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The field strength of the dipole magnets and the curvature of the tunnel.TheDemx27 said:What is the main obstacle/bottleneck in terms of upping the eV's?
Protons with more energy would hit the outer wall of the beam pipe as the magnets are not strong enough to keep them on track.
Tunnel:
For LEP, the limiting factor was acceleration versus synchrotron radiation, so the tunnel was built with 8 curves and 8 straight sections for acceleration. That design is not optimal for the LHC, but making the ring "more circular" to reach a higher energy would have needed a lot of time and money, so they decided to take the old geometry.
Field strength:
The dipole magnets are supposed to be strong enough to handle 7 TeV per proton. Unfortunately, those magnets need "training" - if you send too much current through them, they lose their superconductivity, this is called quench. A quench comes with mechanical changes of the precise geometry of the coils, and it is known that those quenches (usually) increase the current the magnets tolerate. That is repeated until the magnets are trained for the current you plan to run through them. For some magnets, training for 6.5 TeV took a long time, it is unclear if the design value can be reached at all (but they will certainly try).