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The display below shows eight atoms packed together in a cubic arrangement. In the very center of this cube is a hole.
This hole is said to be a **cubic hole** because the hole is surrounded by eight atoms (the coordination number is eight).

Take note that the size of this hole is slightly smaller than that of the atoms surrounding it. 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 *r _{hole}*.

**What is the ratio r_{hole}/r ?**

Derive this ratio using the cubic geometry of the outer (blue) atoms and the laws of trigonometry.

Hint: The Pythagorean theorem will be especially useful: *C ^{2} = A^{2} + B^{2}* .
You will need to apply this theorem twice, once for a side of the cube and once to obtain the main diagonal of the cube.

When you have completed your derivation, you may check your answer.

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Holes in Closest-Packed Structures: Trigonal Tetrahedral Octahedral Cubic

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