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Coordinated Crystals
Age 16 to 18
Challenge Level 
Why do this problem?
This problem is a useful exercise in visualisation and 3D vectors. It provides a natural context for the mathematics and has many extension possibilities. It is an example of a problem where a clear geometric image really facilitates the work with vectors and where the use of the scalar product really facilitates the calculation of the angles.Possible approach
Initially focus on trying to understand the atomic structure.
Encourage discussion and the drawing of diagrams? Share these.
Which ways of thinking about the atomic structure are the simplest
and clearest?
To understand how close the various atoms are to each other
requires clear thinking. It will be easiest to think in terms of
each atom $A$ surrounded by a 'box' of $B$ atoms, in which case it
will be easier to see which distances, and therefore angles, are
possible.
The extension concerning the other crystal configurations is
mathematically very interesting.
You could consider structures well known from chemistry or
encourage students to research the idea following the link from the
problem.
Key questions
What sort of atom lies at the origin?
What is the configuration of all of the $A$ atoms or all of
the $B$ atoms?
What angle is formed between the atom at the origin and its
two closest neighbours?
Possible extension
Consider creating a version of the problem with face-centred
cubic packing, where the first challenge is to determine an
algebraic form of the location of the different atoms.
Possible support
Focus on the central atom and its nearest neighbours only.You may also like
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