Show that close packed hcp and fcc structures have packing fraction 0.74 for
hard spheres. Show that bcc has packing fraction 0.68.
For the cubic lattices, show that the distance between planes is given
by d(hkl) = a/(h2 + k2 + l2)1/2.
For an fcc crystal composed of one atom at each site of the fcc lattice,
show that the (111) planes are are the most dense
planes of atoms possible, and thus that they are the lattice planes that are the
most widely spaced of all planes of atoms in the fcc crystal.