in (b)?15 1.15 Specify the directions of a*, b* and c* in a hexagonal crystal relative to the lattice vectors a, b and c. What is the angle between a* and b*? Determine the magnitudes of a*, b* and c* in terms of the lattice parameters a and c.Using the formula |r*hkl| = 1/dhkl, show that the interplanar spacing of the hkl planes in a hexagonal crystal is given by the formula:Hence show that the planes (hkl), (h, −(h + k), l) and (k, −(h + k), l) have the same interplanar spacing.
16 1.16 Show that in a triclinic crystal the normal to the plane (hkl) is parallel to the vector [uvw] if:Hence, or otherwise, determine a suitable equation to determine the indices of the vector [uvw] parallel to the normal to the plane (hkl) in a triclinic crystal. Hint: it is useful to define an orthonormal set of vectors with respect to which the basis vectors a, b and c of the triclinic crystal are defined.
17 1.17 Using the hint in Problem 1.16, or otherwise, confirm that the volume, V, of the unit cell of a triclinic crystal with cell dimensions a, b, c, α, β and γ is given by the formula:
Suggestions for Further Reading
1 Allen, S.M. and Thomas, E.L. (1999). The Structure of Materials. New York: Wiley.
2 Buerger, M.J. (1963). Elementary Crystallography. New York: Wiley.
3 Buerger, M.J. (1970). Contemporary Crystallography. New York: McGraw‐Hill.
4 Giacovazzo, C. (ed.) (2011). Fundamentals of Crystallography (International Union of Crystallography Texts on Crystallography – 15), 3rd Edition. Oxford: Oxford University Press.
5 Hammond, C. (2015). The Basics of Crystallography and Diffraction (International Union of Crystallography Texts on Crystallography – 12), 4th Edition. Oxford: Oxford University Press.
6 Phillips, F.C. (1971). Introduction to Crystallography, 4th Edition. Edinburgh: Oliver & Boyd.
7 Rousseau, J.‐J. (1998). Basic Crystallography. Chichester: Wiley.
References
1 [1] Miller, W.H. (1839). A Treatise on Crystallography. Cambridge: J. & J.J. Deighton.
2 [2] Whewell, W. (1825). A general method of calculating the angles made by any planes of crystals, and the laws according to which they are formed. Phil. Trans. Roy. Soc. Lond. 115: 87–130.
3 [3] Lewis, W.J. (1899). A Treatise on Crystallography. Cambridge: Cambridge University Press.
4 [4] Aroyo, M.I. (ed.) (2016). International Tables for Crystallography, 6th, Revised Edition, Vol. A: Space‐Group Symmetry, published for the International Union of Crystallography. Chichester: Wiley International.
5 [5] Santoro, A. and Mighell, A.D. (1970). Determination of reduced cells. Acta Crystallogr. A 26: 124–127.
6 [6] Niggli, P. (1928). Handbuch der Experimentalphysik, vol. 7, Part 1: Krystallographische und Strukturtheoretische Grundbegriffe, Chapter 4, §1. Leipzig, Germany: Akademische Verlagsgesellschaft m. b. H.
7 [7] Buerger, M.J. (1963). Elementary Crystallography, pp. 35–45. New York: Wiley.
8 [8] Buerger, M.J. (1970). Contemporary Crystallography, pp. 24–27. New York: McGraw‐Hill.
9 [9] Grimmer, H. (1974). Disorientations and coincidence rotations for cubic lattices. Acta Crystallogr. A 30: 685–688.
10 [10] Altmann, S.L. (1986). Rotations, Quaternions and Double Groups. Oxford: Clarendon Press.
11 [11] Kuipers, J.B. (1999). Quaternions and Rotation Sequences. Princeton: Princeton University Press.
12 [12] Hahn, T. and Looijenga‐Vos, A. (2002). International Tables for Crystallography (ed. T. Hahn) 5th, Revised Edition, Vol. A: Space‐Group Symmetry, published for the International Union of Crystallography, pp. 14–16. Dordrecht: Kluwer Academic Publishers.
13 [13] Frankenheim, M.L. (1842). System der Krystalle. Breslau: Grass, Barth & Co.
14 [14] Bravais, A. (1850). Les systèmes formés par des points. J. Écol. Polytech. 19: 1–128.
Notes
1 1 1 Å = 10−10 m.
2 2 In choosing a unit parallelogram or a unit cell, the crystal is always considered to be infinitely large. The pattern in Figure 1.1a must then be thought of as extending to infinity. The fact that O, X, A and Y in a finite crystal are at slightly different positions with respect to the boundary of the pattern can then be neglected.
3 3 Indices for planes are termed Miller indices after William Hallowes Miller (1801−1880) who popularized the use of these indices through his book A Treatise on Crystallography published in 1839 [1]. Interestingly, Miller was not the first to advocate the use of the (hkl) notation for planes – as Miller himself acknowledged in a preface to his book, the Rev. William Whewell had previously noted that planes can be represented by indices h, k and l in this manner in 1825 [2].
4 4 Formally, the law of rational indices states that all planes which can occur as faces of crystals have intercepts on the axes which, when expressed as multiples of certain unit lengths along the axes (themselves proportional to a, b, c), have ratios that are small rational numbers. A rational number can always be written as p/q, where p and q are integers.
5 5 The name Weiss zone law for this equation was first used by William James Lewis (1847−1926) in the form Weiss's zone‐law on pages 39 and 40 of his 1899 book on crystallography [3]. Lewis's reasoning was that Christian Samuel Weiss (1780−1856) had been the first to call attention to this relationship through his 1804−1806 translation into German of René Just Haüy's 1801 Traité de Minéralogie. Miller knew about this equation as well – it is derived geometrically on page 10 of [1].
6 6 A discussion of whether these should be termed symmetry operators or symmetry elements, the term used in previous editions of this book, is given in Section 1.2.3 of [4]. Current International Union of Crystallography nomenclature is to use the term symmetry operators, reserving the word ‘element’ to define a member of a set, such as a group (Section 2.14). Readers should be aware that both terms are used in practice in the crystallographic literature.
7 7 The volume a · [b × c] is equal to abc sin α cos ϕ, where ϕ is the angle between a and the normal to the plane containing b and c. See also Equation (1.42). If a ≤ b ≤ c and α, β and γ are all <90°, the primitive unit cell described in this manner is a positive reduced form or Type I cell in the nomenclature of Santoro and Mighell [5]; if a ≤ b ≤ c and α, β and γ are all ≥90°, the primitive unit cell is a negative reduced form or Type II cell in the same nomenclature. Both these forms are examples of unit cells termed reduced cells or Niggli reduced cells after Paul Niggli's original work on this topic [6].
8 8 The attention of the reader is, however, drawn to Section 4.6.1 for a discussion on the contemporary definition of crystals in the light of the discovery of quasicrystals and other materials in which there is ordering in an aperiodic manner.
9 9 Of course, this does not preclude local rotation axes having unusual rotation symmetries, for example the occurrence
