Space charge

The Coulomb forces between the charged particles of a high-intensity beam in an accelerator create a self-field which acts on the particles inside the beam like a distributed lens, defocusing in both transverse planes. A beam moving with speed 􀀀 is accompanied by a magnetic field which partially cancels the electrostatic defocusing effect, with complete cancellation at , the speed of light. The effect of this ‘direct space charge’ is evaluated for transport lines and synchrotrons where the number of betatron oscillations per machine turn, , is reduced by 􀀀 . In a real accelerator, the beam is also influenced by the environment (beam pipe, magnets, etc.) which generates ‘indirect’ space charge effects. For a smooth and perfectly conducting wall, they can easily be evaluated by introducing image charges and currents. These ‘image effects’ do not cancel when 􀀀 approaches , thus they become dominant for high-energy synchrotrons. Each particle in the beam has its particular incoherent tune and incoherent tune shift 􀀀 . If the beam moves as a whole, so the centre of mass executes a coherent betatron oscillation, image charges and currents caused by the beam pipe move as well, leading to coherent tune shifts which also depend on the beam intensity. For a realistic beam, the incoherent tune of a given particle depends on its betatron amplitude and position in the bunch, leading to a tune spread (rather than a tune shift) which occupies a large area in the tune diagram of low-energy machines. The ‘space-charge limit’ of a synchrotron may be overcome by increasing its injection energy; various systems which have actually been built are presented.