energy - Could we use particle colliders as fusion generators?

So I know the basic gist is that fusion power's main issue is sustaining the fusion. I also know that there are two methods. The Torus method and the laser method. The torus magnetically contains plasma and heats it with radiation and accelerates the plasma around to make strong enough collisions that protons fuse. The laser method uses 192 lasers and focuses it on tiny frozen hydrogen pellets and aims to initiate fusion each time pellets are dropped.

The though struck me when we could sorta combine the two designs together. The torus doesn't have to worry about making fusion happen at a specific location but it has issues in that the plasma is unevenly heated and leaks. On the other hand, the laser design is extremely complicated in the level of precision needed and would have to repeat this for every pellet. This lead me to think to make something precise and contained at the same time.

I see that particle colliders are able to direct two beams of protons and have them collide at a specific spot with a very precise energy. Couldn't we tune the energy of the two beams of protons to the energy required for them to fuse? We have the ability to smash them into bits, surely we have the ability to have them fuse. (I'm thinking about the type of collider that circles two beams in opposite directions)

It would be at much lower energies than normal colliders and would be very precise and it would be possible to fuse at a specific location that has greater leeway because for protons that missed collision, they'd just circle around again! Thus protons would efficiently be used and very little would be wasted. There wouldn't be problems of plasma leakage because we are focusing them in a thin tight beam.

It seems that this idea has girth, or I feel this way at least, can someone back me up by offering some calculations on how to calculate the efficiency? How would I go about calculating the two circling beams of protons and at what specific velocity would be needed? etc.

Answer

A subtle problem you seem to overlook is that the proton-proton cross section is very small, about 0.07 barns (a barn is $10^{-28}$ square meters) at the LHC energies and not dramatically different at your lower "fusion energies". It means that at the LHC, much like at your dream machine, most of the protons simply don't hit their partners. It is not really possible to focus the proton beams arbitrarily accurately, for various reasons (the uncertainty principle is the truly unavoidable effect: you either localize the beams in the transverse direction, into a "thin pipe", or you specify that the velocity in this transverse direction is zero which is needed if you want to preserve the location "in the thin pipe" in the future, but you can't do both at the same moment). If it were possible, the LHC would be among the first ones that would use the method, to increase the luminosity.

So if you accelerate two beams of protons against each other, an overwhelming majority of them will simply continue in their original motion. (The protons in the LHC have to orbit for half an hour or so – tens of millions of revolutions – before one-half of them collides or disappears.) It costs some energy to accelerate the protons to these energies and you want this energy to be returned from fusion, with some bonus. But the fusion only returns you the energy from the protons that collided (some of them could create helium at your energies but there will always be nonzero probabilities of other final states; it's not a deterministic system that always produces the same final state for a given initial state; quantum mechanics says that the outcomes are random) which is a tiny portion of the protons. So you will be losing most of the energy you invested for the acceleration. Note that the LHC consumes as much energy as the households in Geneva combined and it just produces collisions of protons whose energy is smaller than a joule per pair.

To increase the fraction of the protons that hit their partners, you either need to send them to the collision course repeatedly, like at the LHC, but then you need to pump extra energy to the protons that they lose by the synchrotron radiation (which is always nonzero if the acceleration vector is nonzero, e.g. for all circular paths). Or you will need to dramatically increase the density of the beams.

But if there are many protons in the beam, they will electrically repel each other and you will become unable to focus them for collisions, too. So what you need to do is to electrically neutralize the high-density proton beam and then you have nothing else than the plasma and you face the usual tokamak problems how to stabilize it. Note that the electrons respond totally differently to the external electromagnetic fields than the protons. The LHC uses both electric and magnetic fields to accelerate the protons but to keep the plasma neutral, you must avoid electric fields.

Tokamaks only work with magnetic fields. Whether they will ever become fully working and feasible remains to be seen but the absence of the electric fields implies that they don't have much in common with particle accelerators.