How to draw different zone schemes in the Kronig-Penney model? Kronig-Penney model does not give us an equation of how the electron energy $E$ depends on its wavenumber $k$. Then how do they draw the $E-k$ curves in different zone schemes such as extended, periodic and reduced? If one can explain the method of drawing the extended zone scheme that will solve for my purpose.
 A: $$\cos(ka) = \cos(\alpha (a-b))\cos(\beta b) – \frac{\alpha ^2 + \beta^2}{2\alpha \beta}\sin(\alpha (a-b))\sin(βb) $$
$\alpha ^2 = \frac{2mE}{\hbar^2}$ and $\beta^2 = \frac{2m(E + V_0 )}{\hbar^2}$
When the electron energy increases, values of $\alpha$ e $\beta$ come close. When $\alpha \approx \beta$ the expression:
$\frac{\alpha ^2 + \beta^2}{2\alpha \beta} \approx 1$. And the formula tends to: $$\cos(ka) = \cos(\alpha (a-b))\cos(\beta b)–\sin(\alpha (a-b))\sin(\beta b)$$
But the right side of that equation is $\cos(\alpha(a-b)+\beta b)$. 
One simple solution is: $ka = \alpha (a-b) + \beta b$. When $\alpha \approx \beta$, $k \approx \alpha$, and how energy is proportional to $\alpha^2$, graphics $E$ x $k$ is approximately a parabola. 
But according to Bloch theorem, $k$ is restricted to the Brillouin zone. This can be achieved by $$k = \frac{\cos^{-1}[\cos(\alpha (a - b)+\beta b)] }{a}$$ No matter how energy grows, the values of $k$ oscillates in the interval $\left[- \frac{\pi}{a+b} , \frac{\pi}{a+b}\right]$, what is the Brillouin zone of that periodic frame.
