Update 12_band_structures_in_higher_dimensions.md - polish

src/12_band_structures_in_higher_dimensions.md

@@ -36,13 +36,13 @@ When the Fermi level lies in the band gap, the material is called a semiconducto
 
 In an insulator every single band is either completely filled or completely empty.
 
-How many electrons may an insulator have per unit cell? To answer this we need to integrate the density of states. Integrating $g(E)$ is hard, but integrating $\rho(k)$ is easy.
+How many electrons per unit cell can we expect for an insulator? To answer this we need to know the number of states within an energy band. We can calculate this by integrating the density of states $g(E)$, but this is hard. However, we can easily see how many states there are in an energy band by counting the number of $k-$states in the first Brillouin zone.  
 
 For a single band
 
-$$ N = 2 \int_{BZ}dk_x dk_y dk_z [L\times W\times H] (2\pi)^{-3} = 2 LWH / a^3 $$
+$$ N_{states} = 2 \frac{L^3}{2\pi}^3 \int_{BZ} dk_x dk_y dk_z  = 2 L^3 / a^3 $$
 
-So a single band has 2 electrons per unit cell (because of spin).
+Here, $L^3/a^3$ is the number of unit cells in the system, so we see that a single band can host 2 electrons per unit cell (because of spin). If there are no overlapping bands, a system with 2 electrons per unit cell will therefore be an insulator/semiconductor.
 
 We come to the important rule: