Physics 489 Problem Set 2
Due Tuesday, Sept. 22
(1) Kittel 2.2
(2) Kittel 2.4, diffraction peak in finite-size crystal.
Also: Find a way to graph this function for a few values of M.
(3) Simple example of 3-D Fourier transform: A tetragonal lattice has cell dimensions a×a×c,
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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:
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V (r ) = Vo ∑ j ⎡⎣δ (r − r1 − R j ) − δ (r − r2 − R j )⎤⎦ ,
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with r1 = ( 0, 0, 0 ) and r2 = 1 2,1 2,1 5 .
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[a] What is the general form for the reciprocal lattice vectors, G ?
[b] Find a general form for VG , the Fourier coefficients of the potential. Do this by directly
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computing the Fourier transform. Evaluate the result for the cases G = (1, 0, 0 ) and G = ( 0, 0,1) .
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[c] Show that the G = ( 0, 0, 0 ) coefficient is zero. Explain why one can expect this result for this
case without calculating the Fourier transform.
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Note: The Fourier coefficients of this real potential are complex, and they cannot be converted
to coefficients that are entirely real due to the lack of a point of inversion symmetry. In cases of
simple lattices VG may be considered to be real, as a general property.
(4) In the handout on symmetry, reciprocal lattices and x-ray scattering, in the last figure 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.