Chapter 3

1. Lattice description and crystal structures Chapter 3

2. Content • The crystal lattice • The unit cell • Representation of lattice lines and planes

3. The crystal lattice

4. The crystal lattice

5. The unit cell Unit cell – polyhedra with 3 pairs of parallel faces (parallelepiped) that repeat periodically in 3 dimensions • General rules for choosing the unit cell: • Must have an integral number of formula units (eg.: halite: 1 Na, 1 Cl) • Each corner must be identical (eg.: halite: Cl must occupy all corners) • Express symmetry of the atomic relationship

6. Four basic unit cells • Primitive unit cell • Corners only • Body centered unit cell • Corners and internal midpoint • Face centered unit cell • Corners and midpoint of each face • Base or end centered unit cell • Corners and midpoint of basal planes

7. 14 Bravais lattices Triclinic Monoclinic Orthorhombic Tetragonal Trigonal (Rhombohedral) Hexagonal Cubic (Isometric) (NB: Trigonal = rhombohedral)

8. Direction indices and Miller indices • Direction Indices • Describe the direction of the different axes (lattice lines) which are intersections of two lattice planes • Referring to specific vector square brackets are used: • [uvw]; eg: [010] • Referring to a set of directions triangular brackets are used: • <uvw>; eg <100> • Miller indices • Describe the orientation of different planes by stating which axes are being intersected • When referring to a specific plane, “round” brackets are used: • (hkl); eg: (012) • When referring to a set of planes related by symmetry, then “curly” brackets are used: • {hkl}; eg: {001} • General plane (hkl) and lattice direction [uvw] are perpendicular only for cubic crystals

9. Directional indices • Fig 3.17 Vector (r) = ua + vb + wc  [uvw]

10. Miller Indexes • Miller Indexing is a method of describing the orientation of a plane or set of planes within a lattice in relation to the unit cell.

11. Examples of lattice planes • The (100), (010), (001), (ī00), (0ī0) and (00ī) planes form the faces of the unit cell. • Here, they are shown as the faces of a triclinic (a ≠ b ≠ c, α ≠ β ≠ γ) unit cell . Although in this image, the (100) and (ī00) planes are shown as the front and back of the unit cell, both indices refer to the same family of planes. • It should be noted that these six planes are not all symmetrically related, as when they are in the cubic system

12. Examples of lattice planes • The (101), (110), (011), (10ī), (1ī0) and (01ī) planes form the sections through the diagonals of the unit cell, along with those planes whose indices are the negative of these, eg.: (ī0ī); (ī01); (ī10); (0ī1), . • In the image the planes are shown in a different triclinic unit cell.

13. How to index a lattice

14. How to index a plane

15. How to index a plane

16. The zero index

17. Negative indices

18. How to index a plane

19. Miller Indexes Isometric lattice Hexagonal lattice

20. Atomic positions • Within unit cell

21. Macroscopic symmetries: crystal morphology Chapter 4

22. Contents • Symmetry • Crystal classes • Crystallographic forms

23. Symmetry • Object is symmetrical if: • It can be moved in one of the following ways to appear exactly as before: • Translation • Rotation (n = 1, 2, 3, 4, 6) • Mirror reflection (m = 1, 2, 4) • Inversion (i) • A crystal can be described according to symmetry elements: • Centre of symmetry (inversion) • Axis of symmetry (rotation) • Plane of symmetry (mirror reflection)

24. Point-group symmetry • 32 Point-groups or CRYSTAL CLASSES • Combination of 3 of the symmetry operations in different crystal systems: • Rotation (n = 1, 2, 3, 4, 6  2-, 3- 4- or 6-fold symmetry) • Mirror reflection (m) • Inversion (i) • NB: does not include translation • Therefore just symmetry around specific point - point-groups

25. Crystal forms • All known crystal forms fit into the seven crystal systems. But why don't all crystals in a given set look the same? • Or, stated differently, why can't I learn seven crystal shapes and know all I need to know? • Well, crystals, even of the same mineral, have differing CRYSTAL FORMS, depending upon their conditions of growth. • Whether they grew rapidly or slowly, under constant or fluctuating conditions of temperature and pressure, or from highly variable or remarkably uniform fluids or melts, all these factors have their influence on the resultant crystal shapes, even when not considering other controls.

26. Form of single crystals • Most single crystals has distinctive plane faces and some symmetry. • The actual shape of the crystal will be determined by the availability of crystallising material, and by interference with other crystals, but the angles between the faces will be characteristic of the material and will define an ideal shape. • Gemstones are often single crystals. They tend to be cut artificially to obtain aesthetically pleasing refractive and reflective properties. • This generally requires cutting along crystallographic planes. This is known as cleaving the crystal. A familiar example is diamond, from which decorative stones can be cleaved in different ways to produce a wide range of effects.

27. Simple crystal forms • Consider the following three-dimensional shapes: • Cube: 6 identical squares • Tetrahedron: 4 identical equilateral triangles • Octahedron: 8 identical equilateral triangles • Rhombohedron: 6 identical parallelograms with sides of equal length • These first three shapes are the most important in crystallography, and you should be very familiar with them!

28. Common crystal forms • The symmetry exhibited by real single crystals is determined by the crystal structure of the material. Many have shapes composed of less regular polyhedra, such as prisms and pyramids. • Hexagonal prism: 2 hexagons and 6 rectangles • Square-based pyramid: 4 triangles and a square • Not all single crystal specimens exhibit distinctive polyhedral shapes. Metals, for example, often have crystals of no particular shape at all. • These quartz specimens show a range of shapes typically exhibited by crystals.

29. Crystallographic forms • Regular polyhedra (Fig 4.18) • Cube (Hexahedron) 6 faces • Octahedron 8 faces • Tetrahedron 4 faces • Rhombohedron 6 identical parallelograms with equal sides • Dodecahedron 12 faces • Icosahedron 20 faces • Irregular polyhedra (lower symmetry) • Prism • Pyramid • Bipyramid