S. [20 points] A copper (Cu) atom has one valence electron, while a Zinc (Zn) atom has two. It is known that the crystal structure of Cu metal is face-centered cubic. Such structure is still maintained for partially substitution of Cu by Zn (i.e., replace some Cu atoms by Zn atoms). Generally, substitution is random that will not change the size of primitive cell, hence the BZ can be treated as unchanged. Consider the empty lattice model (i.e., free electron, no distortion for Fermi surface at the BZ boundary), calculate the ratio of Cu:Zn when Fermi sphere starts to touch the BZ boundary. [hint: the reciprocal lattice for fcc is bee. The reciprocal lattice and the first BZ is shown below: ' / NZ 9 7 | 1