Updates degeneracy error and latex error

src/4_sommerfeld_model_solutions.md

@@ -87,7 +87,7 @@ $$
 
 2. 
 $$
-E = \frac{4}{5} (5.2)^{\frac{5/2}} 10^{10} eV
+E = \frac{4}{5} (5.2)^{\frac{5}{2}} 10^{10} eV
 $$
 
 3. 
@@ -170,12 +170,12 @@ ax.xaxis.set_label_coords(1.0, 0.49)
 
 2.The DOS for the positive energies is given by
 $$
-g(\varepsilon) = (2_s + 2_v) 2 \pi \left(\frac{L}{2 \pi}\right)^2 \frac{\varepsilon}{c^2},
+g(\varepsilon) = (2_s \dot 2_v) 2 \pi \left(\frac{L}{2 \pi}\right)^2 \frac{\varepsilon}{c^2},
 $$
 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
 $$
-g(\varepsilon) = (2_s + 2_v) 2 \pi \left(\frac{L}{2 \pi}\right)^2 \frac{|\varepsilon|}{c^2}.
+g(\varepsilon) = (2_s \dot 2_v) 2 \pi \left(\frac{L}{2 \pi}\right)^2 \frac{|\varepsilon|}{c^2}.
 $$
 
 3.$g(\varepsilon)$ vs $\varepsilon$ 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: