Assume that an atom has a radius r and the unit cell is cubic with a side length a. Find the number of atoms per unit cell and the packing fraction for a:
(a) Simple cubic structure, (b) Body-centered cubic structure, and (c) Face-centered cubic structure, (d) Diamond structure.
The packing fraction is the fraction of the volume of the unit cell that is inside the atoms, assuming the atoms are "hard spheres" and that they touch in the direction(s) of the shortest bond(s). Note that the size of the unit cell is determined by both the structure and r, so that there is a characteristic relationship between r and a for each structure. (Hint: the packing fraction is a number. It does not contain r or a.)