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Kronig-Penney model

If one symmetric cell has reflection and transmission amplitudes $r,t$, then
$$
\operatorname{tr}M
=\frac{(t^2-r^2)e^{ika}+e^{-ika}}t.
$$
At a periodic or antiperiodic band edge, with $\operatorname{tr}M=2\sigma$ and $\sigma=\pm1$, setting $z=e^{-ika}$ gives
$$
z^2-2\sigma tz+t^2-r^2=0,
\qquad
z=\sigma t\pm r.
$$


Scattering-amplitude equations for one-dimensional band edges

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 35D / b / Solution

Scattering-amplitude equations for one-dimensional band edges

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Scattering-amplitude equations for one-dimensional band edges

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