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Today I was asked to show that the perpendicular height, $H,$ of a regular tetrahedron is related to the length of a side, $L,$ of the tetrahedron, by $H = \sqrt{(2/3)} \cdot L$. I did this no problem.

I was then asked: Explain how this is relevant in determining a relationship between the unit cell parameters, $a$ and $c$ of a hexagonal-close packed crystal structure: $c = 1.633 · a$.

Could anyone possibly help me out here?

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  • $\begingroup$ Well, if you know what an HCP structure is, then you could have guessed that the said ratio of 1.633 was not put there on someone's whim. $\endgroup$ Feb 11 '20 at 19:17
  • $\begingroup$ No I realise it’s twice of a - I think I know what’s going on but I don’t know what they’re asking for here $\endgroup$ Feb 11 '20 at 20:17
  • $\begingroup$ Imagine the whole construction, with spheres and everything. That 1.633 is not some random decimal number. It must have been derived geometrically, you know, with square roots and stuff. $\endgroup$ Feb 11 '20 at 20:50
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A picture is worth a thousand words (see the two tetrahedra heights making up the vertical c-axis?):

enter image description here

The HCP packing has layers of closest-packed spheres in an ABAB... pattern. The distance between layers is one height of a tetrahedron, and you need two layers to get a repeating structure along c, i.e. the unit cell parameter.

(picture found at https://chemistry.stackexchange.com/a/111921)

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