@@ -230,10 +230,10 @@ 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.
??? hint
Make use of Taylor series.
Make use of a Taylor expansion.
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 for 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 $m$.
...
...
@@ -266,6 +266,6 @@ Due to the alternating hopping energies, we must treat two consecutive atoms as
Note that the coefficients of the trial LCAO's and the periodic boundary condition require that $\left< \Phi \right| H \left| \Phi \right> = N a_n \left< \Phi \right| H \left| \phi_n \right>$ for any $n$. Similar can be done for $\left< \Phi \right| H \left| \Psi \right>$ and $\left< \Psi \right| H \left| \Psi \right>$.
4. Derive the dispersion relation of this Hamiltonian. Does it look like the figure of the band structure shown on the [Wikipedia page](https://en.wikipedia.org/wiki/Peierls_transition#/media/File:Peierls_instability_after.jpg)? Does it reduce to the 1D, equally spaced atomic chain if $t_1 = t_2$.
4. Derive the dispersion relation of this Hamiltonian. Does it look like the figure of the band structure shown on the [Wikipedia page](https://en.wikipedia.org/wiki/Peierls_transition#/media/File:Peierls_instability_after.jpg)? Does it reduce to the 1D, equally spaced atomic chain if $t_1 = t_2$?
5. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands.
6. Derive an expression for the density of states $g(E)$ of the entire band structure.
