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Chapter 4 . Space Charge and Beam Stability

The betatron tunes νz , z = x or y, of transverse oscillations of charged particles in the beam moving with axial velocity c v β = , c being the velocity of light, are mainly determined by the applied focusing forces due to quadrupoles. With finite beam current the tunes are shifted, both by direct space charge and by “image” forces due to induced voltages in the surrounding structure impedances. At relativistic beam energies, the space charge forces are strongly reduced by a factor 2 2 1 β γ − = − (with γ being the particle energy E in units of its rest energy) due to partial compensation of electric and magnetic forces. However, in the Proton Driver at 400 MeV injection energy, γ = 1.426 and the space charge term is then largely dominant.