Chapter 4 Crystallography 4.1 Crystalline substances 75 4.2 Symmetry operations 77 4.3 Two - dimensional motifs and lattices (meshes) 81 4.4 Three - dimensional motifs and lattices 83 4.5 Crystal systems 85 4.6 Indexing crystallographic planes 89 4.7 Twinned crystals 100 4.8 Crystal defects 101 4.9 Polymorphs and pseudomorphs 105 4.1 CRYSTALLINE SUBSTANCES this text, a more detailed treatment of crystal- lography cannot be provided . Crystallography emphasizes the long - range order or crystal structure of crystalline sub- stances. It focuses on the symmetry of crystal- 4.1.1 Crystals and c rystal f aces line materials and on the ways in which their Mineral crystals are one of nature’ s most long - range order is related to the three - dimen- beautiful creations. Many crystals are enclosed sional repetition of fundamental units of by fl at surfaces called crystal faces. Crystal pattern during crystal growth. In minerals, faces are formed when mineral crystals grow, the fundamental units of pattern are molecu- and enclose crystalline solids when they stop lar clusters of coordination polyhedra or growing. Perfectly formed crystals are notable stacking sequences (Chapter 2 ). The ways in for their remarkable symmetry (Figure 4.1 ). which these basic units can be repeated to The external symmetry expressed by crystal produce crystal structures with long - range faces permits us to infer the geometric pat- order are called symmetry operations. In terns of the atoms in mineral crystal struc- addition, crystallography focuses on the tures as well. These patterns inferred from description and signifi cance of planar features external symmetry have been confi rmed by in crystals including planes of atoms, cleavage advanced analytical techniques such as X - ray planes, crystal faces and the forms of crystals. diffraction (XRD) and atomic force micro- Crystallography is also concerned with crystal scopy (AFM). defects, local imperfections in the long - range Mineralogists have developed language to order of crystals. Given the broad scope of describe the symmetry of crystals and the crystal faces that enclose them. Familiarizing students with the concepts and terminology Earth Materials, 1st edition. By K. Hefferan and of crystal symmetry and crystal faces is one J. O’Brien. Published 2010 by Blackwell Publishing Ltd. of the primary goals of this chapter. A second 76 EARTH MATERIALS (a) (b) Figure 4.1 Representative mineral crystals: (a) pyrite; (b) tourmaline. (Photos courtesy of the Smithsonian Institute.) (For color version, see Plate 4.1, between pp. 248 and 249. ) goal of this chapter is to build connections macroscopic crystal. Growth continues in this between crystal chemistry (Chapters 2 and 3 ) manner until the environmental conditions and crystallography by explaining the rela- that promote growth change and growth tionships between chemical composition and ceases. coordination polyhedra and the form, sym- Long - range, geometric arrangements of metry and crystal faces that develop as crys- atoms and/or ions in crystals are produced tals grow. when a fundamental array of atoms, a unit of pattern or motif, is repeated in three dimen- sions to produce the crystal structure. A motif 4.1.2 Motifs and n odes is the smallest unit of pattern that, when When minerals begin to form, atoms or ions repeated by a set of symmetry operations, will bond together, so that partial or complete generate the long - range pattern characteristic coordination polyhedra develop (Chapter 2 ). of the crystal. In minerals, the motif is com- Because the ions on the edges and corners of posed of one or more coordination polyhedra. coordination polyhedra have unsatisfi ed elec- In wallpaper, it is a basic set of design ele- trostatic charges, they tend to bond to addi- ments that are repeated to produce a two - tional ions available in the environment as the dimensional pattern, whereas in a brick wall mineral grows. Eventually, a small cluster of the fundamental motif is that of a single brick coordination polyhedra is formed that con- that is repeated in space to form the three - tains all the coordination polyhedra charac- dimensional structure. The repetition of these teristic of the mineral and its chemical fundamental units of pattern by a set of rules composition. In any mineral, we can recog- called symmetry operations can produce a nize a small cluster of coordination polyhedra two - or three - dimensional pattern with long - that contains the mineral ’ s fundamental com- range order. When several different motifs position and unit of pattern or motif. As the could be repeated by a similar set of symmetry mineral continues to grow, additional clusters operations, we may wish to emphasize the of the same pattern of coordination polyhedra general rules by which different motifs may are added to form a mineral crystal with a be repeated to produce a particular type of three - dimensional geometric pattern – a long - long - range order. In such cases it is useful to range, three - dimensional crystal structure. represent motifs by using a point. A point Clusters of coordination polyhedra are added, used to represent any motif is called a node . one atom or ion at a time, as (1) the crystal The pattern or array of atoms about every nucleates, (2) it becomes a microscopic crystal, node must be the same throughout the pattern and, if growth continues, (3) it becomes a the nodes represent. CRYSTALLOGRAPHY 77 (a) 4.2 SYMMETRY OPERATIONS t1 t2 4.2.1 Simple s ymmetry o perations Symmetry operations may be simple or com- pound. Simple symmetry operations produce repetition of a unit of pattern or motif using a single type of operation. Compound sym- (b) metry operations produce repetition of a unit t1 of pattern or motif using a combination of t two types of symmetry operation. Simple 2 symmetry operations include (1) translation, by specifi c distances in specifi ed directions, (2) rotation, about a specifi ed set of axes, (3) refl ection, across a mirror plane, and (4) inversion, through a point called a center. (c) These operations are discussed below and provide useful insights into the geometry of crystal structures and the three - dimensional t3 properties of such crystals. t1 t1 t1 Translation t2 The symmetry operation called translation t involves the periodic repetition of nodes or 2 motifs by systematic linear displacement. One - dimensional translation of basic design Figure 4.2 (a) Two - dimensional translation at elements generates a row of similar elements right angles (t1 and t2 ) to generate a two- (Figure 4.2 a). The translation is defi ned by the dimensional mesh of motifs or nodes. unit translation vector (t) , a specifi c length (b) Two - dimensional translation (t1 and t2 ) and direction of systematic displacement by not at right angles to generate a two - which the pattern is repeated. Motifs other dimensional mesh or lattice. (c) Three - than commas could be translated by the same dimensional translation (t1 , t2 and t3 ) to unit translation vector to produce a one - generate a three - dimensional space lattice. dimensional pattern. In minerals, the motifs (From Klein and Hurlbut, 1985 ; with are clusters of atoms or coordination polyhe- permission of John Wiley & Sons.) dra that are repeated by translation. Two - dimensional translations are defi ned by two unit translation vectors ( ta and tb or t1 or t1 , t2 and t3 , respectively). The translation and t2 , respectively). The translation in one in one direction is represented by the length direction is represented by the length and and direction of ta or t1 , the translation in the direction of ta or t1 ; translation in the second direction is represented by tb or t2 and second direction is represented by the length the translation in the third direction is repre- and direction of tb or t2 . The pattern generated sented by tc or t3 . The result of any three - depends on the length of the two unit transla- dimensional translation is a space lattice . A tion vectors and the angles between their space lattice is a three - dimensional array of directions. The result of any two - dimensional motifs or nodes in which every node has an translation is a plane lattice or plane mesh . A environment similar to every other node in plane lattice is a two - dimensional array of the array. Since crystalline substances such as motifs or nodes in which every node has an minerals have long - range, three - dimensional environment similar to every other node in order and since they may be thought of as the array (Figure 4.2 a,b). motifs repeated in three dimensions, the Three- dimensional translations are defi ned resulting array of motifs is a crystal lattice . by three unit translation vectors ( ta , tb and tc Figure 4.2 c illustrates a space lattice produced 78 EARTH MATERIALS 1 turn of 360° rotation 2 turns of 180° rotation 3 turns of 120° rotation 1 2 3 4 turns of 90° rotation 6 turns of 60° rotation 4 6 Figure 4.3 Examples of the major types of rotational symmetry (n = 1, 2, 3, 4 or 6) that occur in minerals. (From Klein and Hurlbut, 1985 ; with permission of John Wiley & Sons.) by a three - dimensional translation of nodes or motifs. Table 4.1 Five common axes of rotational symmetry in minerals. Rotation Symbolic Type notation Description Motifs can also be repeated by non- translational symmetry operations. Many One - fold (1 or A1 ) Any axis of rotation patterns can be repeated by rotation (n). axis of about which the motif rotation is repeated only once Rotation (n) is a symmetry operation that during a 360 ° rotation involves the rotation of a pattern about an (Figure 4.3 , (1)) imaginary line or axis, called an axis of rota- tion , in such a way that every component of Two - fold (2 or A2 ) Motifs repeated every the pattern is perfectly repeated one or more axis of 180 ° or twice during a rotation 360 ° rotation (Figure times during a complete 360 ° rotation.