Update src/7_tight_binding_model_sol.md

src/7_tight_binding_model_sol.md

@@ -29,7 +29,7 @@ Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'partic
 
 ### 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: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}}$$ $$ g(\omega) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}\bigg(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2} \bigg) \bigg ] = \frac{L}{\pi a} \frac{1}{\sqrt{\frac{4\kappa}{m}-\omega^2}}$$
+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: $$ v(k) = & \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}}$$ $$ g(\omega) \\ = & \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}\bigg(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2} \bigg) \bigg ] = \frac{L}{\pi a} \frac{1}{\sqrt{\frac{4\kappa}{m}-\omega^2}}$$
 
 ### Subquestion 3
 
@@ -61,3 +61,9 @@ Hint: The group velocity is given as $v = \frac{d\omega}{dk}$, draw a coordinate
 
     ![](figures/dispersion_groupv_dos.svg)
 
+## Exercise 2: Vibrational heat capacity of a 1D monatomic chain
+
+### Subquestion 1
+
+For the energy we have $$U = \int \hbar \omega g(\omega) (n(\omega) + \frac{1}{2})d\omega$$ with $$
+