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This is the complete lecture note for MME501 Analytical Methods for Materials. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
Course code: MME 501 Course title: Analytical Methods for Materials Course credit load: 2 Units Series 1: Crystallography: External Morphology of Crystals, the Law of Constant Angle and Representation by Face Normals • Part 1.1: Crystals and Their External Morphology o Sub Part 1.1.1: The Crystalline State and Crystal Habit o Sub Part 1.1.2: Lattices, Unit Cells and Crystal Systems o Sub Part 1.1.3: Crystallographic Axes, Axial Ratios and Miller Indices • Part 1.2: The Law of Constant Angle o Sub Part 1.2.1: Statement and Origin of the Law o Sub Part 1.2.2: Measurement of Interfacial Angles: Goniometers o Sub Part 1.2.3: Rational Indices, Zones and the Indexing of Faces • Part 1.3: Symmetry of Crystals o Sub Part 1.3.1: Symmetry Elements o Sub Part 1.3.2: Crystal Classes and Point Groups o Sub Part 1.3.3: Crystal Forms and Twinning • Part 1.4: Representation by Face Normals: Projections o Sub Part 1.4.1: Face Normals and the Spherical Projection o Sub Part 1.4.2: The Stereographic Projection o Sub Part 1.4.3: The Gnomonic Projection and Its Applications
Introduction Long before anyone could see an atom, collectors noticed that quartz crystals of every size, from a grain of sand to a boulder, show the same angles between their faces. That simple observation was the first evidence of order inside crystals, and it remains the starting point for every method used to analyse materials. This Series introduces the language of crystallography, from the outward form of crystals to the projections used to represent them. The aim of this Series is to equip you to describe the external morphology of crystals, state and apply the law of constant angle, and explain how crystals are represented by the directions of their face normals. We shall commence with crystals and their external morphology, including lattices and Miller indices. Next, we will explore the law of constant angle and the measurement of interfacial angles. Afterwards, we will turn to the symmetry of crystals, including forms and twins. We will conclude with representation by face normals, using the spherical, stereographic and gnomonic projections. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: describe the external morphology and habit of crystals, • SLO 1.2: describe lattices, unit cells and crystal systems and index faces and directions using Miller indices, • SLO 1.3: state the law of constant angle and explain its origin, • SLO 1.4: measure interfacial angles with goniometers and apply the zone law and the angle formula, • SLO 1.5: identify the symmetry elements of crystals and the 32 crystal classes, • SLO 1.6: describe crystal forms and twinning, • SLO 1.7: explain the representation of crystals by the directions of face normals in spherical and stereographic projection, • SLO 1.8: construct and interpret stereographic and gnomonic projections to measure angles and identify zones.
1.1 Crystals and Their External Morphology 1.1.1 The Crystalline State and Crystal Habit A crystal is a solid in which the atoms, ions or molecules are arranged in a regular pattern that repeats in three dimensions. A single crystal has this pattern throughout, whereas a polycrystal is made up of many small crystals joined at boundaries. When a crystal grows freely from a melt, a solution or a vapour, it develops flat natural surfaces called faces, which meet in straight lines called edges and at points called vertices. This outer shape is the external morphology of the crystal. For a crystal that is a simple polyhedron, the numbers of faces F, vertices V and edges E are related by Euler's relation, F + V = E + 2. A cube has 6 faces, 8 vertices and 12 edges, and an octahedron has 8 faces, 6 vertices and 12 edges, and both satisfy it. The habit of a crystal is its overall shape, such as cubic, prismatic, tabular or needle-like, and it depends on growth conditions such as the temperature, the impurities present and the speed of growth. Worked example: A crystal of halite grown slowly from solution is a cube with 6 faces, 8 vertices and 12 edges, and 6 + 8 = 12 + 2, as Euler's relation requires. Application: The external morphology gives a first clue to the identity of a mineral or crystal and to the conditions under which it grew. Fill in the blank: The overall shape of a crystal, which depends on the conditions of growth, is called its crystal __________. 1.1.2 Lattices, Unit Cells and Crystal Systems The internal order of a crystal is described by a space lattice, an infinite array of points in which every point has identical surroundings, together with a motif, the group of atoms attached to each lattice point. The unit cell is the smallest parallelepiped that builds the whole lattice when it is repeated by translation, and it is described by six lattice parameters, the edge lengths a, b and c and the angles α, β and γ between them. According to the relations between the lattice parameters, there are seven crystal systems: cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic and triclinic. In the cubic system a = b = c and all the angles are 90 degrees, in the tetragonal system a = b ≠ c, and in the triclinic system all the edges and angles differ. When lattice points at the centres of cells and faces are allowed, there are 14 Bravais lattices, three of which are cubic, namely the primitive, body-centred and face-centred lattices. Worked example: Copper has a face-centred cubic lattice with a = 0.361 nanometre, so its unit cell is a cube of edge 0.361 nanometre with lattice points at the corners and at the centre of each face.
