Читать книгу Introduction to Solid State Physics for Materials Engineers - Emil Zolotoyabko - Страница 12

Atomic order in crystals.

Local and translational symmetries.

Symmetry impact on physical properties in crystals.

Wave propagation in periodic media.

Quasi-momentum conservation law.

Reciprocal space.

Wave diffraction conditions.

Degeneracy of electron energy states at the Brillouin zone boundary.

Diffraction of valence electrons and bandgap formation.

In contrast to liquids or gases, atoms in a solid state, in average (over time), are located at fixed atomic positions. The thermally assisted movements around them or between them are strongly limited in space (as for thermal vibrations in potential wells) or have rather low probabilities (as for long-range atomic diffusion). According to the types of the averaged long-range atomic arrangements, all solid materials can be sub-divided into the three following classes, i.e. regular crystals, amorphous materials, and quasicrystals.

Most solid materials are regular (conventional) crystals with fully ordered and periodic atomic arrangements, which can be described by the set of translated elementary blocks (unit cells) densely covering the space with no voids. Nowadays, using the advanced characterization methods, such as high-resolution electron microscopy or scanning tunneling microscopy, it is possible to directly visualize this atomic periodicity (Figure 1.1). Due to the translational symmetry, the key phenomenon – namely, diffraction of short-wavelength quantum beams (electrons, X-rays, neutrons) – takes place. As we show in the following text, sharp diffraction peaks (or spots), which are the “visiting card” of crystalline state, are originated from the quasi-momentum (quasi-wavevector) conservation law in 3D.

In contrary, amorphous materials, being characterized by some kind of short-range ordering, do not reveal atomic order on a long range (Figure 1.2). In other words, certain correlations between atomic positions exist within a few first coordination spheres only and rapidly attenuate and disappear at longer distances. Correspondingly, diffraction patterns taken from amorphs show diffuse features only (called amorphous halo), rather than sharp diffraction peaks.

Figure 1.1 High-resolution scanning transmission electron microscopy image of atomic columns in crystalline GaSb. Cations and anions within dumbbells are separated by 0.15 nm.

Figure 1.2 Structural motifs in silicon dioxide (SiO₂): (a) – ordered atomic arrangement in crystalline quartz; (b) – disordered arrangement in amorphous silica. Large 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.”