Accelerator Physics with Relativity

By Mark, Jack and Frances

(Designing the LHC in an hour and a half)

History of L.H.C

• Built in chosen area because there was already a suitable structure there

• Used to be the large electron/positron collider

• Underground circular tunnel with a circumference of 27km

• In order to accelerate the particle an electric field accelerates in a cavity.

• A magnetic field is applied that induces a force equal to the centripetal force required to orbit correctly.

Finding the non-relativistic velocity of a protonThe energy to which protons are accelerated is 7TeV (1.12x10-6 J)

Without taking into account special relativity:

1.12x10-6 = 1/2mv2

Therefore V = 3.66 x 1010 ms-1

But this is greater than the speed of light indicating that special relativity must influence the velocity.

Finding the relativistic velocity of a protonEk = (γ-1)mc2

γ = E / (mc2) +1

γ = 7.452 x 10 3

γ 2 = 1 /(1 – (v2/c2))From this, the velocity is 2.99999997 x 10 8 (3ms-1 less than the speed of light)

Finding the relativistic centripetal force

As the particles are travelling at speeds we need to use the relativistic centripetal force equation.

Using the results for the velocity from the previous slide, this gives us a required field of 5.4T which is achievable using superconducting magnets.

At the LHC, 8 Rf (Radio frequency) cavities are used to accelerate the protons to close to speed to light, each with an average voltage of 2 MV. With each revolution, the particle experiences the equivalent of a 16MVThe charge of the proton is 1.69*1019 and using the equation for kinetic energy gained by a particle in an electric field is ΔE=qV. This makes the energy equal to 2.56*10-12J per revolution.

The proton makes 11000 revolutions per second and gains energy on each revolution. This gives a theoretical acceleration time of 37 seconds. The real time however is approximately 20 minutes due to physical limitations.

Conclusion•The force required in the magnets is equal to the centripetal for to keep the beam on orbit.•Realistic equations are required to correctly model the motion at high energies. This is cantered around the γ parameter. γ=1/√(1-v2/c2)

And finally it is possible to design the LHC in an hour and a half!