1. RF considerations for FHC E. Shaposhnikova 23.01.2014

2. Questions • Input data • Bunch spacing(s) • Bunch length during physics • Bunch intensity • Output data • Optimum RF frequency • Minimum RF voltage • Longitudinal emittance • Double RF system?

3. Input data (Table 1) • Ringparameters: • Circumference: ~99.7 km • Energy loss per turn @50 TeV: U0=4.6 MeV • Transition energy: γt=120 (Frank Z.) • Beam parameters: • Bunch spacing(s): 5 ns, 25 ns • Bunch length during physics: 8 cm • Bunch intensity: 1.x1011

4. Criteria used • Filling of the RF bucket • maximum momentum filling factor of 0.8 (losses) • minimum filling factor is also important for stability • threshold ImZ/n ~ (filling factor)5/2 (for φs = π) • Single bunch longitudinal stability • Threshold for the loss of Landau damping • Longitudinal impedance budget (inductive part ImZ/n) => assumed to be similar to the LHC ImZ/n = 0.1 Ohm (calculated and measured). Factor 2 is taken as a margin (formfactor for particle distribution,…)

5. RF frequency • 5 ns spacing => n x 200 MHz => 200, 400, 800,… MHz with bucket length = 5, 2.5, 1.25 ns • Bucket length in the presence of synchrotron radiation is reduced by Δφ ~ 2(πU0/V)1/2 • Bunch length of 8 cm => τ4σ = 1.07 ns (Table 1) => 200 MHz or 400 MHz

6. 400 MHz RF Loss of Landau damping Filling factor in momentum

7. 400 MHz or 200 MHz? Longitudinal emittance 400 MHz RF Bunch length and stability 200 MHz RF

8. 200 MHz RF Loss of Landau damping Filling factor in momentum

9. Additional considerations:ring size and harmonic number • 400.79 MHz and bunch spacings of 5 ns, 25 ns hLHC = 35640 = 2x5x4x9x9x11 hSPS = 4620x2= 2x5x4x3x7x11 hFHC = 132930 = 2x5x7x9x211 => C=99.4 km • For rings synchronisation: hSPS/hLHC=7/27 => 7 Trev (LHC) hLHC/hFHC = 9x48/(7x211) => Tsync is 211 times longer hSPS/hFHC= 4x3x11/(9x211) => 132 Trev FCC => replace 211 by 216=12x18 (C=101.75 km)?? Or 204? For spacing of 125 ns (FHeC): 211=> 5x(4x11?) => 5% longer ring (103.7 km) or 0.5% shorter for 210=5x2x3x7

10. Summary 400 MHz • The RF system similar to the LHC (frf=400.8 MHz with Vmax=16 MV/beam) is able to provide at 50 TeVbunches with length of 8 cm (rms) for emittanceof 7.0 eVs (2 σ). In this case bucket area is 13.1eVs (qp= 0.77) andImZ/n for the loss of Landau damping is 0.2 Ohm. • With Vmax=16 MV injected emittance (at 1.7 TeV) should be less than 3 eVs. • Due to the synchrotron radiation damping a controlled longitudinal emittance blow-up (by band-limited RF phase noise) will be required not only during the acceleration ramp but also in the coast. 200 MHz • The 200 MHz RF system has certain advantages, however the design of the SC version doesn't yet exist. • In this RF system for short bunch lengths the longitudinal beam stability will be at the limit and with RF voltage of 10 MV can be recovered (same margin as for in the 400 MHz RF) only for bunches longer than 12 cm (rms). • Another possibility to stabilise the beam is to use a double harmonic RF system (the 400 MHz RF system in addition).