Added solutions for exercise 4 and fixed some typo's

docs/4_sommerfeld_model_solutions.md

@@ -18,10 +18,9 @@ E = \int \limits_0^{\infty} g(\varepsilon) n_{F}(\beta (\varepsilon - \mu)) \var
 $$
 
 #### Question 4.
-The Fermi Surface is the boundary of the fermi sea below which all states are occupied at zero temperature, it respecitvely a line, circle or a sphere in the 1,2 or the 3rd dimension.
+The Fermi Surface is the boundary of the fermi sea below which all states are occupied at zero temperature, it's respecitvely a line, circle or a sphere in the 1,2 or the 3rd dimension.
 #### Question 5.
 If the electrons in the solid can be described by the free electron model, the heat capacity is dominated by the electrons, since $T^3$ decreases faster than $T$ as $T->0$.
-Electrons.
 
 
 ### Exercise 1*: Deriving the density of states for a parabolic dispersion relation.
@@ -34,10 +33,10 @@ $ \Delta k = \frac{2 \pi}{L},  density: $(\frac{L}{2\pi})^d$, where d = 1,2 or 3
 
 #### Question 3.
 $$
-N_{states} = 2_s (\frac{L}{2\pi})^d \int_0^{\varepsilon(\mathbf{k})< \varepsilon_0} n_F(\beta(\varepsilon(\mathbf{k})-\mu))\textrm{d} \mathbf{k}
+N_{states} = 2_s (\frac{L}{2\pi})^d \int_0^{\varepsilon(\mathbf{k})< \varepsilon_0}\textrm{d} \mathbf{k}
 $$
 #### Question 4.
-You need the dispersion relation: $N_{states} = \int_0^{\varepsilon_0}n_F(\beta(\varepsilon-\mu)) \textrm{d} \varepsilon$.
+You need the dispersion relation, the integral boundaries are $\int_0^{\varepsilon_0}$.
 With 
 \begin{align}
 g_{1D}(\varepsilon) &= 2_s \frac{L}{2\pi} \sqrt{\frac{2m}{\hbar^2\varepsilon}},\\