Luminosity of an Ion Collider

A formula is obtained for the luminosity of a collider at the collision of two beams that differ, generally speaking, by their parameters (asymmetric colliders). The formula is valid for counterpropagating and merging beams with coincident longitudinal axes. Three special cases of the formula are considered: collision of two identical axially symmetric bunched beams, collision of a bunch with a coasting beam, and collision of two coasting beams. Collision of intersecting beams is briefly considered, and the method for the luminosity calculation is formulated. The synchronization problem is considered for collisions of asymmetric beams. A method is presented for optimizing parameters of a cyclic collider by minimizing betatron frequency shifts caused by the effect of the space charge of the beams. Numerical examples of luminosity calculations for several types of asymmetric colliders are given.