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From what I was reading, Silicon has a FCC unit cell but they also said that the Silicon atoms form a tetrahedron at 109 degrees from each other. Then they said that the tetrahedron is formed by overlapping of two FCC unit cells ane starting at (0,0,0) and the other starting at (a/4,a/4,a/4) where a is the FCC unit cell dimension. Now, aren't the claims themselves contradictory on the face of it? If I have an FCC unit cell, shouldn't my next cell start at (a,0,0), (0,a,0) and the likes...?? Isn't that the meaning of a unit cell? Shouldn't a lattice be a vectorial repetition of the dimensions of a unit cell? In this case, if I assumed the FCC as my basis vectors, would that get violated if I intended to represent my chrystal lattice with that?

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There are two separate issues here.

Firstly the smallest possible unit cell for a crystal is called the primitive unit cell. However in many cases this is an awkward shape and it's easier to use a bigger unit cell that contains more than one primitive unit cell. The FCC unit cell is one of these non-primitive (I'm not sure what the actual term is) unit cells.

Secondly even the primitive unit cell does not necessarily contain just a single atom. The corners of the primitive cell are marked by atoms that have the same environment, but the silicon structure contains two different environments for the silicon. Have a look at this picture of the silicon structure:

Silicon

If you look at the atom I've outlined in red and the atom I've outlined in green you'll see they are in different environments. If you look at the upper two bonds you'll see that for the red atom the V formed by the bonds projects out of the screen. For the green atom the V lies in the plane of the screen. In effect you get the red atom by rotating the green atom 90º about a vertical axis.

So even the primitive cell contains two silicon atoms, and the compound FCC cell contains four primitive cells. Despite this, every silicon atom is indeed at the centre of a tetrahedron.

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