Problem 1: Crystal Structures (15 Points) Determine the lattice (including the primitive lattice vectors) and basis for each of the structures shown below. In each case, choose the basis with the smallest number of basis elements. Sketch the Wigner-Seitz primitive cell for your specified lattice. Label any distance you need and assume the structure repeats indefinitely as indicated. Calculate the reciprocal lattice vectors for each case and sketch the reciprocal lattice. a) A 1D Copper-oxide chain that repeats along the direction indicated by the dots. Copper Oxygen b) The "Kagome" lattice (repeats in 2D). Each lattice point represents a single atom and the shortest distance between each atom is "a". c) A plane involving three types of atoms in the 2D plane. Each color represents a different type of atom. •