Materials Science on CDROM User Guide
Introduction to Crystallography
Version 2.1
Ann Fretwell, MATTER
Peter Goodhew, University of Liverpool
Assumed Preknowledge
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 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.
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.
Twodimensional crystals can be thought of as the projection of 3D 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 2D 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 nonprimitive types. All 2D 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 2D 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 2D 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.
This section starts by building on the work covered in the 2D Crystallography section
to extend the ideas of unit vectors, unit cells and lattices into three dimensions for
crystals. It is important to complete the 2D 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.
There are three common metallic crystal structures: facecentred cubic (fcc), hexagonal
closepacked (hcp) and bodycentred 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.
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 2D 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
