added minor question on why the Peierls transition occurs

docs/8_many_atoms.md

@@ -306,7 +306,8 @@ As usual, we assume the atomic orbitals to be orthogonal to each other.
 4. Sketch the density of states by graphically constructing it from your sketch of the dispersion (no calculations!)
 5. Calculate the group velocity $v(k)$ for both bands. From the group velocity, calculate the density of states $g(E)$ of the entire band structure and make a plot of it. Check if the plot matches your sketch of subquestion 5.
 6. What are the two possible eigenvectors $[\phi_0 \quad \psi_0]$ at $k=0$? Use this to sketch the two possible wavefunctions $|\Psi_\pm(k=0)\rangle$ as a function of the coordinate $x$ along the chain (assume some peak-shaped atomic orbital such as a Gaussian). Discuss how to understand which wave function has the lowest energy.
-7. Repeat the previous subquestion for $k=\pi/a$. In particular, discuss why the two wavefunctions $|\Psi(k=\pi/a)\rangle$ have different energy when $t_1 \neq t_2$, and why they are degenerate when the $t_1=t_2$.
+7. Repeat the previous subquestion for $k=\pi/a$. Use your sketch to argue why the two wavefunctions $|\Psi(k=\pi/a)\rangle$ have different energy when $t_1 \neq t_2$, and why they are degenerate when the $t_1=t_2$.
+8. Consider your sketch of the density of states. Where is the Fermi energy? Argue why a chain of equidistant atoms can lower its energy by displacing the atoms such that the hoppings become unequal (this is the Peierls transition). Is the same true if each atom contributes 2 electrons?
 
 ### Exercise 3: Atomic chain with 3 different spring constants
 

docs/8_many_atoms_solutions.md

@@ -209,6 +209,9 @@ pyplot.suptitle('Wavefunctions at $k=0$ and $k=\pi/a$');
 pyplot.yticks([],[]);
 ```
 
+8.
+If we have one electron per atom, all states below $E_F = \varepsilon_0$ are filled. Because a band gap appears when the hoppings become different, the total energy of the electrons will be lower than if the hoppings are equal. In contrast, for 2 electrons per atom, all electronic states would be filled with electrons and therefore the total energy would be insensitive to a difference in the hoppings.
+
 ## Exercise 3: atomic chain with 3 different spring constants
 
 1. The unit cell should contain exactly one spring of $\kappa_1$, $\kappa_2$ and $\kappa_3$ and exactly three atoms.