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I have a very specific question which is troubling me. I use a heated disk cathode as an electron emitter. I know that the energy distribution of the electrons emitting from the cathode is

$g(E)=\frac{E}{(kT)^2}e^{-E/kT}$.

The cathode is positioned in the magnetic field so that the cathode normal is parallel to the magnetic field, let's say z-direction. Electrons are thus moving forward in the z-direction and circulate in xy plane. I have a retaining field analyser which is measuring the energy distribution of electrons only in z-direction. I would like to calculate an analytical expression for such distribution but I have some problems so I'm asking for some tips.

Danu
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basketas
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    Is there any kind of accelerating voltage? If there isn't, the velocity distribution of the electrons will change due to the buildup of space charge; and if there is a voltage, it will quickly dominate the velocity distribution... – Floris Jan 12 '16 at 18:07
  • This question and associated answer may be helpful - although it doesn't include the complication of the magnetic field. – Floris Jan 12 '16 at 18:09
  • I don't want to take into account the space charge here. I would just like to derive a distribution in z-direction which is a result of thermionic emission in all directions of the hemisphere. You can look at it as purely mathematical problem. – basketas Jan 12 '16 at 20:27
  • Since the magnetic field in this orientation has no effect on the z component of the velocity and since you are only interested in the energy associated with the z component of velocity, can't you just effectively ignore the magnetic field and say that this problem is equivalent to measuring the energy distribution associated with the z component of electrons being thermally emitted under zero magnetic field conditions? Seems that that shouldn't be too difficult to solve. –  Jan 12 '16 at 21:24
  • Did you look into plasma physics? The movement of charged particles in low density plasmas is well explored. The models should work for you if you remove the second charge component. – CuriousOne Jan 12 '16 at 22:53
  • @Samuel Weir: you're absolutely right. I mentioned magnetic field only to describe my system. I guess the solution is to merge two distributions, thermionic energy distribution and z component of electrons emitted isotropically. But my mathematical capabilities are not strong enough to solve this problem, that's why I'm asking for some tips. – basketas Jan 13 '16 at 10:50
  • @basketas - Unfortunately, my memory of how to transform between different probability distributions is a bit fuzzy. Obviously, if you know g(E) it should be possible to get f(v), the probability distribution for the velocities of the electrons. And once you know f(v), then it should be possible to get h($v_z$), the probability distribution for the z-component of the velocities. I would look up a book on probability distributions. It's probably a fairly basic and straightforward procedure to do the transformations. –  Jan 13 '16 at 18:42

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