I am finding it very hard to understand the implications of the equation obtained for the Kronig Penney Model from Solid State Physics by Kittel. The equation he obtained by using delta potential is
$$\frac{P}{Ka}\sin Ka+\cos Ka=\cos ka$$
where $k$ is a wave vector and
$$K=\sqrt{\frac{2mE}{\hbar^2}}, \quad P=\frac{Q^2ab}{2}, \quad Q=\sqrt \frac{2m(U_0-E)}{\hbar^2} \, .$$
For me, the main problem has been how this equation leads to the origin of bands. Since the values of $\cos (ka)$ lies between -1 and +1, it gives the origin of allowed bands, but what about forbidden gaps or the band gaps.
Best Answer
Consider your equation
$$\frac{P}{Ka}\sin(Ka) + \cos(Ka) = \cos(ka) \, .$$
The right hand side can only ever attain values in the range $[-1,1]$. Therefore, if there is an energy $E$ which causes the left hand side to take a value outside the range $[-1,1]$, then that $E$ can never be realized for any value of $k$. In other words, that $E$ is forbidden and we say it sits in a "band gap".
We can see this graphically. Here I plot the energy $E$ versus the value of the left hand side of the equation for the case
$$a=1, \quad b=10^{-4}, \quad U_0=10^4 \,.$$
The vertical blue line indicates the point at which the left hand side of the equation is $-1$. Any points where we go to the left of that line are forbidden. The red band indicates the range of $E$ over which the left hand side of the equation is outside the range $[-1,1]$. As you can see, there is a band of values of $E$ where no possible $k$ can satisfy the equation. This is a band gap.