Abstract
The Kronig-Penney lattice is a one-dimensional chain of equal segments at the end of which there is a 
function interaction. It predicts for the energy a band structure whose width increases with the energy. These results continue to hold when the functions are replaced by arbitrary potentials, as well as when the problem is generalized to three dimensions giving the well known conduction bands for electrons. In this paper we are interested in what happens to neutrons in crystals. The simplest model would again be a Kronig-Penney lattice, but with the function replaced by boundary conditions at the end of the segments. This approach leads to an R matrix interaction of the type Wigner introduced in his analysis of nuclear reactions. Using Bloch's theorem we solve the problem of the band structure for arbitrary R, but discuss its behaviour only when it has a single pole, a pole and a zero or a picket-fence form. An example with data taken from experiment is presented in the appendix.