Cathode brightness

The total cathode current (before any symmetry reflections) and the average cathode brightness are put in the information box and the ray output file (unless the printing level is very low).

The 'average cathode brightness' is calculated by the program in the following way (and so its definition is not necessarily that given in textbooks):

The ray that starts from the centre of a cathode segment is judged to be 'successful' if it passes the first test plane (as defined by the user). The currents of all the successful rays are summed to give the cathode_current and the areas of the segments from which they come are summed to give the cathode_area. The average cathode brightness is then defined to be

cathode_brightness = cathode_current/(pi*cathode_area).

Its value is given in units of A/(ster.cm^2).

If kT is non-zero (in CPO3DS) the beam brightness is defined to be

beam_brightness = cathode_current/(pi*cathode_area*kT),

where kT is in units of eV. Its value is given in units of A/(ster.cm^2.V). Note that this definition takes no account of aberrations or distortions of the beam, and the value of the brightness might also depend on the position of the first test plane. In the absence of aberrations or other distortions the beam brightness is a constant. Therefore at a potential V that is very much larger than kT/e, the maximum angle theta of the beam will be very much smaller than 1, and so the solid angle will be pi*theta^2 and the beam brightness will be

beam_brightness = beam_current/{pi*theta^2*beam_area*(V+kT/e)}

Taking the beam brightness to be constant then gives the relationship:

Ji = Jo.{(eV+kT)/kT}*theta^2,

where Jo and Ji are the current densities (that is, current per unit area) at the cathode and at potential V respectively. This is the well-known Langmuir formula for the maximum current density attainable at an image formed in a perfect (that is, aberration-free system).