Abstract
The foundations of crystallography are the focus of this chapter. The geometric description of crystals is presented. The distinction of lattice and crystal is emphasized. The notions “translation lattice”, “motif” and “crystal structure” are dealt with. The treatment includes the Bravais translation lattices and the definition of the Miller and Miller-Bravais indices. Extensive discussion is devoted to the (more or less close packed) crystal structures of the elements, substitutional and interstitial solid solutions (including the emergence of ordering (“superstructures”)). Attention is paid to the (X-ray) diffraction analysis of crystal structures. The stereographic projection is introduced, also as a tool for the presentation of texture (preferred orientation) of polycrystals in pole figures and inverse pole figures. Finally, aperiodic crystals, i.e. crystals which do not show translational periodicity, as incommensurately modulated atomic structures and quasicrystals, are discussed. The chapter closes with a consideration on what (actually) defines a crystal.
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Notes
- 1.
This statement is too strong. Slight distortions of a symmetrical appearance may be a prerequisite to achieve the strongest appeal: man and women would not feel attracted to perfectly symmetrical women and man, respectively (here is meant that the midplane through the facial front of men and women would be a perfect mirror plane). This effect may also explain the attractivity of the specifically distorted symmetry, but still “regular” appearance, of some two-dimensional patterns designed by Vasarély, as compared to the perfectly translationally symmetrical, two-dimensional patterns created by Escher, examples of both of which decorate walls in my private home.
- 2.
Metals can also crystallize as a single crystal of macroscopical dimensions. However, usually, we are confronted with polycrystalline specimens, where the individual crystals exhibit a non-regular morphology, as, for example, forced by the (copious) nucleation of crystals in a melt upon cooling and their unconstrained growth until “hit upon” by neighbouring outgrowing crystals (see the "Intermezzo: Making Grain Boundaries Visible" and Fig. 4.2 further on in this section).
- 3.
Many have suggested, at certain instances of time, the completeness of scientific understanding in a certain field of science, and time and again have been proven to be wrong. A typical contemporary example concerns “thermodynamics”: it has often been said that this field has become such mature that nothing of great significance can be added, but current work on the second law of thermodynamics, in systems remote from equilibrium where (local) “order” develops out of “chaos”, or the recent development of “interface thermodynamics” represent activities indicating that “thermodynamics” is “hot”, also today. Even more dramatic, it is recalled that at the end of the nineteenth century the view was generally held that the entire discipline of “physics” was completed. Then, came relativity and quantum theory; so, one is wise not to condemn a certain field of scientific activity to be “dead” or in a rounded-off state. Unfortunately, science policy makers and scientists over and over again step into this trap. Scientific breakthroughs remain unpredictable and cannot be planned by human beings.
- 4.
This remark is restricted to crystals as meant usually: crystals exhibiting long-range translational periodicity (see what follows). So-called aperiodic crystals (see Sect. 4.8 and the epilogue to this chapter) deserve separate treatment.
- 5.
However, the atomic arrangement in an amorphous solid, although lacking long-range order, can exhibit short-range order. For example, in case of amorphous silicon, each individual silicon atom tends to be surrounded by four silicon atoms in tetraedron configuration (cf. the discussion on covalent bonding in Sect. 3.4). Further, if an amorphous solid is composed of atoms of more than two elements, then a tendency can occur for the atoms to be surrounded preferentially by unlike atoms (i.e. A atoms in an A/B mixture would tend to have B atoms as nearest neighbours). So, to describe the atomic structure of an amorphous alloy as completely chaotic or structure-less is an overexaggeration.
- 6.
Considerable confusion occurs regarding usage of the terms (translation) lattice and (crystal) structure. As argued in the above text, the translation lattice is a geometrical abstraction to describe the translational periodicity of the crystal considered. Only upon substitution of the translation-lattice points by the motif (sometimes also called “basis”), the crystal structure is obtained. Unfortunately, in many literature sources, the word lattice is used, whereas (crystal) structure is meant. Against this background, a “highlight” of such conceptual chaos then occurs if one speaks of “lattice defects”, whereas “crystal-structure defects” (also often called “imperfections” or “mistakes”; see Chap. 5) are meant. For an extensive discussion of various such abuses of crystallographic notions, with in fact well-defined meanings, see Nespolo (2019). This is the reason that I (try to) use adjectivally and consistently the words “translation” (or “Bravais translation”) in connection with the substantive “lattice” (to make clear that the crystal structure is not meant), throughout at least this chapter. But I confess that, at other places (where likely no confusion can arise about what is meant; e.g. when the motif is one atom as for many elemental metal crystals), I may have fallen victim to the here discussed sin as well… At many places in the also recent literature, in the fields of materials science, solid-state physics and solid-state chemistry, the word “lattice” in a text can mean either “translation lattice” or “crystal structure”. Although the context usually will make clear the intention of the author(s), a pure and correct use of terms well defined in crystallography must be preferred.