Application: Classification into crystal systems and Bravais lattices is the starting point for describing and identifying any crystalline material. Fill in the blank: In the cubic system all the cell edges are equal and all the angles are __________ degrees. 1.1.3 Crystallographic Axes, Axial Ratios and Miller Indices The faces and directions of a crystal are described relative to crystallographic axes, x, y and z, that are chosen parallel to the edges of the unit cell. The axial ratio, a:b:c, is the ratio of the unit lengths along the axes, with b taken as 1, and it is a constant for a crystal species. A face is defined by its intercepts on the axes, in units of a, b and c, which are called Weiss parameters. Miller indices, (hkl), are the reciprocals of the intercepts, cleared of fractions to give the smallest whole numbers, so a face with intercepts 1, 1 and ∞ has indices (110). A face parallel to an axis has an index of zero for that axis. Directions are written [uvw], and a family of equivalent faces is written {hkl}, so that in a cubic crystal {100} contains six faces, {111} contains eight and {110} contains twelve. Worked example: A face that cuts the axes at 1/2, 1 and ∞ has reciprocals 2, 1 and 0, so its Miller indices are (210). Application: Miller indices give every face and lattice plane a short, unambiguous label that can be used in calculations and in diffraction. Fill in the blank: Miller indices are the __________ of the intercepts of a face on the crystallographic axes. Table 1.1: The seven crystal systems and their Bravais lattices. Crystal System Relations Between the Lattice Parameters Bravais Lattices Cubic a = b = c; α = β = γ = 90 degrees Primitive, body-centred and face-centred Tetragonal a = b ≠ c; α = β = γ = 90 degrees Primitive and body-centred Orthorhombic a ≠ b ≠ c; α = β = γ = 90 degrees Primitive, base-centred, body-centred and face-centred Hexagonal a = b ≠ c; α = β = 90 degrees, γ = 120 degrees Primitive Trigonal (rhombohedral) a = b = c; α = β = γ ≠ 90 degrees Rhombohedral Monoclinic a ≠ b ≠ c; α = γ = 90 degrees, β ≠ 90 degrees Primitive and base-centred Triclinic a ≠ b ≠ c; α ≠ β ≠ γ Primitive
Pilot questions 1.1 1. Define a crystal. (Linked to SLO 1.1) 2. State Euler's relation for a polyhedron. (Linked to SLO 1.1) 3. What is crystal habit? (Linked to SLO 1.1) 4. How many Bravais lattices are there? (Linked to SLO 1.2) 5. Find the Miller indices of a face with intercepts 1, 1 and ∞. (Linked to SLO 1.2) 1.2 The Law of Constant Angle 1.2.1 Statement and Origin of the Law The law of constant interfacial angles, or law of constant angle, states that the angles between corresponding faces on all crystals of the same substance are constant when they are measured at the same temperature. It was first stated by Steno in 1669 from his measurements on quartz, and it was confirmed by Romé de l'Isle for many other minerals. The sizes and shapes of the faces may vary greatly from crystal to crystal, and some faces may
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