Transmission Coefficients and Eigenvalues of a Finite One-Dimensional Crystal

The transmission coefficient for a finite one-dimensional crystal is evaluated and is related to the energy bands of an infinite crystal. It is shown that the transmission coefficient tends to zero in the energy gaps as the length of the crystal increases and that it becomes increasingly oscillatory within the energy bands. The negative eigenvalues are also investigated, and their behavior as the length of the crystal increases is indicated. Several concrete examples are worked out to illustrate the results.