- 7.
Consider the unit cell spanned by the vectors a′ and b′ as indicated in Fig. 4.3b. Evidently, the contents of this unit cell are given by (1) the fractional contributions of the motifs at the corners which totals one motif (each motif at a corner of the unit cell is shared by four unit cells; it should be noted that in general the contributions of the motifs at the four corners are unequal, due to the angle γ not being 90°; cf. Fig. A4.1a and Table A4.1 and their full discussion in the Appendix to this chapter, but do recognize that the discussion in this appendix focusses on the number of (each type of) atoms in the unit cell and not on the number of motifs), plus (2) the fractional contributions of the motifs at the middle of the two sides of the unit cell parallel to b′ (these motifs are shared by two unit cells and the fractional contribution of each of these motifs equals 1/2). Hence, in total, the unit cell contains two motifs.
- 8.
Whereas in science distance units as m, cm, mm, μm, nm, pm,… are generally used, on the basis of international agreement, the unit Å (angstrom) = 0.1 nm = 1 × 10−10 m is still normally used in the field of crystallography, in agreement with a recommendation of the International Union of Crystallography (IUCr), recognizing that the size of and distances between atoms in crystal structures are of the order of 1 Å. For example, see the use of distance units in the leading journals of this field, as Acta Crystallographica, Journal of Applied Crystallography and Zeitschrift für Kristallographie.
- 9.
The trigonal crystal system has also been designated as rhombohedral crystal system; but see footnote 10.
- 10.
Regarding the nomenclature for crystal systems and Bravais translation lattices, the advice of the International Union of Crystallography has been adopted: The adjective rhombohedral is used here to designate a specific Bravais translation lattice; the adjective trigonal is reserved for the crystal-structure types (corresponding to a specific collection of space groups; cf. Sect. 4.1.2) sharing the unit-cell parameter prescriptions as indicated for the trigonal crystal system (see also footnote 9). Further, see Hammond (2001) and Schwarzenbach (1996).
- 11.
This convention certainly did and does contribute to the confusing use of the term “lattice” where “(crystal) structure” is meant, as discussed extensively and criticized in footnote 6.
- 12.
A set of hkl with a common divisor has relevance in the discussion of the diffraction by crystals. See the discussion of the so-called Laue indices, with hkl replaced by HKL, in Sect. 4.5.
- 13.
More correctly: two identical possibilities: the third atom can be attached to either the top (one side) or bottom (opposite side) of the already contacting pair of spheres.
- 14.
Note that the stacking sequence BA.BA.BA.etc. does not produce a mirror structure of the stacking sequence AB.AB.AB.etc., but is fully identical to it.
- 15.
As indicated in Sect. 4.2.1.2 for the cubic close packed (Cu type, f.c.c.) structure, a spacing change for the stacks of close packed layers perpendicular to any of the four <111>-directions leads to a distortion rendering the structure non-cubic. In contrast, for the hexagonal close packed (Mg type) structure, a distortion of the ideal (hexagonal) close packed structure by changing c/a does not lead to a change of crystal symmetry.
- 16.
A Wigner-Seitz cell contains one atom and Wigner-Seitz cells can be arranged such that they fill space completely. Hence, according to conjectures as those by Hooke and Haüy discussed in the begin of this chapter, the Wigner-Seitz cell can be considered as a building unit of a crystal (see Fig. 3.28). Yet, from a crystallographic point of view, a Wigner-Seitz cell is not a primitive unit cell because it is not defined by translation vectors of the lattice.
- 17.
The last described situation happens at so-called phase boundaries and triple points of a single component system, with the combination of atoms in the chemical formula for the compound considered taken as the “component” (cf. Sect. 7.5.1).
- 18.
However, elasticity theory has been applied successfully to describe quantitatively the change of the (average) lattice parameter of, and its variation in, a matrix containing misfitting precipitates (i.e. inclusions much larger than a single atom; see E.J. Mittemeijer, P. van Mourik, ThH de Keijser, Philos. Mag. A 43, 1157–1164 (1981), E.J. Mittemeijer, A. van Gent, Scripta Metallur. 18, 825–828 (1984) and J.G.M. van Berkum, R. Delhez, ThH de Keijser, E.J. Mittemeijer, Phys. Status Solidi (a) 134, 335–350 (1992)). An overview of changes of the overall lattice parameter in specimens by elastic accommodation of misfitting, coherent and incoherent (see at the end of Sect. 5.3) precipitates is provided by T. Akhlaghi, S.R. Steiner, E.J. Meka, Mittemeijer, J. Appl. Crystallogr. 49, 69–77 (2016). See also the “Intermezzo: Coherent and incoherent interfaces versus coherent and incoherent diffraction” at the end of Sect. 5.3
- 19.
