My write-up had an inequality for the pressure in the vacuum chamber, suggesting that you can decrease the pressure indefinitely below that given by the mean free path = path length, although qualitatively, you shouldn't do that because it would decrease the availability of hydrogen for generating ions. I needed to figure out the exact pressure - this is usually done experimentally because there are so many parameters at work: volume flow rate of hydrogen, temperature, molar gas constant, ionization efficiency of filament etc. Intuitively, I believed that it should be exactly at equality.
At first, there seemed to be a differential equation at work:
- Increasing the pressure decreases the mean free path and hence increases the collision factor and hence decreases the beam current produced.
- Increasing the pressure increases the number of hydrogen molecules available for ionization.
I spent many non-work hours, say, sitting in the car, contemplating this at the back of my mind. Today, I figured that you just needed to model the collected beam, then maximize the equation for the collected beam. It beats writing a DE.
After 3 hours of rough work, I finally proved it! Integral form of Beer-Lambert law, Lagrange multipliers, and voila, proof. Modelling the current was difficult, solving it was elegant (:
At first, there seemed to be a differential equation at work:
- Increasing the pressure decreases the mean free path and hence increases the collision factor and hence decreases the beam current produced.
- Increasing the pressure increases the number of hydrogen molecules available for ionization.
I spent many non-work hours, say, sitting in the car, contemplating this at the back of my mind. Today, I figured that you just needed to model the collected beam, then maximize the equation for the collected beam. It beats writing a DE.
After 3 hours of rough work, I finally proved it! Integral form of Beer-Lambert law, Lagrange multipliers, and voila, proof. Modelling the current was difficult, solving it was elegant (:
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