Fast electrons passing a beam of slow electrons at a right angle

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• Philip Koeck
In summary: The phase shift of the fast electrons is completely deterministic and proportional to the projected electrostatic potential produced by the slow beam.In summary, when fast electrons interact with a beam of slower electrons, the phase shift is predictable and due to the electrostatic forces between the two beams.

Philip Koeck

I'm wondering what happens when fast electrons (100 - 300 keV) interact with a beam (diameter 1 to 10 μm) of slower electrons (1 to 10 keV), which is at right angles to the trajectories of the fast electrons. The beam of slower electrons is relatively dense with 1 to 10 electrons per μm line density.
I'm mainly interested in the phase shift or modulation of the fast electrons depending on how close they pass the beam of slow electrons.
I've simulated this in two rather classical ways and get very similar results, but I'm uncertain whether experimental results would be very different due to QM effects.

The two simulations I did are:

1. The slow beam is treated as a stationary and continuous charge distribution whereas the fast electrons are treated as a plane and monochromatic wave.
This is pure wave optics, completely ignoring any particle nature.
The phase shift of the fast electron wave is of course completely deterministic and proportional to the projected electrostatic potential produced by the slow beam.

2. Every electron is treated as a classical charged particle and the trajectories are calculated using Runge-Kutta. The slow electrons form a beam, whereas the fast electrons pass this beam at various distances.
The electrostatic potential traversed by each fast electron is summed up and the result agrees very well with simulation 1. Very little randomness in the phase shift of the fast electrons is introduced by the particle nature of the slow beam, but some of the slow electrons do get knocked out of their original trajectory quite a bit.

Is there a more quantum mechanical simulation one could do and would one expect very different results?
Could the phase shift of the fast electrons become very unpredictable in reality?

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Philip Koeck said:
I'm wondering what happens when fast electrons (100 - 300 keV) interact with a beam (diameter 1 to 10 μm) of slower electrons (1 to 10 keV), which is at right angles to the trajectories of the fast electrons. The beam of slower electrons is relatively dense with 1 to 10 electrons per μm line density.
I'm mainly interested in the phase shift or modulation of the fast electrons depending on how close they pass the beam of slow electrons.
So the beams are not intersecting? If that's the case, there is no mechanical (electro-mechanical) interaction/scattering, but rather it would be strictly electromagnetic. I would imagine there are electrostatic (two negatively charged regions deflecting each other), then v x B (or I x B) effects, . . . . and I'd have to think of others. I haven't looked at interacting beams in quite a while, and then it was more about colliding beams or beams in plasmas.

When I read fast electrons, I was thinking in the MeV range, since that it the domain of some current work.

berkeman and Philip Koeck
Astronuc said:
So the beams are not intersecting? If that's the case, there is no mechanical (electro-mechanical) interaction/scattering, but rather it would be strictly electromagnetic. I would imagine there are electrostatic (two negatively charged regions deflecting each other), then v x B (or I x B) effects, . . . . and I'd have to think of others. I haven't looked at interacting beams in quite a while, and then it was more about colliding beams or beams in plasmas.

When I read fast electrons, I was thinking in the MeV range, since that it the domain of some current work.
The fast electrons are in a transmission electron microscope. Typical energies are 100 to 300 keV.
I'm trying to find out what would happen if you use a beam of slower electrons (1 to 10 keV, orthogonal to the fast electrons) as an electron optical element, such as a phase plate or a beam splitter for holography.

The fast electrons can intersect the slow electron beam, but since the slow beam only contains 1 to 10 electrons per μm the probability of a head on collision is low (classically speaking).
Of course this thinking doesn't make sense in the first simulation where I treat the fast electrons as a plane wave.

The beam current of the fast electrons is much smaller than that of the slow, so one could say that the fast electrons fly by one at a time (classically speaking).

In both simulations I only consider electrostatic forces and potentials. The Lorentz force should be negligible since the beam current of the slow beam is very small, about 10 μA.