Solutions lecture 7

src/8_many_atoms_sol.md

@@ -17,7 +17,7 @@ pi = np.pi
 
 ### Subquestion 1
 
-Accoustic branch corresponds with (-) in the equation given in the lecture notes. Use the small angle approximation $\sin(k) \approx k$ to ease calculations. For the taylor polynomial take $\omega^2 = f(x) = f(0) + f'(0)k + f''(0)k^2$ (some terms vanish). You should find: $$|v_g| = \sqrt{\frac{\kappa a^2}{2(m_a+m_2)}}$$
+Accoustic branch corresponds with (-) in the equation given in the lecture notes. Use the small angle approximation $\sin(k) \approx k$ to ease calculations. For the Taylor polynomial take $\omega^2 = f(x) \approx f(0) + f'(0)k + f''(0)k^2$ (some terms vanish, computation is indeed quite tedious). You should find: $$|v_g| = \sqrt{\frac{\kappa a^2}{2(m_a+m_2)}}$$
 
 ### Subquestion 2
 
@@ -25,3 +25,19 @@ Optical branch corresponds with (+) in the equation given in the lecture notes.
 
 ### Subquestion 3
 
+Density of states is given as $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$. $\frac{dN}{dk} = \frac{L}{2\pi}$ since we have 1D. $\frac{dk}{d\omega}$ can be computed using the group velocity: $$\frac{dk}{d\omega} = \bigg(\frac{d\omega}{dk} \bigg)^{-1} = (v_g)^{-1}$$
+
+## Exercise 2: the Peierls transition
+
+### Subquestion 1
+
+A unit cell consits of exactly one $t_1$ and exactly $t_2$ hopping.
+
+### Subquestion 2
+
+Using the hint we find:
+$$ E \phi_n = \epsilon \phi_n + t_1 \psi_n + t_2 \phi_{n-1} $$
+$$ E \psi_n = \epsilon \psi_n + t_2 \phi_{n+1} + t_2 \psi_n $$
+
+
+