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I wonder what the benefits of having a cyclotron at 4 levels are. Why do they split the proton beam into 4 bunches, accelerate them and merge them into one before the next accelerator?

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The most stringent requirements for the Proton Synchrotron Booster (PSB) come from the intensity side. In particular it needs to fill the Proton Synchrotron (PS) with as many particle as possible wasting as little time as possible. This is because as the beam coasts at injection energy in the partially filled PS, it suffers a lot from collective effects, especially space charge, giving some intensity loss.

At the time of the design study, three design options were identified for the PSB: a Rapid Cycling Synchrotron (RCS), two interlaced rings at $600~$MeV (Twin Accelerator Ring Transfer (TART)) or a system of four superposed rings at $800~$MeV [1]. Eventually the choice was made for the latter, I guess following estimates of cost and risk.

Having a machine made up of four (vertically-stacked) rings allows delivering four time more particles per unit time, being de facto four machines which simultaneously fill up the PS. With some clever magnet design, combining together the four rings, quite a lot of equipment cost can be saved while maintaining a very compact machine.

Note that the next PSB upgrade, foreseen by the LHC Injector Upgrade (LIU) and being deployed right now, consists in increasing its max energy. This will in turn allow for an increase of the PS injection energy, mitigating the nasty effects discussed above and, in the end, providing more intensity for the luminosity upgrade of the LHC.

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