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I am trying to get my head around scaling of an element in the world I'm working with, so this question may give me some perspective, that I in no way have.

If we had working room temperature ambient pressure superconductors (a la LK99 type) when the "Large Hadron Collider" was built. This I am certain would decrease the complexity of the device but would it change the overall design significantly? Ie: would the trackway be shorter or just smaller as a whole?

I do not know how much of the design of the LHC is a result of the limitation of conventional electromagnets, because of heat/power consumption, to bend the particle beams and still attain the desired energy levels.

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An acceptable answer does not need to be exact, just detailed enough to convey a sense of scale.

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    $\begingroup$ Given that the LHC makes extensive use of superconductors this question fails the did basic research check. $\endgroup$
    – sphennings
    Jan 6 at 3:34
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    $\begingroup$ @sphennings I specified room temperature superconductors. Sorry if my omission of specifying superconducting when I mentioned conventional electromagnet. Confused you. I assumed it was obvious. $\endgroup$
    – Gillgamesh
    Jan 6 at 4:09
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    $\begingroup$ Note that the LHC was built inside the tunnels of the LEP. The economical reasons are obvious: don't dig a whole new tunnel if you already have one. But that means that even if we suddenly had had science-fictional technology at the time the LHC was built, we would have needed a very strong justification if we wanted to motivate digging a larger tunnel. $\endgroup$
    – Stef
    Jan 6 at 12:02

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  1. The electromagnets of the Large Hadron Collider are superconducting. Yes, they need cooling with liquid helium. Yes, this increases the complexity of the machine and creates points of failure. Were room-temperature superconductors available, the LHC would have been cheaper to build and more reliable in operation. That's all.

    In total, about 10,000 superconducting magnets are installed, with the dipole magnets having a mass of over 27 tonnes. About 96 tonnes of superfluid helium is needed to keep the magnets, made of copper-clad niobium-titanium, at their operating temperature of 1.9 K (−271.25 °C), making the LHC the largest cryogenic facility in the world at liquid helium temperature. LHC uses 470 tonnes of Nb–Ti superconductor. (Wikipedia, s.v. Large Hadron Collider)

  2. The ring is so large because the protons are accelerated at 0.99999999 c, only some 3.1 m/s less than the speed of light. (For fun, the Lorentz factor of a proton moving at maximum speed in the LHC is about 6,930!)

    When a charged particle undergoes an acceleration, it emits electromagnetic radiation and thus loses some energy. The larger the radius of the synchrotron accelerator the smaller the energy losses due to the curvature of the path. Powerful synchrotron accelerators have to be large because the maximum energy of the accelerated particles depends directly on the radius of the path. (The accelerated particles reach maximum energy when the energy losses due to the curvature of the path become equal to the additional energy imparted by the accelerating electromagnets in each revolution around the ring.)

Bonus! The lamented American Superconducting Super Collider which was at one time being built in Texas would have used a ring three times as large, which would have allowed it to reach an energy about 6 times as high as the LHC. Sadly, it was not to be.

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    $\begingroup$ So the limiting factor is the actual deflection of the particles not the power needed to contain them around the curve of the track. Vital missing (one of!) piece of the puzzle. Also Bonus. I remember following the SSC in Popular Science / Mechanics way back. But CERN is pushing for its grandson to be built by 2040. The Future Circular Collider (FCC). Maybe we will get our black holes then. $\endgroup$
    – Gillgamesh
    Jan 5 at 20:20
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    $\begingroup$ "Were room-temperature superconductors available, the LHC would have been cheaper to build and more reliable in operation. That's all.": and note that this requires not only superconductors that operate at room temperature, but ones that can do so under the magnetic fields required. Critical temperature isn't the only parameter of interest. $\endgroup$ Jan 5 at 21:37
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    $\begingroup$ But surely if it had been cheaper to build, they would have built a bigger one, rather than give back part of their budget? I feel like that's what the question is going for. $\endgroup$ Jan 6 at 14:25
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    $\begingroup$ @profane-tmesis.info: Like all megaprojects, the LHC started with a certain budget and then grossly overran it. There was no point in time when somebody was handed down nine billion dollars and told to build the largest particle accelerator he could with that money. There was an initial design and cost estimation, the budget was then approved by CERN and friendly research agencies, construction started, the budget ran out and a new estimation was made, new money was approved, construction continued, the money ran out, a new estimation was made etc. etc. $\endgroup$
    – AlexP
    Jan 6 at 14:50
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    $\begingroup$ @profane-tmesis.info: The diameter was fixed from the very beginning, because they did not dig a new tunnel; they re-used the tunnel which had been used for the Large Electron-Positron Collider (1989–2001). If they had had to add the cost of tunnel digging to the effort then the LHC would have never been built at all. $\endgroup$
    – AlexP
    Jan 6 at 17:24
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As well as the parameters mentioned in AlexP's answer, the critical magnetic field is also a significant limiting factor (and for the LHC it's the limiting factor, in combination with the size of the tunnel), when working at a given radius of accelerator. Basically, for any superconductor, if it is in a large enough magnetic field, it stops being a superconductor, and Bad Things happen. Assuming the energy you can put into the beam is unlimited (not an unreasonable assumption, the synchrotron losses from the LHC beams are only about 7 kW), and your radius is fixed by the size of the tunnel you have, the limiting factor becomes the magnetic field you can apply before your superconductors stop superconducting. The NbTi superconductors used in the LHC can generate fields up to about 10 Tesla, the LHC running at its maximum energy requires a field of 8.3 Tesla, which is quite close to this limit. Type II superconductors (which so far all high temp superconductors are) have much higher critical fields, up to about an order of magnitude higher than type Is like NbTi.

So: you could perhaps get approximately 10 times more energy in your beam if you could perfectly replace the NbTi in the LHC with a type II room temp superconductor, or equivalently reduce the radius by a factor of 10 at the same energy (though the synchrotron losses would increase by a factor of 100 since synchrotron radiation is proportional to $r^{-2}$). This is all only indirectly related to the temperature of the superconductors, it's really about type I vs type II. For a nice discussion of all the technical limits to current particle accelerators which isn't too technical in itself, see this CERN publication.

Also, a note on a confusion you may have: because keeping the beam contained ("deflecting the particles") only requires applying a force perpendicular to the direction of travel, it does not actually require any power to be transferred out of the magnets (since energy is only transferred by forces with a component acting along the direction of motion). You have to put a lot of energy into generating the magnetic fields in the first place, but once they're there, if it's a perfect superconductor, they require no power to maintain.

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