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Materials Science on CD-ROM User Guide

Introduction to Crystallography

Version 2.1

Ann Fretwell, MATTER
Peter Goodhew, University of Liverpool

Assumed Pre-knowledge

This module has been developed to introduce the topic of Crystallography. Before starting this module, it is assumed that the student is familiar with the basics of vector addition and subtraction.

Module Structure

The module consists of five sections:

Crystals

Crystals are usually associated with having naturally developed, flat and smooth external faces. It has long been recognised that this evidence of external regularity is related to the regularity of internal structure. Diffraction techniques are now available which give much more information about the internal structure of crystals, and it is recognised that internal order can exist with no external evidence for it.

2-D Crystallography

Repeating patterns can be described in terms of their symmetry. Similarly, the regularity of the internal structure of crystals can be described formally in terms of symmetry elements. In this section we consider the symmetry and the categorisation of patterns and crystals in just two dimensions to simplify the ideas presented. Two-dimensional crystals can be thought of as the projection of 3-D crystals onto a plane. Reflection symmetry is demonstrated and then rotation symmetry is explained. Exercise pages allow the user to identify reflection and rotation symmetry in 2-D shapes. The combination of reflection and rotation symmetry to give the ten plane point symmetry groups follows.

Two dimensional patterns and crystals can be built up from a single unit of pattern called a basis, by repeated translations, due to two vector operators, known as unit vectors. Unit vectors outline a repeated area of a pattern called a unit cell. Repeated translations of unit vectors mark out a grid of identical points in space, called a lattice. The definition of a lattice is very important and is given as: a repeating pattern of points, each point having the same surroundings. The concept that patterns and crystals can be built by associating an identical basis in the same way with each point of a lattice is demonstrated.

Unit cells can be divided into primitive and non-primitive types. All 2-D lattices can be categorised into one of five classes of plane lattices. The unit cells for each of these five lattices are shown. By clicking on these unit cells, the user is shown the symmetry elements that are present in these five lattice types. An exercise page follows which allows the user to select the correct lattice type for several 2-D representations of crystals.

Finally, this section deals with translational symmetry elements, in this case the glide line. When all the symmetry types are combined with the five plane lattice type, then the 17 plane groups are created. All 2-D patterns and crystals can be assigned to one of these plane groups. The 17 groups are given and by clicking on the group name an example of a pattern is given. A flow chart follows which gives the user the opportunity to identify the plane group of patterns given in the additional question.

3-D Crystallography

This section starts by building on the work covered in the 2-D Crystallography section to extend the ideas of unit vectors, unit cells and lattices into three dimensions for crystals. It is important to complete the 2-D section first. The convention of labelling crystal axes is explained and this is reinforced by a drag and drop labelling exercise. All crystals can be assigned to be in one of seven crystal systems.depending on the shape of their unit cell. Seven distinct unit cells exist, the shape of these unit cells is determined by the symmetry of the crystal system. Within each of the crystal systems, different lattices are possible. There are 14 different lattices, known as the Bravais lattices. By clicking on the crystal system name, the Bravais lattices are illustrated.

This section then addresses symmetry elements in three dimensions. Mirrors lines become mirror planes, and objects can have more than one rotation axis in different directions. These symmetry elements all pass through the centre of the object, through a single point. This combination of symmetry elements is called the point symmetry group for an object.

Additional kinds of symmetry arise in three dimensions, which do not occur in two dimensions. These are centres of symmetry and inversion axes. These are demonstrated graphically. In total there are 32 different combinations of symmetry elements giving the 32 point symmetry groups. These point groups are distributed amongst the 7 crystal systems and are also known as the 32 crystal classes. This section then looks at symmetry elements including translation and describes glide planes and screw axes.

If the 14 Bravais lattices and 32 point groups are combined and we consider the translational symmetry elements of glide planes and screw axes, then there are 230 different possible combinations. These 230 space groups describe all the possible different spatial arrangements of symmetry in crystals.

Crystal structures

There are three common metallic crystal structures: face-centred cubic (fcc), hexagonal close-packed (hcp) and body-centred cubic (bcc). This section first looks at how these structures can be built by packing together atoms of the same size. Unit cells are presented for these three structures which can be rotated by the user. The user is instructed to try to find the rotation axes of these structures. This section finishes with a consideration of the packing density of these structures.

Indexing Directions and Planes

It is often necessary to state the direction in a crystal. A method is described which shows how to calculate the Miller index for a direction in a 2-D lattice. It is important to be able to identify the unit vectors in the lattice of the crystal, since the Miller index expresses direction in terms of ratios of these vectors. This method is easily extended to three dimensions. Exercises allow the user to see if they have understood the method shown. This section then looks at how to index planes in crystals and their lattices. Again an exercise page follows to check that this method has been understood. Finally the user is introduced to the four index Miller Bravais system for indexing planes and directions in hexagonal systems.

Bibliography

The student is referred to the following resources in this module:

Hammond, C., The Basics of Crystallography and Diffraction, Oxford University Press, 1997

Hammond, C., An Introduction to Crystallography, Oxford University Press, 1972

Kelly, A. and Groves, G.W., Crystallography and Crystal Defects, Longman, 1970

 

The University of Liverpool
Copyright of The University of Liverpool 2000