Stochastic Coulomb interactions limitation to brightness of electron sources

It is well known that stochastic Coulomb interactions cause a displacement of electron trajectories in a beam, resulting in an increase of the size of a focused spot. The magnitude of the effect can be calculated by Monte-Carlo simulation, or approximated by various equations. Usually these equations can only be applied in field free sections of an instrument. For an electron source, we would project the displaced beam trajectories back to the virtual source plane and add the broadening to the size of the virtual source. In that way, the reduction of the source brightness can be calculated (for the application of the brightness concept for very small sources see). However, the electrons in the source are accelerated in a non-uniform field. An estimate of the Coulomb effect can still be obtained by cutting the beam in thin slices and adding the effects in the individual slices. For Schottky sources, it has been shown that stochastic interactions indeed have an effect on the maximum obtainable brightness. (To perform a correct calculation,) one needs to know both the total current in the beam and the shape of the beam envelope, since these determine the current density in the slices. These two parameters both depend on the gun configuration: the distance from cathode to anode, the shape of the cathode, the temperature and workfunction of the cathode and the anode extraction voltage. We have recently combined three simulation programs that used to be separate: the calculation of the electric field and electron trajectories in an electron gun, the quantum mechanical calculation of the emission current density from the cathode for the calculated field, and the calculation of the Coulomb effects. So now if we change the extractor voltage, we automatically get a new current and a new beam envelope, which makes the calculation of the Coulomb effects more realistic.