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**Hexagonal Close Packing (Why do I get c/a = 3^1/2 ??)**

## Homework Statement

For the hexagonal close packed unit cell, show that c/a = 1.6333. Where c is the height of the cell, and a is the length of the sides of the base. (See picture below)

## Homework Equations

http://img254.imageshack.us/img254/5609/hcpqr6.th.jpg [Broken]

Note: There's a typo in the picture above: [itex]c = h^2 - a^2[/itex] should be [itex] c^2 = h^2 - a^2[/itex].

## The Attempt at a Solution

[tex]

h = 4r , \

\\ a = 2r , \

a^2 + c^2 = h^2 \Rightarrow c^2 = h^2 - a^2 = 16r^2 - 4r^2 = 12r^2

\Rightarrow c = 2 \sqrt{3}r \Rightarrow

\frac{c}{a} = \frac{2 \sqrt{3}r}{2r} = \sqrt{3} \approx 1.732 \neq 1.633

[/tex]

Can someone at least give me a hint at what I did wrong? It just seems like I am certainly correct, unless the angle between a and c is not [itex]90^o[/itex] (which I really doubt)

Thanks!

**Tangential Latex Question:**Does anyone know why \\ is not producing a new line for me? Thanks.

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