Klein tunnelling and the Klein paradox

The Klein paradox is reassessed by considering the properties of a finite square well or barrier in the Dirac equation. It is shown that spontaneous positron emission occurs for a well if the potential is strong enough. The vacuum charge and lifetime of the well are estimated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Klein tunnelling is a property of relativistic wave equations, not necessarily connected with particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, it is expected that as Z increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling.