic diffraction pattern is diffraction halos.
It can be found that the essential difference between amorphous structure and crystals lies in the LRO, and the similarity lies in the high SRO.
SRO means that amorphous atoms only have a high degree of local correlation, which is the result of the strong chemical bonds between the nearest neighbor (including the next neighbor) atoms to maintain as a fixed component solid. This makes the short-range structure of amorphous materials similar to that of crystals, so the SRO is considered as the structural feature of amorphous materials. A large number of simulations and experiments show that the short-order scale of amorphous crystals should be less than 1 nm.
Common diffraction methods, such as X-ray diffraction and selective electron diffraction, are based on LRO. Therefore, it is difficult to directly obtain short-range informations and images of amorphous material. To investigate the average structure information such as the radial distribution function (RDF), the analysis of atomic ordering of amorphous structures is generally based on the fitting results of diffraction (electron, neutron, etc.) or spectroscopy (X-ray fine structure absorption spectroscopy, nuclear magnetic resonance spectroscopy, etc.).
In 2011, Chen Mingwei and Akihisa Inoue [8] of Northeastern University used the most advanced spherical aberration correcting transmission electron microscopy (AC-TEM) technology to reduce the diameter of the coherent electron beam to 3 Å for nanobeam electron diffraction (NBED) analysis to replace traditional selective area electron diffraction (SAED). It was found in the experiment that under the 3.6 Å electron beam, the diffraction pattern of amorphous material showed obvious patterning (Figure 1.4). For the first time, the order of the atomic neighbor and the next-nearest neighbor structure of the amorphous alloy was observed.
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