Clear as crystal

Unearth the beautiful mathematics of symmetry whilst investigating the properties of crystal lattices

Age

16 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Crystals can be represented mathematically by infinite lattices of points occupied by atoms or ions.The symmetry properties of crystal lattices are physically very important and mathematically very fascinating. In this problem we investigate the symmetries of these lattices mathematically.

A crystal symmetry operation is a transformation which when applied to the vector positions of the ions causes the following:

1. The points in space occupied by atoms or ions before and after the transformation are identical.

2. Each atom or ion in the crystal shifts onto the position of an identical atom or ion.

3. The distance between any neighbouring pairs of atoms or ions is unchanged before and after the transformation.

Which of the following are sometimes always or never true? If always or never, give a proof. If sometimes, give an example where it works and an example where it does not. You might want to focus your attention on BCC or FCC packing, although feel free to invent mathematical lattices of your own.

a) A rotation about a given point is a symmetry.

b) A reflection through a plane which does not pass through any of the lattice points is a symmetry.

c) A shear which maps the lattice onto itself is a symmetry.

d) For a crystal lying on an integer lattice, the translation by $(l/2, m/2, n/2)$ is a symmetry, where $l, m, n$ are integers.

e) Repeated application of the same symmetry will eventually restore the crystal to its original state.

f) If $T_1({\bf v})$ and $T_2({\bf v})$ are both symmetry operations then the combination $T_1(T_2{\bf v}) $ is a symmetry operation.

g) If neither $T_1({\bf v})$ nor $T_2({\bf v})$ is a symmetry operation then $T_1(T_2({\bf v}))$ is not a symmetry operation.

h) Application of a symmetry operation leaves at least one point fixed.

i) Application of a symmetry operation leaves exactly $3$ points fixed.

Can you invent any mathematical lattices with unusual symmetry properties?

Teachers' Resources

Why do this problem

This problem gives students a chance to explore mathematically the very important physical idea of symmetries of crystal lattices, and encourages students to consider the properties that lattices with different types of symmetry would need. Issues concerning group theory will naturally emerge.

This problem lends itself to collaborative working, both for students who are inexperienced at working in a group and students who are used to working in this way.

Many NRICH tasks have been designed with group work in mind. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.

Group working

Allocating these clear roles (Word, pdf) can help groups to work in a purposeful way - success on this task should be measured by how effectively the members of the group work together as well as by the solutions they reach.

To start with group work introduce the four group roles to the class. It may be appropriate, if this is the first time the class have worked in this way, to allocate particular roles to particular students. If the class work in roles over a series of lessons, it is desirable to make sure everyone experiences each role over time.

For suggestions of team-building maths tasks for use with classes unfamiliar with group work, take a look at this article and the accompanying resources.

Explain the task and make it clear that everyone needs to be ready to share what they did with the rest of the class at the end of the session.

You may want to make Zome, calculators, spreadsheets, graphing software, squared or graph paper, poster paper, and coloured pens available for the Resource Manager in each group to collect.

While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together. This is a good way of highlighting the mathematical behaviours you want to promote, particularly with a challenging task such as this.

You may choose to focus on the way the students are co-operating or the focus might be mathematical.

Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.

We assume that each group will record their diagrams, reasoning and generalisations in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:

Every group is given a couple of minutes to report back to the whole class. Students can seek clarification and ask questions. After each presentation, students are invited to offer positive feedback. Finally, students can suggest how the group could have improved their work on the task.
Everyone's posters are put on display at the front of the room, but only a couple of groups are selected to report back to the whole class. Feedback and suggestions can be given in the same way as above. Additionally, students from the groups which don't present can be invited to share at the end anything they did differently.
Two people from each group move to join an adjacent group. The two "hosts" explain their findings to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.

Possible approach

Start by sharing with students the three parts of the definition of a crystal symmetry operation. Allow them some time to make sense of the three statements, and then discuss how each statement fits in with their idea of what is meant by symmetry.

Once an understanding of crystal symmetries is reached, share the nine statements about crystal symmetries. These could be divided among the class with groups of students working to make sense of different statements or all groups could be allowed to select from all of the statements those that seem most tractable or interesting. Groups should use examples of lattices and transformations to justify their decisions that the statements are sometimes, always or never true. Encourage students to focus on the properties a lattice and a transformation would need to have in order to satisfy each statement.

After everyone has had time to consider and work on the statements, bring the class together to share findings. Students could be invited to explain their thinking on each statement, drawing diagrams of lattices to support examples they have considered.

Finally, the last part of the problem invites students to create examples of mathematical lattices with interesting symmetry properties - it is possible to describe lattices where there are no possible translation symmetries but there are reflection or rotation symmetries, for example.

Key questions

Can you create lattices where each transformation is a symmetry?

Can you create lattices where each transformation is not a symmetry?

Possible extension

In considering statement g), challenge students to find an example where the two transformations are of different types, for example, combining a translation with a reflection to form a glide reflection.

Possible support

Coordinated Crystals is a challenge in visualising crystals from a description of their lattice structure, and thinking about the separation and angle between atoms.

Crystal Symmetry looks at the symmetries of a particular crystal structure under certain matrix transformations.

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