Static potential accelerator question

[artis]

From what I know accelerators that use cavities like LHC for example pass the protons multiple times around in order for the cavities to accelerate them at each pass to a higher energy, since they can't accelerate the protons to an energy high enough with just one pass.
So the protons pass through the same potential gradient but multiple times gaining energy each time if what I get is true. The cavities used for this synchrotron action are RF powered cavities

If this is so then I have an obvious question, why can't a static potetial straight or circular accelerator that uses a static DC potential do the same?
I read that static accelerators are limited by their breakdown voltage , yes sure but can't one just use a certain voltage and then have multiple static cavities/capacitors which are arranged one after another with some distance in between?I guess what I'm asking is can the same potential difference (100kV for example) be repeated again and again and the proton exiting from each previous accelerating gap would have a velocity which would be increased again by each next gap?Or is the reason to this that in RF cavities the proton arrival is timed with the field such that the field is in phase and accelerates them but when they have already entered the cavity while in a static potential cavity there would be a sort of stray field which would first somewhat decrease the speed of the arriving protons before increasing it?

 

[shjacks45]

There was a linear accellerator that had hundreds of RF amplifiers along ~100ft. Old, big vacuum tubes for RF Amps. Not DC but phase change between amplifiers. More for high volume not high energy protons. Circular cyclotron would be easier to maintain vacuum and only need one amplifier. Light is kinda RF; has anyone used lasers to accelerate protons?

 

[bobob]

If this is so then I have an obvious question, why can't a static potetial straight or circular accelerator that uses a static DC potential do the same?
I read that static accelerators are limited by their breakdown voltage , yes sure but can't one just use a certain voltage and then have multiple static cavities/capacitors which are arranged one after another with some distance in between?

Sure, you could do that, but that turns out to be the rf cavities because...

I guess what I'm asking is can the same potential difference (100kV for example) be repeated again and again and the proton exiting from each previous accelerating gap would have a velocity which would be increased again by each next gap?

Consider a beamline made of metal tubing segments separated by some gap. Fix the voltage to be 100kV higher than the previous tube. When a charged particle enters the tube, it drifts through the tube until it sees the electric field at the next gap. The next tube must be 100kV higher in voltage. To keep the electric field large, the gap must be small. Figure out what the voltage of the last tube must be to accelerate a proton to 1 GeV. Neglect any relativistic effets. The force at each gap is $\vec{F} = q\vec{E}$ so that $\vec{a} = (q/m_p) \vec{E}$ and to simplify things further take the electric field to just be the voltage divided by the gap length.

Or is the reason to this that in RF cavities the proton arrival is timed with the field such that the field is in phase and accelerates them but when they have already entered the cavity while in a static potential cavity there would be a sort of stray field which would first somewhat decrease the speed of the arriving protons before increasing it?

Well parts of that are true, but it's not the answer to your question. Imagine two tubes and for simplicity, one is held at ground potential and the other has an AC voltage of 100kV (referenced to ground). The particle enters the first tube and is accelerated by 100kV, i.e., the voltage on the tube is negative with respect to ground. Since the metal tube is free of any electric field, the AC voltage can swing positive during the time the proton is inside the tube bcause the proton sees no electric field there. Now, when it reaches the end of that tube the voltage is positive with respect to the next tube across the gap which is at ground. So, the proton gets kicked into the tube at ground potential, thereby gaining velocity from the accelerations through the 100kV potential difference each time it passes through a tube.

In a real accelerator, the AC voltage is connected across the two tubes so that when the proton reaches the gap, it sees the full peak to peak voltage instead of one-half of that had one of the tubes been connected to ground. That is your rf cavity. The AC voltage is at rf frequencies.

As to your question about phase, the frequency of the rf is critical to ensuring that at each gap the two tubes have the correct polarity. For the proton, the tube it's exiting has to be positive with respect to the tube it's entering or it will be slowed down. It also has to be accelerated enough to meet timing timing criteria when it reaches the next gap. If you think of the full peak to peak voltage as a reference, so that $V = -100 cos(\omega t)$ kV, then $V_{pp}$ occurs when $\omega t = 0, 2\pi,...$. The voltage difference is less than that for different angle. The largest deviation from the full peak to peak voltage that still results in the particle being accelerated is called the phase acceptance, i.e., $V_{gap} >V_{0} cos(\omega t \pm \theta)$ at the time $t$ the proton arrives at the gap.

 

[bobob]

There was a linear accellerator that had hundreds of RF amplifiers along ~100ft. Old, big vacuum tubes for RF Amps. Not DC but phase change between amplifiers. More for high volume not high energy protons. Circular cyclotron would be easier to maintain vacuum and only need one amplifier. Light is kinda RF; has anyone used lasers to accelerate protons?

Not to my knowledge, but it has been done with electrons, but not in the way I understand you are thinking about it. Look up "Wakefield Accelerator." The laser is used to "punch a hole" through a medium (plasma) such that there are a lot of ionized atoms/molecules along the "hole" (or "cavity" to be more accurate), that the electron will pass through. Those ions provide a huge electric field compared with what you can get with an rf cavity, by orders of magnitude. Since the ions are heavy compared with an electron, you can think of the ions as being fixed with the electrons being accelerated toward them. You get something like 1 GeV electrons in a distance of a few cm. This would obviously not (in any way that comes to mind) work for protons or any positively charged particle.