Physics 617 Problem Set 2
Due Weds, Feb. 10
(1) Ashcroft-Mermin #4.6
(2) [a] For the simple cubic lattice of cube edge a, show that the lattice planes indexed by Miller
indices (h, k, l) are separated by,
a
d=
2
h + k 2 + 2
[b] For the orthorhombic lattice, with perpendicular cell edges a, b, and c, find the corresponding
relationship for d in terms of the Miller indices. [You can do this using the cell-intercept method;
for that the rule for direction cosines may be helpful: the sum of the three squares for any given
vector direction is equal to 1.]
(3) Simple example of 3-D Fourier transform: A tetragonal lattice has cell dimensions a×a×c,

with c corresponding to a3 according to standard convention (same as problem 1). The periodic
potential consists of two Dirac delta functions in each cell:

  
  
V (r ) = Vo ∑ j ⎡⎣δ (r − r1 − R j ) − δ (r − r2 − R j )⎤⎦ ,


with r1 = ( 0, 0, 0 ) and r2 = 1 2,1 2,1 3 .

[a] What is the general form for the reciprocal lattice vectors, K ?
[b] Find a general form for VK , the Fourier coefficients of the potential.

[c] Show that the K = ( 0, 0, 0 ) coefficient is zero. Explain why one can expect this result for this
case without calculating the Fourier transform.
[d] What other reciprocal lattice points also correspond to Fourier coefficients that are zero? Can
you identify a pattern in k-space?
(
)
(4) Ashcroft-Mermin #6.5
(5) In the handout on x-ray scattering, in figure 3 the visible x-ray reflection lines are shown
schematically for the simple cubic, BCC, and FCC lattices. The highest-order reflections shown
are: (311) for SC, (332) for BCC, and (440) for FCC. In each case find the next visible reflection
line.