513.001 Molecular and Solid State Physics

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Length of the largest wave vector

(a) Find the length of the largest wave vector in the first Brillouin zone of a simpe cubic lattice measured from k = 0 (the center of the Brillouin zone, Γ) that is parallel to a cube edge [100]* Γ-X, parallel to a cube body diagonal [111]* Γ-R, and parallel to a cubic face diagonal [110]* Γ-M.

(b) Find the length of the largest wave vector in the first Brillouin zone of a fcc lattice measured from k = 0 (the center of the Brillouin zone Γ) to any point on the Brillouin zone boundary. (This is more difficult than part (a)).

Hint: For a face centered cubic lattice, you should start with the primitive lattice vectors in real space and construct the primitive lattice vectors in reciprocal space. From the lattice vectors in reciprocal space you can determine the first Brillouin zone.

The notation [u,v,w]* indicates a direction in reciprocal space whereas [u,v,w] indicates a direction in real space.