I was trying to use the paschen's law to calculate the breakdown voltage for our vacuum chamber. Our pressure is 10^-8 torr, and distance was 3mm. However, I cannot do it because Paschen only worked at higher pressures (>several Torr).
Any other way that I can calculate the breakdown voltage in low pressure?
When you get near perfect vacuum, the surface condition of your electrodes becomes the dominant factor in determining the breakdown. Any asperities on the cathode will become field emitters. If there is any gas left, it will be ionized by these field emissions. The ions will accelerate and hit the cathode, causing local heating, greater emission, and runaway. This can result in a "spit" - where the HV tank will discharge across the gap. Depending on the capacitance of your HV supply, this may result in significant surface damage - or it may result in erosion of the asperity, and thus a lowering of the field emission. The latter process is called "seasoning".
As you raise the voltage (before breakdown) you should be able to measure the increase in field emission current; the slope of that curve (Plotted as a Fowler-Nordheim plot) can be used to estimate the $\beta$ value which is an indication of the surface smoothness. As you condition the surface, $\beta$ will decrease.
I'm afraid that in this regime there are no simple analytical solutions...
