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Fundamentals of Crystalline Materials

M.C. Escher (1959)

M.C. Escher (1959)

The beauty of natural crystals as caused by the regular polyhedral shape, their symmetry, beautiful color, brightness, and, in some cases, clarity has been fascinating human beings all the time. Nowadays, a wide range of modern materials can be synthesized or grown artificially as crystals. The broad spectrum of crystalline materials includes metals, semiconductors, superconductors, ceramics, polymers, organic molecular crystals, proteins, and so on. First of all, we have to answer the following fundamental questions: What are the characteristic features of the crystalline state? What is a crystal? What defines the degree of crystallinity?

1.1 Crystalline State

Our fundamental knowledge about crystals on a macroscopic scale goes back to the early extensive studies of the morphology of natural crystals, the minerals. Historically, these studies of external faces and the measurement of the precise angles between them were the key for the derivation of the fundamental laws of morphological symmetry of crystals. However, we should be aware of the fact that the regular polyhedral shape of a crystal (form or a combination of them) is, of course, a major characteristic macroscopic feature of crystals but not a proper feature for their definition. Both natural as well as synthetic crystals will grow only with an idiomorphic shape under definite growth conditions. Figure 1.1 shows examples of individual gypsum crystals that illustrate the variety of different crystal “habits,” which define the general shape. The collection of synthetic single crystals shown in Figure 1.2 were grown using a growth technology that do not allow for a polyhedral growth of the crystals.

Figure 1.1 (a–c) Gypsum crystals shown in the world-renowned mineral collection “terra mineralia” in castle Freudenstein Freiberg/Saxonia, Germany (www.terra-mineralia.de).

Figure 1.2 Collection of synthetic single crystals (Leibniz Institute for Crystal Growth Berlin, Germany).

The most fundamental macroscopic properties of crystals are

• homogeneity
• anisotropy
• symmetry.

Macroscopic homogeneity means the crystal is chemically and physically uniformly built. We can consider it as a continuum. The physical properties in different volume elements will show equal characteristics in parallel directions. It is obvious that this treatment is an abstraction and approximation that is applicable only at the macroscopic scale. Anisotropy means for us directional dependence of physical properties. All crystals are anisotropic with respect to some of their physical properties. When we try to determine the thermal expansion of a calcite crystal along the threefold c-axis and perpendicular to it, we will measure different values of different signs. Along the c-axis, we will find an expansion of the calcite crystal, and perpendicular to it, the crystal shows the anomaly of thermal dilatation with increasing temperature. Anisotropy of crystals does not mean that all crystal properties have to show a different physical behavior in different directions. For example, cubic crystals are optically isotropic. They will, therefore, show neither polarization nor double refraction.

In general, the concept of symmetry is a key to the description of crystals. When we consider “symmetry” as one of the main macroscopic features of crystals, we mean the symmetry concept (symmetry operations and symmetry elements) for describing external forms of crystals, the morphological symmetry of crystals. It is quite clear that there is a basic correspondence between the morphological symmetry (outer symmetry) and the structural symmetry (inner symmetry) of crystals. The fundamentals for the description of the outer and inner symmetries of a crystal will be outlined in Sections 1.2–1.4.

Strictly speaking, all the fundamental macroscopic features of crystals such as homogeneity, anisotropy, and morphological symmetry are the result of the internal order of the crystals at the microscopic level. It should be noted that a periodic arrangement of building units (atoms, groups of atoms, ions, and molecules) of crystals was already predicted by scientists from their comprehensive studies of macroscopic properties of crystals long before von Laue, Friedrich, and Knipping confirmed the periodic order of crystals experimentally by their famous X-ray diffraction experiments in 1912. A year later, father William H. Bragg and son William L. Bragg solved the first crystal structures (NaCl, KCl, CaF2, ZnS, FeS2, NaNO3, and CaCO3) from X-ray data (for comprehensive historical survey of crystallography, see, e.g., [1–4]). Nowadays, crystal structure analysis by means of X-ray and electron and neutron diffraction is well developed and routinely applied. Furthermore, modern microscopic techniques such as high-resolution transmission electron microscopy (HRTEM) allow the direct imaging of the atomic arrangement in crystalline structures. Figure 1.3a shows a HRTEM image of a (100) oriented GaAs crystal and the corresponding electron diffraction pattern. For the applied imaging conditions and the specimen thickness given, the white spots represents the projected atomic rows along the [100] zone axis. We can easily construct a lattice were the nodes are occupied by atoms, reflecting their periodic arrangement. The corresponding diffraction pattern (Figure 1.3b) consists of sharp spots (Bragg peaks) situated also on a lattice. This is the so-called reciprocal lattice of the crystal.

Figure 1.3 HRTEM micrograph of [001] oriented GaAs (a) and the corresponding electron diffraction pattern (b).

With respect to the macroscopic features, we can simply define crystals as homogeneous anisotropic solids. These solids are composed of a three-dimensional periodic arrangement of matter, which forms the microscopic structure of the crystal. As we will show in Sections 1.2 and 1.4, the periodic order can mathematically be described by translation lattices. A “decoration” of the lattice points with matter (atoms, ions, and molecules) generates then the crystal structure. So far, such a definition of crystals is strictly connected with order and periodicity. Periodicity means a periodic infinite repetition of some basic structural unit in all directions by translation. The macroscopic feature of homogeneity is only strictly fulfilled when we consider an infinite space lattice with identical lattice points and identical surrounding. The definition given above within the framework of “classical crystallography” describes what we mean with the term ideal crystal. The following questions arise when we are dealing with real crystals, which are finite: What are the boundary conditions that allow for the application of the symmetry concept of an ideal crystal for the description of real crystals? How has one to define ordered structures that lack three-dimensional periodicity within our concept of crystalline matter?

A real crystal is always finite. When we have large-sized crystals, the deviations from the infinity of the underlying lattice concept are negligible. The separation of the matter decorating the lattice points is in the order of 10−8 cm (1 Å). Thus, for a crystal size of 1 cm, we have ∼108 periodically arranged atoms. However, our real crystal may contain local deviations of chemical composition and various kinds of crystal imperfections (crystal defects) of different dimensions, which may disturb or even destroy the symmetry of our crystal.

A real crystal consisting ideally of only one grain (a continuous lattice without any grain boundary) is called a perfect single crystal. Practically, all single crystals are imperfect crystals, containing various kinds of crystal defects. A crystal containing a few grain boundaries is still a single, however imperfect, crystal. Contrary to this, a polycrystal is composed of many crystallites (grains) of different size and orientations with an equal probability for all possible orientations. The entity of crystallite orientations is called texture. The different stages of crystallinity from single crystalline via a texture to polycrystalline can be determined by means of diffraction as shown in Figure 1.4.

Figure 1.4 Correlation of real structure of crystals and the corresponding diffraction patterns.

Another important classification criteria for real crystals is connected to their grain size, where the terms microcrystalline (diameter of the grains, d > 1 µm), subfine grain-sized crystals (d < 1 µm), and nanocrystalline (d < 100 nm) are...