1. A certain solid has a simple cubic Bravais lattice with a cubic primitive cell of sides a. Each primitive cell contains two different atoms: atom A located at (0,0,0) and atom B located at (a/2)(1,1,1). Let the atomic form factors be f_A(K) = 1/|Ka| and f_B(K) = 0.4/|Ka|, in arbitrary units. (Clearly, K is non-zero here.)
(a) List the 4 shortest, inequivalent, non-zero reciprocal lattice vectors. Sets of equivalent vectors can be mapped into one another by cubic symmetry operations, e.g., (1,0,0), (-1,0,0), (0,1,0), etc. From each equivalent set, please select a member which has non-negative k_x, k_y and k_z components.
(b) Calculate the geometrical structure factor for each of the four vectors listed in (a).
(c) Calculate the relative intensity of the diffraction spots corresponding to each of the four vectors listed in (a). Normalize your answers so that the intensity arising from the shortest vector is unity.
(d) Imagine performing a powder diffraction experiment on this material. Assume that a = 2.3 Angstroms and the X-ray source has a wavelength of 1.2 Angstroms. Calculate the four smallest non-zero angles, phi, at which scattering cones will be generated.
2. A two-dimensional metal has 3 electrons/unit cell in the valence bands.
(a) Calculate the free-electron Fermi wavevector in units of pi/a.
(b) Assuming a square lattice, construct the first three Brillouin zones and draw the free-electron Fermi surface in the extended zone scheme. (You do not need to produce a scale drawing. However, your diagram should be sufficiently detailed that it is clear which zones are cut by the Fermi surface. If you wish, you can use the circle drawn on the back of this paper to represent the Fermi surface, in which case you should adjust the rest of the digram to roughly the right size.) The circle would appear on the back of the exam question sheet. You can draw your own when you try out this question.
(c) Redraw each branch of the free-electron Fermi surface in the reduced zone scheme, and superimpose a sketch of the Fermi surface in the presence of a weak periodic potential. (Make sure the free-electron and nearly-free-electron surfaces are clearly labeled.)
(d) Imagine the periodic potential is made much stronger. Explain the likely effect on each branch of the Fermi surface.