Re: How Electron Traps Work

Hi Richard,

So, you'd like to build a small electron accelerator and storage ring? That's a pretty big project. Such an accelerator has five important parts: there's an electron source; a vacuum pipe and vacuum pumps; there's a uniform magnetic field---we call it the "bending field"---to steer the electrons around in a circle; there are, optionally, focusing and steering magnets; and there's an electric field source which accelerates the electrons. All of these things are quite complicated. That doesn't mean you cannot build them at home; our own Mad Scientist Fred Niell built cyclotrons at home while in high school, and explains how he did it at http://www.niell.org/.

The betatron, synchrotron, cyclotron, etc., all have these same components; the only difference is in how you *control* the magnetic and electric fields. For a cyclotron or betatron, you keep the bending field constant, which means that the particle's orbit will change as it picks up more and more energy. A cyclotron uses an electric field to accelerate the particle; a synchrocyclotron uses an energy-dependent electric field; a betatron oscillates the magnetic field. A synchrotron can't tolerate changes in the particle orbits, so every time the particles pick up energy, you have to increase the bending magnet fields. In practice, this leads to very different choices for the shapes of the magnets, but the question of "how does this accelerator control its fields" is more fundamental.

I'll try to answer your questions one by one. However, since I don't quite know what your goals are, my answers may be off the mark. Let me know (via MadSci) if you need more-specific help with whatever you're up to.

Is a betatron the best to contain electrons (Compared to a cyclotron, synchrocyclotron, etc)?

That depends what you are trying to do. If you want to make very high energy electrons, above 10-20 MeV each, it is best to use a linac or a synchrotron. Cyclotrons and synchrocyclotrons are not very good for electrons; these accelerators work best when the particle's velocity is low. Since the electron is so low mass, even a (say) 100 keV electron is already moving at 1/2 the speed of light. If you want lots of few-MeV electrons, the betatron is best. If you don't care about high energies, but simply want a strong electron beam, just use an electrostatic linear accelerator.

My second is that how strong is the electric and magnets in the betatron/Synchrotrons?

From looking at pictures of betatron magnets, I would guess that tend to be around 0.1-0.2 Tesla. Betatrons do not have separate electric fields at all. High-energy synchrotrons often use very strong magnets, up to 7 or 8 Tesla, and the strongest electric fields possible in a vacuum tank---several megavolts per meter. There is a simple relation between the magnetic field, a particle's momentum, and the radius of the particle orbit. If you write the bending field B in Tesla, the radius R in meters, and the particle's momentum p in GeV/c, and your particles will obey the equation

p = 0.3 B R

(The momentum p, according to relativity, is beta * gamma * m, or (m*v)/(c*sqrt(1 - v^2/c^2)), where the mass m is in GeV/c^2.)

My third is what is the highest capacity of electrons in the betatron (not MeV but number)?

Accelerators usually cite how much electric current they carry. The highest numbers I have heard for a synchrotron is about 2 amperes; betatrons may go about that high. An ampere of current is 6x10^19 electrons per second, so you can figure out how *many* electrons are being stored if you know their velocity and the size of the ring.

My fourth is does the speed of the electrons moving in a circle make a difference?

Yes! An electron's energy and momentum depend on its speed; the momentum, in particular, tells you how strong a magnet you need to keep the electron going around your accelerator. If you're trying to *do* something with the electrons, like crash them into a metal and make x-rays, they *each* need enough energy to do their intended task.

My fifth question is that if you have a betatron/Synchrotrons that has say 10 electrons in it, what does it mean that it has x MeV (In relation to amount of electrons in the system).

No. Electron volts (eV) and mega-electron-volts (MeV) are a measurement of how much kinetic energy each electron has. An electron that crosses a 9 volt potential---say, one that jumps across the terminals of a 9-volt battery in vacuum---will pick up 9 eV of kinetic energy. An electron that crosses a 5-megavolt potential, in a large Van de Graff machine, will pick up 5 MeV. If you accelerate 100 electrons in the same Van de Graff, they will pick up 5 MeV *each*.

My sixth question is that how long can I contain the electrons before I direct them to the target?

That depends entirely on how well-built your machine is. Unless you have been very careful with the steering & focusing systems, the electrons will eventually crash into the walls of the beam pipe. If you don't have a strong enough bending magnet, they will crash into the walls when their energy gets too high. If you don't have a good enough vacuum system, they will crash into gas molecules. The best electron accelerators---the multi-million-dollar storage rings used by X-ray laboratories---can store their beams for at least a day.

My last question is what is the energy lost by radiative energy loss, called synchrotron radiation (loss per time and electrons in system).

Synchrotron radiation is complicated; you should consult a physics textbook before trying to figure anything out in detail. There is a good calculation on the web here.

I'm sorry that I can't give you more specific help, because I don't know what the goal of your project is. If you simply want "lots of electrons", and you really don't care about their energy, forget about accelerators---just use a piece of wire! Your 1500 watt toaster or hair dryer probably has 12 amperes, or nearly 10^20 electrons per second, moving through it. Arc welding uses even higher currents---actually making the electrons jump through the air, so you can think of a welder's arc as a low-energy (a few eV per electron), high-current (a great many electrons per second) linear accelerator. Many x-ray generators are, indeed, simply high-voltage sparks crashing into a piece of metal.

If you are going to experiment with this sort of thing, please be careful. Even a very small accelerator is likely to need a beefy power supply---which can stop your heart faster than you can figure out what's going on.

Cheers,

-Ben