Electrons on parade?

[Philip Koeck]

Does a beam of electrons become more monochromatic as it propagates?
Can you cool an electron beam noticeably?

I'm considering a very narrow beam of electrons with, for example an initial width in the order of 100 nm, and an energy around 1 keV and with a very small divergence angle (maybe 0.01 mrad). The average distance between neighbouring electrons is in the order of 1000 nm.
The electrons are initially emitted by a thermionic source (at slightly random intervals and with a range of energies).
Let's say that everything surrounding this beam is cooled to 77 K.

Would the electrons give off enough energy (due to acceleration in the field of the neighbours) so that the energy spread of the beam is noticeably reduced after propagating about 10 cm?

Some numbers I think are roughly correct:
A 1 keV electron has a speed of about 20 000 000 m/s
A thermal electron at 77 K should have a rms speed of about 60 000 m/s
The energy spread of electrons emitted by a heated tungsten filament is about 3 eV so the spread of velocities should be about 1 000 000 m/s.

So cooling to 77 K would make a big difference.

 

[mfb]

Radiation from random close encounters in the beam will be negligible in your setup. At 1000 nm the electrostatic potential is just ~1 meV, you need to go below 1 nm to even start having relevant collisions (that's not yet relevant radiation!).

Can you cool an electron beam noticeably?

In a synchrotron over many revolutions, as higher energy electrons will emit more synchrotron radiation. That won't get you to 77 K, however, unless we are very loose with the "beam" definition.

What is the application?

 

[Philip Koeck]

Radiation from random close encounters in the beam will be negligible in your setup. At 1000 nm the electrostatic potential is just ~1 meV, you need to go below 1 nm to even start having relevant collisions (that's not yet relevant radiation!).In a synchrotron over many revolutions, as higher energy electrons will emit more synchrotron radiation. That won't get you to 77 K, however, unless we are very loose with the "beam" definition.

What is the application?

I'm working on an electron gun that produces a low energy electron beam with a high beam current (up to 10 microA). This beam is supposed to function as a phase plate for a transmission electron microscope.

The optics of this gun uses only electrostatic lenses, which suffer a lot from chromatic aberration.

I'm just trying to get a feeling for how the energy spread of the beam develops while the electrons fly the length of the gun (about 30 - 40 cm in total).

If there was some simple way of considerably reducing the energy spread of electrons in a beam that would obviously be very interesting.

 

[Vanadium 50]

I don't see how the energy spread can cool through drifting.

In any event, there are many parameters that influence this, and the actual design matters. I'd recommend looking at a text like Carey, The Physics Of Charged Particle Beams.

 

[Philip Koeck]

I don't see how the energy spread can cool through drifting.

Not due to drifting. It's when an electron oscillates in the field built up by the electrons in front of and behind it that it gives off EM radiation and loses energy.

 

[Vanadium 50]

I don't think this cools it. This sounds like what's called space-charge and it is a source of heating and loss But look it up in Carey.

 

[Vanadium 50]

Think about it this way. Electrons repel. So they are going to want to fill the phase space volume available, not reduce it.

 

[Philip Koeck]

Think about it this way. Electrons repel. So they are going to want to fill the phase space volume available, not reduce it.

It might be missing something or making wrong assumptions but let's discuss a very artificial model system. I think the physics should be correct anyway.

Let's say we've managed to produce a beam that's 1-dimensional so the electrons are really on an perfect line.
As before they are emitted slightly randomly with a range of velocities (same numbers as before).
There's no external force on the electrons in the z-direction (the beam direction), only the forces between the electrons.
Due to oscillations in the field of neighbouring electrons each electron gives off EM-radiation, but this is symmetric in the z-direction on average over time and over all the electrons.
This means that the total momentum of the electrons is conserved.
On the other hand the total kinetic energy has to decrease due to the emitted radiation.

The only way I can make sense of all that is that the velocity variation decreases whereas the average velocity stays the same.

Is something fundamentally wrong with my argument, apart from practical feasibility?

(I am trying to get hold of the book by Carey.)

 

[Vanadium 50]

II pointed you to a reference. The best thing to do is to read it, not to argue that you don't have to.

And what is going on with PF? There are at least four threads going on right now where the OP's position is "I'm not going to show you the calculation, or even do the calculation, but to all the people who can do it: you're wrong!"

 

[Philip Koeck]

II pointed you to a reference. The best thing to do is to read it, not to argue that you don't have to.

And what is going on with PF? There are at least four threads going on right now where the OP's position is "I'm not going to show you the calculation, or even do the calculation, but to all the people who can do it: you're wrong!"

I gave a detailed description of my qualitative thinking in my previous post.
I'm not claiming it's correct, but I believe it's worth discussing.
Isn't that what PF is about?

 

[Vanadium 50]

Actually, it's not detailed. What is your emittance? What is your gradient? How are you focusing, etc. etc. etc.

But the point is that you need to do the calculation to determine if you have enough cooling. Badgering us is not going to make this work, no matter how much you do it.

Read the book. Do the calculation. See what you get.

 

[Philip Koeck]

Actually, it's not detailed. What is your emittance? What is your gradient? How are you focusing, etc. etc. etc.

But the point is that you need to do the calculation to determine if you have enough cooling. Badgering us is not going to make this work, no matter how much you do it.

Read the book. Do the calculation. See what you get.

I thought you said the beam would heat (post 6) and I couldn't make sense of that.
That's why I switched to an ideal situation and a qualitative thought experiment.
From post 6 on my question wasn't whether the cooling is enough.
My new question was whether the beam cools at all or whether it heats.
In post 8 I discuss why I think it should cool rather than heat.

The details I give are the beam current, electron energy and energy spread (post 1). For the idealized situation in post 8 that's all I can specify.

At the moment I really just want to understand whether the energy spread increases (heating) or decreases (cooling).

 

[Vanadium 50]

I think the beam will heat. If the cooling is negative, it's heating.

It's clear you don't believe me. Fair enough. Do the calculation.