The display below shows three atoms packed together in a closest-packed arrangement. In the very center of this structure is a hole. This hole is said to be a trigonal hole because the hole is surrounded by three atoms (the coordination number is three).
Take note that the size of this hole is quite small. If an atom is sufficiently small, it can fit into this hole. Click on the "Show Atom in Hole" to insert a small atom (colored red) in the hole.
What is the largest atom that can fit into the hole without pushing the outer (blue) atoms apart?
The radius of the outer atoms is designated r. The radius of the hole (this is the radius of the largest sphere that can fit in the hole) is represented by rhole.
What is the ratio rhole/r?
Derive this ratio using the triangular geometry of the outer (blue) atoms and the laws of trigonometry.
Hint: The law of cosines will be particularly useful: C2 = A2 + B2 - 2 A B cos c
When you have completed your derivation, you may check your answer.
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