@@ -229,8 +229,7 @@ Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes w
1. Find the magnitude of the group velocity near $k=0$ for the _acoustic_ branch.
2. Show that the group velocity at $k=0$ for the _optical_ branch is zero.
3. Derive an expression for the density of states $g(\omega)$ in the _optical_ branch.
4. Make a plot of your expression of $g(\omega)$ found in 3. Does the plot look like the bar diagram of the density of states of the optical branch in the lecture notes?
3. Derive an expression for the density of states $g(\omega)$ for the _acoustic_ branch and small $ka$. Make use of your expression of the group velocity in 1.
#### Exercise 2: atomic chain with 3 different spring constants
Suppose we have a vibrating 1D atomic chain with 3 different spring constants alternating like $\kappa_ 1$, $\kappa_2$, $\kappa_3$, $\kappa_1$, etc. All the the atoms in the chain have an equal mass $m$.
