Sommerfeld figures

src/4_sommerfeld_model.md

@@ -33,7 +33,32 @@ $$
 k_x=\frac{2\pi p}{L},\ k_y=\frac{2\pi q}{L},\ k_z=\frac{2\pi r}{L}\ \Rightarrow\ \varepsilon=\frac{2\pi^2\hbar^2}{mL^2}\left( p^2+q^2+r^2 \right)
 $$
 
-![](figures/free_electron.svg)
+```python
+kf = 3;
+extrapol = 1.1;
+ks = np.arange(-kf, kf+1);
+kcont = np.linspace(-extrapol*kf, extrapol*kf, 200);
+
+Edis = ks**2;
+Econt = kcont**2;
+
+fig = pyplot.figure();
+ax = fig.add_subplot(111);
+ax.plot(kcont, Econt, 'b-');
+ax.plot(ks, Edis, 'k.', markersize=10);
+for i in range(2*kf + 1):
+    ax.plot([ks[i], ks[i]], [0.0, Edis[i]], 'k:');
+ax.set_xlim(-3.75, 3.75);
+ax.set_ylim(0.0, 11);
+
+ax.set_xlabel(r"$k \enspace \left[ \frac{2 \pi}{L} \right]$");
+ax.set_ylabel(r"$\varepsilon$");
+
+ax.set_xticklabels([""] + ks.tolist() + [""]);
+ax.set_yticks([]);
+
+draw_classic_axes(ax, xlabeloffset=.6);
+```
 
 Comparable to phonons, but: electrons are _fermions_.
 
@@ -97,7 +122,34 @@ For copper, the Fermi energy is ~7 eV. It would take a temperature of $\sim 70 0
 
 The total number of electrons can be expressed as $N=\frac{2}{3}\varepsilon_{\rm F}g(\varepsilon_{\rm F})$.
 
-![](figures/fermi_level.svg)
+```python
+kf = 3.0;
+extrapol = 4.0/3.0;
+kfilled = np.linspace(-kf, kf, 500);
+kstates = np.linspace(-extrapol*kf, extrapol*kf, 500);
+
+Efilled = kfilled**2;
+Estates = kstates**2;
+
+fig = pyplot.figure();
+ax = fig.add_subplot(111);
+ax.plot([kf, kf], [0.0, kf*kf], 'k:');
+ax.plot(kstates, Estates, 'k--');
+ax.plot(kfilled, Efilled, 'b-', linewidth=4);
+ax.axhline(kf*kf, linestyle="dotted", color='k');
+
+ax.set_xticks([kf]);
+ax.set_yticks([kf*kf + 0.4]);
+ax.set_xticklabels([r"$k_F$"]);
+ax.set_yticklabels([r"$\varepsilon_F$"]);
+
+ax.set_xlabel(r"$k$");
+ax.set_ylabel(r"$\varepsilon$");
+
+ax.set_xlim(-kf*extrapol, kf*extrapol);
+ax.set_ylim(0.0, kf*kf*extrapol);
+draw_classic_axes(ax, xlabeloffset=.6);
+```
 
 The bold line represents all filled states at $T=0$. This is called the _Fermi sea_. Conduction takes place only at the _Fermi surface_: everything below $\varepsilon_{\rm F}-\frac{eV}{2}$ is compensated.