accelerator physics - Why is the LHC circular and 27km long?

The LHC in Geneva is a circular accelerator, 27 km long - why is it like that ?

Answer

The LHC is a synchrotron, that is, a accelerator with a magnetic field confining the orbit on a circular path and using RF accelerating cavities to accelerate the particles.

The voltage provided by the cavities is limited (the order of MV) and thus a linear accelerator cannot achieve such high energies (of the order of the TeV) (although some projects of TeV linear collider are in development, CLIC and the ILC) because it would be extremely long. The idea is thus to have a circular path, the particle going through the cavities at each turn and gaining a small amount of energy each turn.

To have this circular path, we use a magnetic field, it does not accelerate the particles, but it provides a force perpendicular to the motion, thus allowing to bend the trajectory and to obtain a circular orbit.

Why does it have to be that long ? A fundamental relation for the synchrotrons is:

p = q B r

where p is the particle momentum, q is the charge of the particle, B is the magnetic field and r is the radius of curvature.

We can then see that to have a high momentum (and energy) we need a high magnetic field and a large radius.

In the LHC, the magnetic field is already at the limit of what a superconducting magnet can achieve (almost 8.5 T).

The LHC then needs a very large tunnel. For that it reuses the tunnel of the LEP which was also a synchrotron, but for electrons.

In that case the size of the tunnel is not really given by the same reasoning. We need to take into account the synchrotron radiation: any accelerated charge radiates energy in the form of a EM radiation: "light".

But the amount of radiation goes with the inverse of the fourth power of the mass, the electron being very light they emit a large amount of radiation. For protons, this effects is almost negligible, that's why it does count for the LHC.

But for LEP (and LEP gave it's tunnel to the LHC) this was the main limitation to the achieved energy. And to obtain a high energy, the larger the tunnel the better, because the amount of radiation decreases with the bending angle of the dipole magnets, meaning that a large radius leads to lower radiation.

Finally, the size of precisely 27 km was chosen for geographic consideration: the tunnel is between the Jura Mountains and the Leman lake, this implies strict constraints in the civil engineering.