Solutions lecture 7

src/7_tight_binding_model_sol.md

+```{python initialize=true}
+import matplotlib
+from matplotlib import pyplot
+
+import numpy as np
+
+from common import draw_classic_axes, configure_plotting
+
+configure_plotting()
+
+pi = np.pi
+```
+
 # Solutions for lecture 7 exercises
 
 ### Exercise 1: Lattice vibrations
@@ -14,6 +27,32 @@ Hint: The lectures concerns atom vibrations, so what will a phonon be?
 
 Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'particles' are phonons?
 
-2. 
+2.
+
+Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ with $E=\hbar\omega$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}} = a\sqrt{\frac{\kappa}{m}}\cos(\frac{ka}{2})$$ $$ g(\omega) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \bigg ] = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$
+
+3.
+
+```python
+##pyplot.subplot(1,2,1)
+##k = np.linspace(-pi/2, pi/2, 300)
+##pyplot.plot(k, np.cos(k))
+##pyplot.xlabel('$k$'); pyplot.ylabel('$v(k)$')
+##pyplot.xticks([-pi/2, 0, pi/2], [r'$-\pi/2$', 0, r'$\pi/2$'])
+##pyplot.yticks([0, 1], [0, r'$2\sqrt{\frac{\kappa}{m}}$'])
+##pyplot.hlines([1], -pi/2, 0, linestyles='dashed')
+
+pyplot.subplot(1,2,2)
+w = np.linspace(-0.95, 0.95, 300)
+g = 1/np.sqrt(1-w**2)
+pyplot.plot(w, g)
+pyplot.xlabel(r'$\omega$'); pyplot.ylabel('$g(w)$')
+pyplot.xticks([-1, 0, 1], [r'$-2\sqrt{\frac{k}{m}}$', 0, r'$2\sqrt{\frac{k}{m}}$'])
+pyplot.yticks([0, 1], [0, r'$\frac{L}{2\pi a}\sqrt{\frac{\kappa}{m}}$'])
+pyplot.hlines([1], -pi/2, 0, linestyles='dashed')
+```
+
+4.
+
+
 
-Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$ with $E=\hbar\omega$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}} = a\sqrt{\frac{\kappa}{m}}\cos(\frac{ka}{2})$$ $$ g(k) = \frac{L}{2\pi}\frac{d}{d\omega} \big (\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \big ) = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$