@@ -228,6 +228,10 @@ $\rightarrow \rho_{\rm R}(k)=\frac{L}{2\pi}$, which is lower than for the case o
...
@@ -228,6 +228,10 @@ $\rightarrow \rho_{\rm R}(k)=\frac{L}{2\pi}$, which is lower than for the case o
Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes with 2 different masses (can be found below [here](#more-degrees-of-freedom-per-unit-cell)).
Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes with 2 different masses (can be found below [here](#more-degrees-of-freedom-per-unit-cell)).
1. Find the magnitude of the group velocity near $k=0$ for the _acoustic_ branch.
1. Find the magnitude of the group velocity near $k=0$ for the _acoustic_ branch.
??? hint
Make use of Taylor series.
2. Show that the group velocity at $k=0$ for the _optical_ branch is zero.
2. Show that the group velocity at $k=0$ for the _optical_ branch is zero.
3. Derive an expression for the density of states $g(\omega)$ for the _acoustic_ branch and small $ka$. Make use of your expression of the group velocity in 1.
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.
