513.001 Molecular and Solid State Physics

MIT 8.231 Physics of Solids, 2.5

Indium crystallizes in a body centered tetragonal lattice. For the conventional unit cell the lattice parameters are a = 3.233 Å and c = 4.938 Å and the basis is (0 0 0), (½ ½ ½).

The primitive translation vectors are

a₁ = a/2 x + a/2 y + c/2 z

a₂ = a/2 x + a/2 y - c/2 z

a₃ = a/2 x - a/2 y - c/2 z

a) Make a careful sketch of the conventional unit cell. What is its volume?

b) Make a careful sketch of the primitive unit cell defined by the above vectors. What is its volume?

c) Make a careful sketch of the Wigner-Seitz cell. Note that c > √2a.

d) Find the primitive translation vectors of the reciprocal lattice. What lattice type is the reciprocal lattice?

e) Find and make a carefully labeled sketch of the first Brillouin zone.

f) How many atoms are there in the primitive unit cell of this structure? How many atoms are there in the basis associated with the primitive cell? What does this imply about the structure factor for each point in reciprocal space? Test your hypothesis by finding an expression for the structure factor referred to the cartesian coordinates of the conventional tetragonal cell. Evaluate it at a few points on your reciprocal lattice.