Updates solutions

src/4_sommerfeld_model_solutions.md

@@ -8,9 +8,9 @@ $$
 E = \int \limits_0^{\infty} g(\epsilon) n_{FD}(\beta (\epsilon - \mu)) \epsilon \mathrm{d} \epsilon
 $$
 
-3. ?
+3. The electronic heat capacity $C_e$ approaches $3N k_B$.
 
-4. Thermal smearing is too significant and we can not accurately approximate the fraction of the excited electron through triangles anymore. Thus the Sommerfeld expansion breaks down.
+4. Thermal smearing is too significant and we can not accurately approximate the fraction of the excited electron with triangles anymore. Thus the Sommerfeld expansion breaks down.
 
 5. Electrons.
 
@@ -167,11 +167,9 @@ ax.xaxis.set_label_coords(1.0, 0.49)
 ```
 
 2. The DOS for the positive energies is given by
-
 $$
 g(\epsilon) = (2_s + 2_v) 2 \pi \left(\frac{L}{2 \pi}\right)^2 \frac{\epsilon}{c^2},
 $$
-
 where $2_s$ is the spin degeneracy and $2_v$ is the valley degeneracy.
 If we account for the negative energies as well, we obtain
 $$
@@ -179,7 +177,6 @@ g(\epsilon) = (2_s + 2_v) 2 \pi \left(\frac{L}{2 \pi}\right)^2 \frac{|\epsilon|}
 $$
 
 3. $g(\epsilon)$ vs $\epsilon$ is a linear plot. Here, the region marked by $-k_B T$ is a triangle whose area gives the number of electrons that can be excited:
-
 $$
 \begin{align}
 n_{ex} &= \frac{1}{2} g(-k_B T) k_B T\\