Solutions lecture 7

src/7_tight_binding_model_sol.md

@@ -8,7 +8,7 @@ Hint: Normal modes have the same function as $\mathbf{e}_1,\mathbf{e}_2,\mathbf{
 
 Hint: The lectures concerns atom vibrations, so what will a phonon be?
 
-??? {Major Hint}
+??? 'Spoiler'
 
     What's the question's title?
 
@@ -16,4 +16,4 @@ Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'partic
 
 2. 
 
-Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ with $E=\omega \hbar$. So we find: $$v = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{sin(ka){\sqrt{1-cos(ka)}}}
+Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$ with $E=\omega \hbar$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{sin(ka){\sqrt{1-cos(ka)}}}$$ $$ g(k) = \frac{L}{2\pi}