Evaluation of Longitudinal Single-Bunch Stability in the SPS and Bunch Optimisation for AWAKE

We examine the longitudinal single-bunch stability of proton beams in the SPS using Matrix Equations for LOngitudinal beam DYnamics calculations (MELODY). We determine the minimum stable bunch length achievable at a bunch intensity of Np = 4.0 × 1011, and focus on its application to AWAKE. Our calculations use the expected SPS operational limits following LS2. We find that the shortest bunch length obtainable without bunch rotation is τ4σ = 0.99 ns, which requires maximising the RF voltage amplitudes. Bunch rotation can reduce τ4σ significantly: to 0.77 ns without controlled emittance blow-up and to 0.84 ns with blow-up. Optimisation of phase and voltage jumps can improve the symmetry of the bunch shape and can further decrease the bunch length to 0.71 ns and 0.78 ns, respectively. However, this option requires modifications of the 800 MHz Low Level RF system.