1A Crystal

Crystallography is based on a specific symmetry which is called translationalsymmetry. The exactmeaning of translational symmetry will be clearer when readingthis book. On an intuitive level we can define a crystal as a 3D object densely built ofrepeating structural units located at fixed distances from their neighbors, thedistances being unchanged across the crystal volume. This picture is a result ofscientific developments over hundreds of years.

In fact, attempts to understand the internal structure of materials had beenundertaken already in ancient times. Most renowned is the speculative atomictheory of ancient Greeks (primarily by Leucippus and Democritus). At the end of themiddle ages and, especially, during the Renaissance period, people began torecognize the importance of measurements for establishing scientific concepts.Perhaps Dutch jewelers were the first to understand that diamond stone is builtdifferently from a piece of amber. However, the first quantitative results are dated by1669, when Danish professor of anatomy Nicolas Stenon, the court physician ofDuke of Toscana, published a paper, in which he established the first law ofcrystallography – the law of constant angles. While crystals of the same material candiffer by their external shape and size, the angles between the corresponding crystalfaces are always identical. In his study, Stenon had measured the angles betweenfaces forming the habits (typical shapes) of large natural crystals of quartz (SiO2)and hematite (Fe2O3).

The next important step was accomplished only in 1784, by French mineralogistRen�e Ha€uy, who established the second law of crystallography – the law of rationalparameters. According to Ha€uy, if certain edges of a crystal are taken as the axes ofcoordinate system, than the ratios of segments cut by two dissimilar crystal faces onany axis are always found to be rational fractions. In other words, the segmentsmentioned can be expressed as integer numbers of some elementary lengths. In fact,the exact meaning of this law was understood later in the context of so-called Millerindexes (see Chapter 3), which were introduced only in 1839. Anyway, it was a greatdiscovery, since on that basis one canfirmly conclude that the crystal consists of somerepeating blocks, the integer number of which within crystalline specimens provides

Basic Concepts of Crystallography, Emil Zolotoyabko.� 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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the law of rational parameters. The presence of elementary blocks assembledwithoutvoids assure 3D periodicity on a scale dictated by the block size. Ha€uy assumed thatelementary blocks, which form crystals, are small polyhedrons. Therefore, longbefore modern powerful methods of structural analysis, by measuring the crystalhabits only, the spatial periodicity of crystalline structureswas deduced.Nowadays, byusing high-resolution transmission electron microscopy or scanning tunnelingmicroscopy, the periodic atomic networks within crystals or on crystal surface canbe directly visualized (see Figures 1.1 and 1.2, respectively).

In fact, it is amazing that such giant numbers of atoms or molecules (�1023) aredisposed at equal distances from each other, and it is worth discussing very briefly thereasons behind that. Strictly speaking we have no comprehensive quantitativedescription of such cooperative phenomenon. We know only that the interatomicpotential consists of two terms of opposite signs reflecting short-distance repulsionbetween ion cages and long-distance attraction between ions via electron �glue�.Competition between these two contributions provides theminimumof the potentialat some fixed distance, rd (see Figure 1.3), which could be considered as theinteratomic distance in a crystal. However, crystal formation is a cooperative actionof billions and billions of atoms and we are still far from being able to calculate thesecollective effects with high enough accuracy.

Nevertheless, thermodynamics predicts that, at least at zero absolute temperature,T¼ 0, the structurally ordered state is most favorable. According to the third law ofthermodynamics, the entropy, S, of the system tends to zero, when T ! 0. In turn,

Figure 1.1 High-resolution transmission electron microscopy image of atomic columns in theInAs/GaSb superlattice.

6j 1 A Crystal

Figure 1.2 Scanning tunneling microscopy image (29.7� 29.7 nm) of the periodically arrangedatomic rows on the cleavage surface of the InAs/GaSb superlattice.

Repulsion

Po

ten

tial

rd

Distance, r

Attraction

Figure 1.3 Typical behavior of interatomic potential as a function of the interatomic distance, r.Sharp ascent left-side from the equilibrium point, rd, is due to repulsive forces between ions, whileattractive forces result in the tile slowly approaching zero potential at large distances, r� rd.

1 A Crystal j7

entropy, S, is related to the number of possible system configurations, N, by theBoltzmann equation:

S ¼ k lnN ð1:1Þwhere k is the Boltzmann constant. It follows from Equation 1.1 that in order to haveS¼ 0 at T¼ 0, the value ofN should be one, that is, the single (unique) configurationof the system is realized. It is evident that this corresponds to the ordered state, sinceany deviation from order towards disordered state could be realized in manyequivalent ways. In fact, if we have a number of spatially ordered atoms in oursystem, tiny shift of each of them means certain disorder and thus the number ofdisordered configurations grows rapidly with the number of atoms involved.

If temperature T> 0, thermodynamics allows different configurations of thesystem to exist, including disordered ones. Practically, we know that at roomtemperature some solid materials exist in the amorphous (i.e. structurally disor-dered) form, which is possible, at least, from the entropy point of view. However, theamorphous form of a material is the exception rather than the rule, since it isassumed that even at finite temperature there are crystalline forms with lower freeenergy. If so, the amorphous state is always metastable, that is, it seemingly existsbecause of the insignificant diffusion rates or due to some stabilizing factors, such asfor example, a high concentration of impurity atoms or other lattice defects,inhomogeneous strain fields, and so on.

Nonetheless, most solid materials are crystals, and we begin by describing theirsymmetry properties.

8j 1 A Crystal