Crystals such as silicon have only one kind of atom, but they occur in two different orientations. We can model this behavior in one dimension by using a line of masses which are identical, but with two different spring constants, alternating, which we take as C₁ and C₂.

(A)    give the equations of motion for this line of masses, assuming that each can only move in the direction of the line

(B)    find an expression for the dispersion relationship w(k)    (you do not need to simplify it or solve for w or k)

(C)    how many optical phonon branches are there in this model

(D)    what is the k vector of the highest frequency phonon in this model

(E)    what is the frequency of the highest frequency phonon in this model