6638
/www/nrich/html/content/id/6638/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<ul id="stemLinks">
<li><a href="http://nrich.maths.org/6574">Warm-up</a></li>
<li><a href="http://nrich.maths.org/6874">Try this next</a></li>
<li><a href="http://plus.maths.org/content/power-groups">Read: mathematics</a></li>
<li><a href="http://en.wikipedia.org/wiki/Crystal_structure">Read: science</a></li>
<li><a href="http://plus.maths.org/content/disorderly-quasicrystals">Explore further</a></li>
</ul>
<div> </div>
<br></br>
<p><span style="font-style: italic;">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.</span><br></br>
<br></br>
A <span style="font-style: italic;">crystal symmetry operation</span> is a transformation which when applied to the vector positions of the ions causes the following:<br></br>
<br></br>
1. The points in space occupied by atoms or ions before and after the transformation are identical.<br></br>
<br></br>
2. Each atom or ion in the crystal shifts onto the position of an identical atom or ion.<br></br>
<br></br>
3. The distance between any neighbouring pairs of atoms or ions is unchanged before and after the transformation.<br></br>
<br></br>
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.<br></br>
<br></br>
a) A rotation about a given point is a symmetry.<br></br>
<br></br>
b) A reflection through a plane which does not pass through any of the lattice points is a symmetry.<br></br>
<br></br>
c) A shear which maps the lattice onto itself is a symmetry.<br></br>
<br></br>
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.<br></br>
<br></br>
e) Repeated application of the same symmetry will eventually restore the crystal to its original state.<br></br>
<br></br>
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.<br></br>
<br></br>
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.<br></br>
<br></br>
h) Application of a symmetry operation leaves at least one point fixed.<br></br>
<br></br>
i) Application of a symmetry operation leaves exactly $3$ points fixed.<br></br>
<br></br>
Can you invent any mathematical lattices with unusual symmetry properties?<br></br>
</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem</h3>
<div>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.</div>
<br></br>
<div>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.</div>
<br></br>
Many NRICH tasks have been designed with group work in mind. <a href="http://nrich.maths.org/7011&part=">Here</a> we
have gathered together a collection of short articles that outline
the merits of collaborative work, together with examples of
teachers' classroom practice. <br></br>
<h3>Group working</h3>
<div>Allocating these clear roles (<a href="/content/id/2290/Roles.doc">Word</a>, <a href="/content/id/2290/Roles.pdf">pdf</a>) 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.</div>
<br></br>
<div>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.</div>
<br></br>
<div>For suggestions of team-building maths tasks for use with
classes unfamiliar with group work, take a look at this <a href="http://nrich.maths.org/6933&part=">article</a> and the
accompanying resources.</div>
<br></br>
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.<br></br>
<br></br>
You may want to make Zome, calculators, spreadsheets, graphing
software, squared or graph paper<a href="http://nrich.maths.org/6676&part="></a>, poster paper, and
coloured pens available for the Resource Manager in each group to
collect.<br></br>
<br></br>
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.<br></br>
<br></br>
You may choose to focus on the way the students are co-operating
or the focus might be mathematical.<br></br>
<br></br>
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.<br></br>
<br></br>
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:<br></br>
<ul>
<li>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.<br></br>
</li>
<li>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.<br></br>
</li>
<li>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.</li>
</ul>
<br></br>
<h3>Possible approach</h3>
<div>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.</div>
<br></br>
<div>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.</div>
<br></br>
<div>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.</div>
<br></br>
<div>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.</div>
<h3>Key questions</h3>
<div>Can you create lattices where each transformation is a
symmetry?</div>
<div>Can you create lattices where each transformation is not a
symmetry?</div>
<br></br>
<h3>Possible extension</h3>
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.<br></br>
<h3>Possible support</h3>
<div><a href="http://nrich.maths.org/6574&part=">Coordinated
Crystals</a> is a challenge in visualising crystals from a
description of their lattice structure, and thinking about the
separation and angle between atoms.</div>
<div><a href="http://nrich.maths.org/6507&part=">Crystal
Symmetry</a> looks at the symmetries of a particular crystal
structure under certain matrix transformations.</div>
<br></br>
<br></br>
<br></br></mdoxml>
2
5
0
0
0
0
1
Clear as crystal
Unearth the beautiful mathematics of symmetry whilst investigating
the properties of crystal lattices
Applications
chemistry
Using, Applying and Reasoning about Mathematics
Investigations