Update src/8_many_atoms_sol.md

src/8_many_atoms_sol.md

@@ -81,7 +81,37 @@ Notice that the red shaded area is not a part of the *Band structure* anymore!
 
 ### Subquestion 5
 
-Both $v_g$ and $m_{eff}$ can be calculated using the respective 
+Both $v_g$ and $m_{eff}$ can be calculated using the respective formulae given in the previous lectures
+
+### Subquestion 6
+
+We know $g(\omega) = \frac{dN}{dk} \frac{dk}{d\omega} = \frac{L}{2\pi} \frac{1}{v_g}$ with $v_g$ from the previous subquestion. Graphically the density of states looks accordingly:
+
+```python
+pyplot.subplot(1,2,1)
+k = np.linspace(-2*pi, 2*pi, 400)
+t1 = 1;
+t2 = 1.5;
+pyplot.plot(k, -(t1+t2)*np.cos(k/2),'r',label='1 atom dispersion')
+pyplot.plot(k[199:100:-1],-(t1+t2)*np.cos(k[0:99]/2),'r--',label='1 atom dispersion with folded Brillouin zone')
+pyplot.plot(k[299:200:-1],-(t1+t2)*np.cos(k[300:399]/2),'r--')
+pyplot.plot(k, np.sqrt(t1**2 + t2**2+2*t1*t2*np.cos(k)),'b',label='2 atom dispersion')
+pyplot.plot(k, -np.sqrt(t1**2 + t2**2+2*t1*t2*np.cos(k)),'b')
+
+pyplot.xlabel('$ka$'); pyplot.ylabel(r'$E-\epsilon$')
+pyplot.xlim([-2*pi,2*pi])
+pyplot.xticks([-2*pi, -pi, 0, pi,2*pi], [r'$-2\pi$',r'$-\pi$', 0, r'$\pi$',r'$2\pi$'])
+pyplot.yticks([-t1-t2, -np.abs(t1-t2), 0, np.abs(t1-t2), t1+t2], [r'$-t_1-t_2$',r'$-|t_1-t_2|$', '0', r'$|t_1-t_2|$', r'$t_1+t_2$']);
+
+pyplot.subplot(1,2,2)
+w = np.sqrt(t1**2 + t2**2+2*t1*t2*np.cos(k) - np.sqrt(t1**2 + t2**2+2*t1*t2*np.cos(k)
+pyplot.hist(w,50, orientation='horizontal',ec='black',color='b');
+pyplot.xlabel(r'$g(\omega)$')
+pyplot.ylabel(r'$\omega$')
+pyplot.yticks([],[])
+pyplot.xticks([],[])
+pyplot.tight_layout();
+```