The Kronig-Penny model of a solid crystal contains an
infinite array of repulsive delta functions.
This potential has a new symmetry,
that a translation by the lattice spacing leaves the problem unchanged.
The probability distributions must therefore have this symmetry
The general solution in the region
By matching the boundary conditions and requiring that the probability be periodic, we
derive a constraint on
similar to the quantized energies for bound states.
The graph below shows as a function of . If this is not between -1 and 1, there is no solution, that value of and the corresponding energy are not allowed.
This energy band phenomenon is found in solids. Solids with partially filled bands are conductors. Solids with filled bands are insulators. Semiconductors have a small number of charge carriers (or holes) in a band.
Jim Branson 2013-04-22