1. Introduction to Crystallography
Practical I - A
Introduction to Crystallography

2. The crystal lattice

3. The unit cell General rules for choosing 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

4. Choosing the unit cell

5. 14 Bravais lattices Triclinic Monoclinic Orthorhombic Tetragonal
Trigonal (Rhombohedral)
Hexagonal
Cubic (Isometric)
(NB: Trigonal = rhombohedral)

6. Crystallographic axis
One of three lines (sometimes four, in the case of a hexagonal crystal), passing through a common point, that are chosen to have definite relation to the symmetry properties of a crystal, and are used as a reference in describing crystal symmetry and structure.
The crystallographic axes are imaginary lines that we can draw within the crystal lattice. 
These will define a coordinate system within the crystal.
For 3-dimensional space lattices we need 3 or in some cases 4 crystallographic axes that define directions within the crystal lattices. 
Depending on the symmetry of the lattice, the directions may or may not be perpendicular to one another, and the divisions along the coordinate axes may or may not be equal along the axes. 
The lengths of the axes are in some way proportional to the lattice spacing along an axis and this is defined by the unit cell.

7. Crystal symmetry
Symmetrically arranged faces reflect the internal arrangement of atoms. The symmetry can be described according to three symmetry elements:
Centre of symmetry
A central point which is present when all faces or edges occur in parallel pairs on opposite sides of the crystal.
A point, within a crystal, through which any straight line also passes through two points on the edge of the figure at the same distance from the centre but on opposite sides.
The centre of symmetry at a point (0,0,0) operates on any point (x,y,z) to give an identical point at (-x,-y,-z).
Axis of symmetry
A line about which a crystal may be rotated through 360°/n until it assumes a congruent position (identical image is seen); n may equal 2, 3, 4 or 6 – depending on the number of times the congruent position is repeated, resulting in 2-fold (diad), 3-fold (triad), 4-fold (tetrad) and 6-fold (hexad) axes.
Plane of symmetry (also mirror plane)
A plane by which the crystal may be divided into two halves which are mirror images of each other.
Videos

8. Crystallographic classification system
Using the elements of symmetry discussed above, crystallographers have recognized
32 Crystal classes (point groups)
Classified based on three symmetry operations
6(7) Crystal systems
Classified based on lattice parameters (a, b, c and α, β, γ)
Symmetry is highest (high symmetry) in the cubic system, where many elements are repeated, and lowest (low symmetry) in the triclinic system, where only a centre of symmetry may be present (i.e. there may be no plane or axis of symmetry).

9. Crystal forms (230 space groups)
All known crystal forms fit into the above 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.
Video

10. Practical
Classify your own examples

11. Concept of a lattice and description of crystal structures
Sources:
HR Wenk and A Bulakh, 2004, Minerals: Their Constitution and Origin, Univ Press, Cambridge
© DoITPoMS, University of Cambridge

12. Introduction
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.
Miller Indices were developed by William Hallowes Miller.
These indices are useful in understanding many phenomena in materials science, such as explaining the shapes of single crystals, the form of some materials' microstructure, the interpretation of X-ray diffraction patterns, and the movement of a dislocation , which may determine the mechanical properties of the material.

13. How to index a lattice

14. How to index a lattice

15. How to index a plane

16. How to index a plane

17. How to index a plane

18. The zero index

19. Negative indices

20. Parallel planes
Lattice planes can be represented by showing the trace of the planes on the faces of one or more unit cells. The diagram shows the trace of the () planes on a cubic unit cell.

21. Bracket Conventions
In crystallography there are conventions as to how the indices of planes and directions are written. When referring to a specific plane, “round” brackets are used:
(hkl)
When referring to a set of planes related by symmetry, then “curly” brackets are used:
{hkl}
These might be the (100) type planes in a cubic system, which are (100), (010), (001), (ī00) (0ī0) and (00ī). These planes all “look” the same and are related to each other by the symmetry elements present in a cube, hence their different indices depend only on the way the unit cell axes are defined. That is why it useful to consider the equivalent (010) set of planes
Directions in the crystal can be labeled in a similar way. These are effectively vectors written in terms of multiples of the lattice vectors a, b, and c. They are written with “square” brackets:
[UVW]
A number of crystallographic directions can also be symmetrically equivalent, in which case a set of directions are written with “triangular” brackets:
<UVW>

22. 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

23. 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.

24. Practical work Q1: Determine the Miller Indices for plane A and B

25. Practical assignment Q2: Construct the faces indicated here on the different faces and assign Miller indices to each one (A to L)

26. Draw your own lattice planes
The following link shows you simulations generating images of lattice planes as you enter a set of Miller indices (each index between 6 and -6) in the following format: (1;-2;0) es/lattice_draw.php

27. ANSWERS

28. Answers: Q1:
A: 112
B: 221

29. Answers Q2:

31. Worked examples
The figure below is a scanning electron micrograph of a niobium carbide dendrite in a Fe-34wt%Cr-5wt%Nb-4.5wt%C alloy.
Niobium carbide has a face centred cubic lattice.
The specimen has been deep-etched to remove the surrounding matrix chemically and reveal the dendrite.
The dendrite has 3 sets of “arms” which are orthogonal to one another (one set pointing out of the plane of the image, the other two sets, to a good approximation, lying in the plane of the image), and each arm has a pyramidal shape at its end.
It is known that the crystallographic directions along the dendrite arms correspond to the < 100 > lattice directions, and that the direction ab labelled on the micrograph is [10ī]