Emittance growth due to crab crossing imperfections in electron–ion colliders

To achieve luminosities ranging from 10 33 to 10 35 cm - 2 s - 1 , future electron–ion colliders require very small beam sizes, together with a large number of bunches. The strong focusing necessary to achieve β -functions as low as 20 cm can only be achieved by placing low- β quadrupoles as close as possible to the interaction point (IP). With the beam energies of the ion and electron beam differing by more than an order of magnitude, beams need to be separated before encountering the first ion low- β quadrupole, which would otherwise greatly defocus the lower-energy electron beam. This beam separation can be achieved by either deflecting the electron beam away from the ion beam by means of magnetic dipole fields, or by introduction of a crossing angle large enough as to provide sufficient separation at the first ion beam quadrupole. While the former scheme results in large amounts of synchrotron radiation generated near or even inside the detector volume, the latter significantly reduces the luminosity due to the relatively large ion bunch length. This luminosity reduction can be overcome by application of a crab crossing scheme, thus re-instating head-on collisions. However, due to the high ion beam energy, crab cavity RF voltages in the range of tens of megavolts are required, scaling linearly with the RF wavelength. It is therefore desirable to choose the crab cavity RF frequency as high as possible, thus reducing the required voltage. This paper examines the effect of the finite crab cavity RF wavelength on beam dynamics in the proposed electron–ion colliders eRHIC [1], ELIC [2], and LHeC [3].