One common way to form an electron beam is by a thermionic cathode (i.e. the process of thermionic emission), which works by heating a metal so that electrons with randomized energies tend to overcome the work function of the metal and boil off.
The emission current as a function of cathode temperature is provided by the Richardson-Dushman equation, though at higher temperatures space-charge effects dominate and limit the current to a threshold defined by Child’s law (i.e. space-charge rather than temperature limited case). In Schottky emission, a field in front of the cathode lowers the potential barrier, thereby increasing emission; it is described by the Schottky equation, which reduces to the Richardson-Dushman equation at low fields. At higher fields, extended Schottky emission occurs, involving tunneling effects (like in Field Emission). . In cold Field Emission, unlike thermionic emission, electrons are tunneled from the surface with a very high field and no elevated temperature (Fowler-Nordheim effect).
There is a distribution of energies, normal and transverse, of electrode emitted from the cathode, baesd on the tail of the Fermi-Dirac distribution exceeding the work function, which for thermionic emission is roughly centered at E = k T, for Boltzmann constant k and cathode temperature T.  . (See also chromatic aberration.)
NoteThis page is abridged from the full SIMION 8.1.1 "Supplemental Documentation" (Help file). The following additional sections can be found in the full version of this page accessible via the "Help > Supplemental Documentation" menu in SIMION 8.1.1:
- Field Emission
- Thermionic Emission, virginia.edu, Surface Science course.
- Cathode emissions in CPO software for other cathode handling capabilities
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