Update 4_sommerfeld_model.md - polish

src/4_sommerfeld_model.md

@@ -27,12 +27,12 @@ $$
 \mathcal{H}=\frac{ {\bf p}^2}{2m}=-\frac{\hbar^2}{2m}\left( \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} \right)\ \Rightarrow\ \varepsilon=\frac{\hbar^2}{2m}\left( k_x^2+k_y^2+k_z^2 \right)
 $$
 
-If we assume periodic boundary conditions, $\psi(x,y,z)=\psi(x+l,y+L,z+L)$, it follows that the allowed wavenumbers are given by
+If we impose periodic boundary conditions, $\psi(x,y,z)=\psi(x+l,y+L,z+L)$, it follows that the allowed wavenumbers are given by
 
 $$
 k_x=\frac{2\pi p}{L},\ k_y=\frac{2\pi q}{L},\ k_z=\frac{2\pi r}{L}\ \Rightarrow\ \varepsilon=\frac{2\pi^2\hbar^2}{mL^2}\left( p^2+q^2+r^2 \right).
 $$
-(this is exactly analogous to phonons, see lecture 2 - Debye model)
+(this quantization of k-space is exactly analogous to that for phonons, see lecture 2 - Debye model)
 
 ```python
 kf = 3;
@@ -44,7 +44,7 @@ Edis = ks**2;
 Econt = kcont**2;
 
 fig = pyplot.figure();
-fig.set_size_inches(2, 2)
+#fig.set_size_inches(2, 2)
 ax = fig.add_subplot(111);
 ax.plot(kcont, Econt);
 ax.plot(ks, Edis, 'k.', markersize=10);
@@ -62,16 +62,13 @@ ax.set_yticks([]);
 draw_classic_axes(ax, xlabeloffset=.6);
 ```
 
-Comparable to phonons, but: electrons are _fermions_.
+The difference with phonons is that electrons are _fermions_, implying that there are only 2 electron states (due to spin) per $k$-value.
 
-- Only 2 (due to spin) allowed per $k$-value
-- Fill up from the lowest energy until you run out of electrons
-
-$\rightarrow$ Calculate when you are out of electrons $\rightarrow$ _Fermi energy_.
+Given the number of electrons in a system, we can now fill up these states starting from the lowest energy until we run out of electrons, at which point we reach the _Fermi energy_.
 
 ![](figures/fermi_circle_periodic.svg)
 
-In order to compute the density of states, we need to perform an integration of k-space. Assuming three dimensions and spherical symmetry (the dispersion in the free electron model is isotropic) we find for the total number of states:
+To compute the density of states, we need to perform an integration of k-space. Assuming three dimensions and spherical symmetry (the dispersion in the free electron model is isotropic) we find for the total number of states:
 
 $$
 N=2\left(\frac{L}{2\pi}\right)^3\int{\rm d}{\bf k}=2 \left(\frac{L}{2\pi}\right)^34\pi\int k^2{\rm d}k=\frac{V}{\pi^2}\int k^2{\rm d}k ,