Fully relativistic treatment of electron-optical image formation based on the Dirac equation

The theory of image formation for an electron microscope is based on the non-relativistic Schrödinger equation, whereas present-day electron microscopes operate with acceleration voltages of the order of one hundred to several hundreds of kilovolts, in which case relativistic effects become important. We present a fully relativistic theory of image formation, based on the appropriate Dirac equation. It is shown that, within certain approximations, always valid for today's electron microscopes, a very simple expression for the current density can still be derived in terms of wave functions that are solutions to the relativistically covariant Klein Gordon equation. The following paper presents the analysis of the often stated possibility to obtain the relativistically correct current density from the non-relativistic current density just by replacing the values of the non-relativistic momentum by the correct relativistic expression