Here, the following remark is in order. At the applied nitriding temperature (<600 °C) and pressure (usually 1 atm), these iron-nitride phases are not equilibrium phases: they are prone to decomposition in iron and nitrogen gas (see the "Intermezzo: The Fe–C and Fe–N phase diagrams" at the end of Sect. 9.5.2.1). As a consequence these iron-nitride compound layers can contain an amount of porosity due to the precipitation of nitrogen gas during nitriding, particular in the “oldest” part of the compound layer (i.e. the surface adjacent part). Such porosity generally has a negative effect on the mechanical properties (perhaps with exception for the case of friction under lubrication). This leads to dedicated nitriding treatments to minimize or even to avoid this porosity, or one removes mechanically the porosity affected surface adjacent part of the compound layer after the nitriding treatment.
- 20.
W. L. Bragg (“Sir Lawrence Bragg”) was the son of W. H. Bragg. Both, father and son, have contributed, separately and in cooperation, enormously to the field of the diffraction of X-rays by crystals. However, it was the son who first derived what is now known as “Bragg’s law”. That it had to be him, and not his father, may be due to the initial inclination of the father to focus on a particle-like, rather than a wave-like character of the X-rays. The joint results by father and son Bragg constitute an impressive example of the fruitful effect of family ties for the progress of science. Another such example is provided by the Burgers brothers, W. G. and J. M. (see the “Intermezzo: a historical note about the Burgers vector” in Chap. 5). The fascinating, early history of X-ray crystallography has been recorded in two books: (1) J.M. Bijvoet, W.G. Burgers, G. Hägg (eds.) Early Papers on Diffraction of X-rays by Crystals (Published for the International Union of Crystallography by A. Oosthoek’s Uitgeversmaatschappij N.V., Utrecht, The Netherlands, Volume I, 1969 and Volume II, 1972) and (2) P.P. Ewald, Fifty Years of X-ray Diffraction (Published for the International Union of Crystallography by A. Oosthoek’s Uitgeversmaatschappij N.V., Utrecht, The Netherlands, 1962)
- 21.
For example, for a cubic crystal, the (100), (010), (001), (−100), (0−10) and (00−1) sets of lattice planes are equivalent, the precise (hkl) notation being dependent on how the crystal axes have been defined for the individual crystal. With the notation {hkl}, i.e. the use of the braces, it is indicated here that any set of equivalent lattice planes of {hkl} type can contribute to the reflection concerned.
- 22.
The scattering power of a unit cell can even be zero for a certain HKL; i.e. this HKL reflection does not occur in the diffraction pattern. One then speaks of “systematic extinction”. In a sense, this effect is artificial: it depends on the unit cell chosen and occurs only for non-primitive unit cells in which more than one motif (cf. Sect. 4.1.1) is present such that the waves scattered by the various motifs in the unit cell interfere destructively for specific HKL’s (again: note that the HKL notation of a reflection depends on the choice of unit cell) and as a consequence the unit cell has zero scattering power for these reflections. Example: consider the f.c.c. unit cell relevant for many metals (Cu, Al, etc.). This is a non-primitive unit cell containing 4 motifs (= 4 identical metal atoms; cf. Sect. 4.2.1.2). For this choice of unit cell, it can be shown that the 100 reflection is extinguished. This is no longer true, if, for example, departing from the f.c.c. unit cell, with a random distribution over the atomic sites of the crystal structure of Cu and Au atoms of a solid solution of the composition AuCu3, the distribution of the Cu and Au atoms becomes ordered (cf. bottom left of Fig. 4.35). Then, the 100 reflection is no longer extinguished (see Fig. 4.36a, b).
- 23.
The adjective “standard” refers to a SGP of a low index plane of the crystal. Such standard SGPs are available for the various crystal systems (e.g. see Johari and Thomas 1969), but, of course, can also be generated (simply) by available (commercial and free) software.
- 24.
A pole figure should not be referred to using the Laue indices, HKL, of the reflection used for measurement of the pole figure (See Sect. 4.5 for the distinction between Miller and Laue indices). For example, for a f.c.c. crystal structure, the notion “200 pole” (HKL = 200) is meaningless, in contrast with “{100} pole” (hkl = 100). In case of the example considered, a {100} (hkl = 100) pole figure can be measured employing a 200 (HKL = 200) reflection (as the 100 (HKL = 100) reflection is extinguished; see footnote 22 in Sect. 4.5). Violations of this ruling occur frequently in the literature, i.e. within the framework of the example discussed, often “{200} pole figures” are published....
- 25.
Electron backscatter diffraction (EBSD) is usually carried out in a scanning electron microscope (SEM; see Sect. 6.9). The electrons impinging on the surface (tilted with respect to the incident electron beam) and penetrating surface adjacent material of a crystalline solid, after backscattering may be diffracted, according to Bragg’s law (Eq. (4.9)), by lattice planes inclined with respect to the surface of the specimen. Such diffracted electrons may escape from the surface of the material over a depth ranging till, say, 10–40 nm. These thus back-scattered and diffracted electrons can produce a diffraction pattern on a detector (screen). This diffraction pattern reveals so-called Kikuchi bands where each band corresponds to one set of diffracting lattice planes. The (mostly computerized) interpretation of this diffraction pattern (pattern of Kikuchi bands) leads to determination of the crystal orientation in the specimen frame of reference.
- 26.
It can be shown that the (X-ray) diffraction pattern of a crystal can be conceived as the Fourier transform of its (electron) density. (The mathematical operation “Fourier transform” cannot be introduced in this book, but that does not obstruct understanding the essence of this footnote) Thus, it has been proposed to define “long-range positional order” as equivalent with the occurrence of sharp peaks in the Fourier spectrum of the object. In this way, one would have given an operational definition of “long-range positional order”, and by avoiding a reference to an experimental diffraction pattern but instead relying on a mathematical operation to be applied on the object considered, some obscurity in the definition of a crystal in the sense discussed here could be avoided (Lifschitz 2007).
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Appendix: How to Deal with Atoms at Unit-Cell Boundaries
Appendix: How to Deal with Atoms at Unit-Cell Boundaries
The composition and the density of a unit cell are equal to the composition and density of the whole crystal. Thus, the information about geometry (a ≡│a│, b ≡│b│ and c ≡│c│, α, β, γ) and contents (fractional coordinates 0 ≤ x, y, z < 1; compare also Table 4.1) of the unit cell suffices to calculate the composition and the density of massive, solid crystalline material, provided the grain-boundary density is insignificant (Thus, this statement need not hold for nanocrystalline materials; i.e. for crystal/grain sizes less than 100 nm).
A crystal structure can be represented graphically by a unit cell of that crystal structure, displaying atoms with fractional coordinates in the range 0 ≤ x, y, z ≤ 1 (note that for drawing the unit cell the second “≤” is not a “<” like above). Upon counting the atoms on the basis of a unit-cell drawing, a problem arises for atoms located at fractional coordinates with either x = 0 (or 1) or y = 0 (or 1) or z = 0 (or 1), i.e. for atoms somewhere at the bounding faces or edges or at corners of the unit cell. Two examples of such cases are shown in Fig. 4.65a, b. Table 4.8 lists the fractional coordinates of the atoms observed in these unit-cell drawings.
a Unit cell of the two-dimensional crystal structure of the hypothetical compound “BlGrWh”. b Unit cell of the three-dimensional crystal structure of the physical compound CsCl. The fractional coordinates x and y in (a) and x, y and z in (b) have also been indicated. c A Cs atom, of the crystalline compound CsCl, at a corner of the unit cell considered in (b), contributes to the eight unit cells sharing this corner
The drawn unit cell of the CsCl structure in Fig. 4.65b shows 8 Cs atoms but only 1 Cl atom (see Table 4.8). The drawn unit cell may thus suggest erroneously that the composition of this ionic compound is given by the formula Cs8Cl, whereas the true composition is represented by the formula CsCl. The discrepancy is resolved if one recognizes that the Cs atoms, having their centres of mass located at the corners of the unit cell, also contribute to/are part of the adjacent, neighbouring unit cells. Hence, adopting the Cs atoms as solid spheres, only a fraction 1/8 of each sphere is a part of the drawn unit cell, as every corner of the unit cell is equally (cf. footnote a to Table 4.8!) shared by 8 unit cells (see Table 4.8 and Fig. 4.65c). The fractional contribution of an atom that would lie on an edge of the unit cell is 1/4, because every edge of the unit cell is shared by 4 unit cells. The fractional contribution of an atom that would lie on a face of the unit cell is 1/2, since every face of the unit cell is shared by 2 unit cells.
Thus, it now simply follows that the unit cell as drawn in Fig. 4.65b contains 8 × 1/8 Cs = 1 Cs atom and 1 Cl atom, in agreement with the true composition of the compound.
In general, if unit cells are considered, as in drawings, with all atoms with fractional coordinates in the range 0 ≤ x, y, z ≤ 1, then, for counting the atoms in the unit cell, and if any of their fractional coordinates equals 0 or 1, one has to sum them according to their partial contributions to the unit cell considered. This complication is obviously avoided if one only considers/draws all atoms with fractional coordinates in the range 0 ≤ x, y, z < 1 (note the use of “<” instead of “≤”); then all atoms considered/drawn contribute fully to the unit cell considered/drawn. But such graphical presentations are not made usually.
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Mittemeijer, E.J. (2021). Crystallography. In: Fundamentals of Materials Science. Springer, Cham. https://doi.org/10.1007/978-3-030-60056-3_4
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