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 Ú 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
... rons where the number of betatron oscillations per machine turn, É, is reduced by ¡É. 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 Ú approaches , thus they become dominant for high-energy synchrotrons. Each particle in the beam has its particular incoherent tune É and incoherent tune shift ¡É. 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.