Physics 573: Problem set 1

1. Calculate the c/a ratio for a hexagonal close-packed lattice.

2. (a) What are the reciprocal lattice vectors of a hexagonal lattice (not HCP)?
   (b) What is the volume of the Brillouin zone?
   

3. A one-dimensional monatomic crystal has lattice spacing a. The total scattering amplitude is F = Σ_n exp{-in(a·Δk)}.
   (a) Perform the sum over N lattice points and compute |F|² = sin²{N(a·Δk)/2}/sin²{(a·Δk)/2}.
   (b) The scattering maxima obtain where the denominator vanishes or where (a·Δk) = 2πl, where l is any integer. Estimate the width of the scattering peak by assuming a small fluctuation in Δk so that (a·Δk) = 2πl + ε.

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