add solutions to Sommerfeld exercises

src/4_sommerfeld_model_solutions.md

@@ -89,9 +89,11 @@ $$
 \Delta E = \frac{\pi^2}{6}(k_B T)^2\frac{\partial}{\partial \epsilon}\left(\epsilon g(\epsilon)\right)\bigg|_{\epsilon=\epsilon _F}
 $$
 
-4, 6.
+5.
 
-Check the source code written in python for solving integral using midpoint rule
+$C_v = 1.6713.10^6 eV/K$
+
+4, 6.
 
 ```python
 mu = 5.2
@@ -123,7 +125,11 @@ dEplus = np.sum(integral(elements, 1000+dT))*((1e6 - 0)/len(elements)) - 0.8e10
 dEmin = np.sum(integral(elements, 1000-dT))*((1e6 - 0)/len(elements)) - 0.8e10 * 5.2**(5./2)
 
 CV = (dEplus - dEmin) / (2*dT);
+
+print('dE = {:.4e} eV'.format(dE))
+print('Cv = {:.4e} eV/K'.format(CV))
 ```
+Check the source code written in python for solving integral using midpoint rule.
 
 ### Exercise 4: graphene