Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko
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open circles and black filled circles indicate oxygen and silicon atoms, respectively.
Quasicrystals in some sense occupy a niche between crystals and amorphs. They have been discovered in the beginning of 1980s by Dan Shechtman during his studies (by electron diffraction) of the structure of rapidly solidified Al–Mn alloys. Quasicrystals can be described as fully ordered, but non-periodic arrangements of elementary blocks densely covering the space with no voids. An example of filling the 2D space in this fashion, by the so-called Penrose tiles (rhombs having smaller angles equal 18° and 72°), is shown in Figure 1.3. Amazingly that despite the lack of the long-range translational symmetry, quasicrystals, like regular crystals, also produce sharp diffraction peaks (or spots), their positions being defined by the quasi-momentum conservation law in high-dimensional space (higher than 3D, see Section 1.1). In this high-dimensional space (hyperspace), quasicrystals are periodic entities, their periodicity being lost when projecting them onto real 3D space.
Figure 1.3 Dense filling of 2D space by spatially ordered, though non-periodic Penrose tiles (b). Fivefold symmetry regions (regular pentagons) are clearly seen across the pattern. Elemental shapes composing this tiling, i.e. two rhombs with smaller angles equal 18° (blue) and 72° (red), are shown in the (a).
In 1992, based on these findings, the International Union of Crystallography changed the definition of a crystal toward uniting the regular crystals and quasicrystals under single title with an emphasis on the similarity of diffraction phenomena: “A material is a crystal if it has essentially a sharp diffraction pattern. The word essentially means that most of the intensity of the diffraction is concentrated in relatively sharp Bragg peaks, besides the always present diffuse scattering.” In 2011, Dan Shechtman was awarded Nobel Prize in Chemistry “for the discovery of quasicrystals.”
1.1 Crystal Symmetry in Real Space
Across this book, we will focus on physical properties of regular crystals, amorphs and quasicrystals being out of our scope here. Thinking on conventional crystals, we first keep in mind their translational symmetry. As we already mentioned, the long-range periodic order in crystals leads to translational symmetry, which is commonly described in terms of Bravais lattices (named after French crystallographer Auguste Bravais):
The nodes, rs, of Bravais lattice are produced by linear combinations of three non-coplanar vectors, a1, a2, a3, called translation vectors. The integer numbers in Eq. (1.1) can be positive, negative, or zero. Atomic arrangements within every crystal can be described by the set of analogous Bravais lattices.
Classification of Bravais lattices is based on the relationships between the lengths of translation vectors, |a1| = a, |a2| = b, |a3| = c and the angles, α, β, γ, between them. In fact, all possible types of Bravais lattices can be attributed to seven symmetry systems:
Triclinic: a ≠ b ≠ c and α ≠ β ≠ γ;
Monoclinic: a ≠ b ≠ c and α = β = 90°, γ ≠ 90°; in this setting, angle γ is between translation vectors a1 (|a1| = a) and a2 (|a2| = b); whereas the angles α and β are, respectively, between translation vectors a2^a3 and a1^a3;
Orthorhombic : a ≠ b ≠ c and α = β = γ = 90°;
Tetragonal: a = b ≠ c and α = β = γ = 90°;
Cubic: a = b = c and α = β = γ = 90°;
Rhombohedral: a = b = c and α = β = γ ≠ 90°;
Hexagonal: a = b ≠ c and α = β = 90°, γ = 120°.
A parallelepiped built by the aid of vectors a1, a2, a3 is called a unit cell and is the smallest block, which being duplicated by the translation vectors allows us to densely fill the 3D space without voids.
Translational symmetry, however, is only a part of the whole symmetry in crystals. Atomic networks, described by Bravais lattices, also possess the so-called local (point) symmetry, which includes lattice inversion with respect to certain lattice points, mirror reflections in some lattice planes, and lattice rotations about certain rotation axes (certain crystallographic directions). After application of all these symmetry elements, the lattice remains invariant. Furthermore, rotation axes are defined by their order, n. The latter, in turn, determines the minimum angle,
To deeper understand why some rotation axes are permitted, while others not, let us consider the covering of the 2D space by regular geometrical figures, having n equal edges and central angle,