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
+## Exercise 1: Lattice vibrations
 
-1.
+### Subquestion 1
 
-Hint: Normal modes have the same function as $$\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$$ in $$\mathbb{R}^3$$
+Hint: Normal modes have the same function as $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$ have in $\mathbb{R}^3$.
 
 Hint: The lectures concerns atom vibrations, so what will a phonon be?
-??? Final Hint
+
+??? hint "Major hint"
 
     What's the question's title?
 
 Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'particles' are phonons?
 
-2. 
+### Subquestion 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: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}}$$ $$ g(\omega) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \bigg ] = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$
+
+### Subquestion 3
+
+```python
+pyplot.subplot(1,2,1)
+k = np.linspace(-pi+0.01, pi-0.01, 300)
+pyplot.plot(k, np.sin(k)/(np.sqrt(1-np.cos(k))));
+pyplot.xlabel('$k$'); pyplot.ylabel('$v(k)$');
+pyplot.xticks([-pi, 0, pi], [r'$-\pi/2$', 0, r'$\pi/2$']);
+pyplot.yticks([-np.sqrt(2), 0, np.sqrt(2)], [r'$-2\sqrt{\frac{\kappa}{m}}$', 0, r'$2\sqrt{\frac{\kappa}{m}}$']);
+pyplot.tight_layout();
+
+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([0.5, 1], [0, r'$\frac{L}{2\pi a}\sqrt{\frac{\kappa}{m}}$']);
+pyplot.tight_layout();
+pyplot.show()
+```
+
+### Subquestion 4
+
+Hint: The group velocity is given as $v = \frac{d\omega}{dk}$, draw a coordinate system **under** or **above** the dispersion graph with $k$ on the x-axis in which you draw $\frac{d\omega}{dk}$.
+
+??? hint "Plot for density of states"
+
+```python
+k = np.linspace(0,pi/2,1000)
+w = 4*np.sin(k) + np.sin(3*k)
+pyplot.hist(w,50, orientation='horizontal',ec='black')
+pyplot.ylabel(r'$\omega$')
+pyplot.xlabel(r'$g(\omega)$')
+```
 
-Group velocity is given as $$v=\hbar^-1 \frac{\partial E}{\partial k}