Solutions lecture 7

src/8_many_atoms_sol.md

@@ -56,12 +56,23 @@ The dispersion is given by: $$ E = \epsilon \pm \sqrt{t_1^2 + t_2^2 + 2t_1t_2\co
 
 ```python
 pyplot.figure()
-k = np.linspace(-pi, pi, 300)
-pyplot.plot(k, np.sqrt(5+2*2*np.cos(k)),'b')
-pyplot.plot(k, -np.sqrt(5+2*2*np.cos(k)),'b')
-pyplot.plot(k, np.cos(k/2),'r')
+k = np.linspace(-2*pi, 2*pi, 400)
+t1 = 1;
+t2 = 1.5;
+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.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.xlabel('$ka$'); pyplot.ylabel(r'$E-\epsilon$')
-pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$'])
-pyplot.yticks([-1, 0, 1], ['$E_0-2t$', '$E_0$', '$E_0+2t$']);
+pyplot.xlim([-2*pi,2*pi])
+pyplot.xticks([-2*pi, -pi, 0, pi,2*pi], [r'$-2ka$',r'$-ka$', 0, r'$ka$',r'$2ka$'])
+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.vlines([-pi, pi], -3, 3, linestyles='dashed');
+pyplot.hlines([-np.abs(t1-t2), np.abs(t1-t2)], -2*pi, 2*pi, linestyles='dashed');
+pyplot.fill_between([-3*pi,3*pi], -np.abs(t1-t2), np.abs(t1-t2), color='red',alpha=0.2);
+
+pyplot.legend(loc='lower center');
 ```