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According to Marder, Condensed Matter Physics, Chapter 2:

Within the planes normal to the vector [1,1,1], the atoms of an fcc lattice lie in a two dimensional triangular lattice

However, he does not provide a proof of this claim. How would one go about showing this?

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2 Answers 2

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The easiest way is to simply construct the (111) face and then it will be self-evident.

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The vector (1,1,1) is normal to a plane x+y+z=c. The nearest lattice point is (+1,-1,0) from this. There are six of these, and it is easy to show the distance from eg +1,-1,0 to +1,0,-1 or +1,0,-1 to 0,0,0 are equal, and thus it must be a triangular lattice.

The same vector in N dimensions (1,1,1,...), produces a simplex-based lattice, is usually how one studies these.

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