Circular accelerators vs Linear Accelerators

[barnflakes]

Can somebody show me the physics of why it's better to have a circular accelerator than a linear accelerator in terms of size? I know that a linear accelerator needs to be a lot bigger than a circular accelerator to produce the same energy, but I can't think of how to show this using physics equations. If anybody could show me I'd be very grateful.

ps. this is for a presentation I'm writing for A-level students, not for homework.

 

[vanesch]

In a linear accelerator, the particles see the accelerating field only once, and the particles that didn't interact are lost. As such, a linear accelerator will have much lower overall luminosity (each time, you need to produce new particles to be accelerated) and will generally not be able to reach the same energies.

In a circular accelerator, the particles see the accelerating field (in the cavities) each time they turn around, so they can be accelerated step by step. Also, the particles remain in the accelerator for many hours, so you only "use up" those that are genuinly interacting (most don't).

However, for electrons, a circular accelerator causes bremsstrahlung, which makes the beams loose energy again, so from a certain amount of acceleration onward, you will loose as much as you win by the accelerating field. A linear accelerator doesn't have that problem. And for protons, you need very strong magnetic fields to curve the path of the protons, requiring a huge cryogenic system and all that.

 

[barnflakes]

So in theory, a truly massive linear accelerator is better? How big would a linear accelerator need to be to produce the same amount of energy that you would produce in say, the LHC, of 27km radius? I want to include some relevant mathematics, so is there a couple of simply formulas I could show them to give them an idea? (I'm thinking along the lines of F = ma here).

 

[vanesch]

So in theory, a truly massive linear accelerator is better? How big would a linear accelerator need to be to produce the same amount of energy that you would produce in say, the LHC, of 27km radius? I want to include some relevant mathematics, so is there a couple of simply formulas I could show them to give them an idea? (I'm thinking along the lines of F = ma here).

Well, in as much as SLAC is a reference, where you have 50 GeV for 3 km, given that the LHC goes up to 7 TeV, that's more than 100 times more energy, so that would be something of 300 km.

Very rough estimation.

 

[Rajini]

Hi,
Usually building such facilities are complicated..It depends on the particle to be accelerated..for e.g., if you consider electrons...Initially they are accelerated in linac (linear acc.) and then sent to circular storage rings..That means after sending the electrons (may be up to 10 GeV) into circular rings..these electrons stays/circulates/accelerated in these storage rings for few hours (say ~12 hrs)..after 12 h electrons are again injected!
PS: Storage ring is under ultra high vacuum and electrons are usually accelerated (0.9 of c) using magnets/wigglers/undulators etc..(mainly for producing high intense/brilliance X-ray).

gluck
Rajini

 

[Vanadium 50]

So in theory, a truly massive linear accelerator is better?

"Better" in what way? For that matter, what do you mean by "theory"? These are engineering design choices. Which is better, a bus, a truck or a sports car? It depends on what you want to do with them, no?

I'd like to make two more comments - circular accelerators don't have to be superconducting. In fact, since the strongest magnets are conventional, there are even advantages to doing that. This disadvantage is operational costs: removing the heat generated by the current from the magnets is not cheap - nor is powering them in the first place.

The other one is that rarely in circular proton accelerators do you use up the beam in collisions. More typically the beam size increases (and the density decreases) to the point where one might as well start over. The beam spot is about the size of a human hair, and the beam travels billions of miles during a fill. Any small deviation from the reference orbit becomes important with those scales.