Problem: Copper Oxide Layers

The common building blocks for most high temperature (high ) superconductors   are copper oxide layers, as depicted in Figure 1.6. Assume the distance between copper atoms (filled circles) is a. For simplicity let us also assume that in the third dimension these CuO layers are simply stacked with spacing c, and there are no other atoms in the crystal. In first approximation the layers have a four-fold symmetry; the crystal is tetragonal.

  
Figure 1.6: CuO lattice

  (a) Sketch the Bravais lattice and indicate a possible set of primitive vectors for this crystal. What is the unit cell, and what is the basis?    

(b) In LaCuO one discovers, at closer inspection, that the CuO lattice is actually not flat, but that the oxygen atoms are moved a small amount out of the plane (``up" or ``down") in an alternating fashion (in Figure 1.7, a + means up and a - means down). What is the primitive cell and lattice spacing for this crystal? What is the reciprocal lattice?   Describe (qualitatively) what happens in the X-ray diffraction pattern as the distortion is decreased gradually to zero.

  
Figure 1.7: Distorted CuO lattice