SLAC - PUB - 246 December 1966

The space-charge limit on the beam current which can be usefully stored in a colliding-beam storage ring is determined by the current density in the interaction region and, in the case of a thin, flat ribbon beam, by the value of PY., the vertical P-function there. The quantity which is limited is the dimensionless parameter , JP r 2 6 a constant Y where J is the current density in the interaction region, re the classical electron radius, and y the energy in rest-mass units. In the static "Amman-Ritson" limit, P appears because of its presence in frequency-shift Zrmulae. 1 By the same token, PY appears in coherent-instability limits when the only damping mechanism present is Landau damping by the non-linearity of the force between beams. ance of the form (1) in the incoherent "Courant" limit is on a more speculative basis,3 but it is probably warranted. The luminosity is proportional to the product of the current in one beam and the current density in the other. Evidently, if the stored current is limited, as for example, by the radiofrequency power available, the current density should be increased to the space-charge limit by adjusting the beam dimensions in the interaction region. If, on the other hand, there is no limitation on beam current, the limitation on luminosity will be set by the aperture that can be filled at the interaction region. In the former case, there is a clear gain to be got from decreasing Py in the interaction region, and, in the latter case, there may be a gain, depending on other details of the storage ring. All high-energy storage rings now contemplated will be current-limited at high energies because of the rf power required for synchrotron radiation, so it is advantageous to design the magnet lattice to produce a small value of Py at the interaction region. In the design of cyclic particle accelerators in the past, structures yielding abnormal P-functions, i. e., p-functions differing greatly from R/v have been studiously avoided; primarily because the large values of P, which tend to accompany the small values, aggravate the tolerances on the magnetic guide field, and, in an accelerator no significant advantage accrued from abnormal values of P. That this was not the case in storage rings, where one beam is the target for the other beam, was realized by the French storage-ring group which provided in the design …