Possible Beam Parameters in Double RF Operation of the CERN LHC

The LHC operates using a 400 MHz SC RF system. A 200 MHz NC RF system was foreseen in the LHC Design Report to improve beam capture and the bare resonators were manufactured, but never installed. Later the second harmonic RF system was proposed to cure longitudinal beam instabilities in the absence of a dedicated wideband feedback system in the LHC. For nominal intensities the longitudinal beam stability is ensured by controlled emittance blow-up during the acceleration ramp. Recently slow growing instabilities were observed at the end of long fills at 6.5 TeV as bunches shrink due to synchrotron radiation damping. For High Luminosity LHC twice higher intensities should be kept stable with new equipment installed in the ring. Additional motivations for a second RF system in the LHC have also been considered. Operation with an extra RF system is limited by the required RF configuration (phase between the two RF systems) and longitudinal beam stability. In this work requirements for the double RF systems are analyzed together with a possible range of longitudinal beam parameters