problem 1: Tides of solid ground and beam energy stability

a) Tidal forces (mostly gravity of moon and sun) create tides in the ocean but also deform the ‘solid’ earth. Establish a formula relating the beam energy change (assuming ultra-relativistic beam, i.e. ΔE/E = Δp/p) to a local change in the ground elevation (i.e. local change in earth radius).

b) find out by how much the beam energy of the ALS shifts assuming an average tidal elevation change of solid ground of about 25 cm (it really is that large!), an ALS beam energy of 1.9 GeV, an ALS circumference of 196.8 m and an ALS momentum compaction factor of 0.0008.

c) Compare this to the largest collider, LEP (predecessor of LHC) at CERN in Geneva, Switzerland. Its circumference is 27 km and the momentum compaction factor was roughly 0.0001. What is the energy change due to tides in that case? Does it scale with the circumference? Why/Why not?

d) Find the precision with which the Z₀ mass and width was measured at LEP and compare to the effect of tides in c above. Conclusions?
 
problem 2: In designing a single turn, single kicker injection system, we have the choice of locating our kicker at two different positions, let us call those locations A and B. For location A, the betatron phase advance µ from the septum position is the usually optimal π/2 and the horizontal beta function at the septum is 4m. For B, µ is (5* π)/8, and βs is 8m. The beta function at the kicker βk is the same as the one at the septum in both cases.
 
a) Where would you place the kicker in order to minimize the necessary kick angle θ and therefore the required kicker magnet strength – location A or B? Why?

b) If the required beam stay clear at the septum imposes a distance between the injected beam trajectory and the closed orbit of x_S=2 cm, what is the required kicker strength θ to completely remove any remaining angle after passing the kicker (i.e. to put the injected beam onto the closed orbit)?

Assume the beta function at the septum and the kicker is 5 m and the phase advance is π/2 for this case.

c) Based on the dependence of the size and divergence of the beam on the lattice functions:

σ = √βε  with β the beta function and ε the (invariant) emittance

σ' = √γε  with γ = (1+α²)/β, α' = −β/2
 
describe qualitatively why placing an injection kicker at a location with high beta function (and correspondingly small alpha function) reduces the required kicker strength.

d) Similarly, give your qualitative explanation, why placing a septum magnet with a septum sheet of a given, finite thickness at a location with high beta function is advantageous.