(a) In lecture, we learned that the choice of unit cell is far from unique. In discussing FCC lattices, for instance we could choose a trigonal primitive unit cell, a1=2a(y^+z^);a2=2a(x^+z^);a3=2a(x^+y^), or a conventional cubic unit cell, a1=ax^a2=ay^a3=az^, with a 4 atom basis. Calculate the density of lattice points in k-space for these two choices of unit cell. Hint: Use what you learned in Problem Set 2, Question 3, in order to reduce the amount of calculations. Explain in a few words why you should expect that the primitive unit cell has a lower density of reciprocal lattice points than a non-primitive unit cell with a basis. (b) Since the Laue condition is satisfied when the scattering vector K=kk is equal to a reciprocal lattice vector, why is the diffraction pattern independent of the choice of the unit cell even though they have different densities of lattice points in k-space? In your answer, sketch the two lattices from part (a) in k-space, with one fitted inside the other. Label the reciprocal lattice points on your sketch obtained from the real-space primitive lattice vectors according to Miller indices defined by the primitive unit cell