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.