5. Summary

We have demonstrated the possible existence of zeros in the bremsstrahlung matrix element by calculating the sign of this matrix element in two limiting cases - the soft photon limit, and the fast incident electron limit - and showing it need not be the same. Since the soft photon bremsstrahlung is related to elastic scattering, zeros in the soft photon bremsstrahlung are related to zeros in the elastic scattering matrix elements, in some circumstances observed as Ramsauer-Townsend minima. We have demonstrated that Levinson's theorem can be invoked to identify situations in which zeros in the soft photon radial matrix element, and therefore in the radial matrix element away from the soft photon limit, must exist. Similarly, we have used the extension of Levinson's theorem to the case of ionic species to identify situations where such zeros must exist in ions. We have argued that zeros in free-free matrix elements can be observable if the energy of either (or both) the incident or outgoing electron is sufficiently small.