Single beam crab dynamics

In an electron-positron machine in which the beams cross at an angle, the characteristics of head on collisions can be preserved by tilting the bunches. The crab angle is created by driving head and tail in opposite directions at a location of zero dispersion or by introducing a head tail energy difference at a location of finite dispersion. The authors observe that bunches are tilted at any point in the machine where the dispersion is non zero and the longitudinal phase space ellipse is not erect. A crab angle is therefore generated at the interaction point as long as it is not a symmetry point of the longitudinal focusing and the dispersion is finite. The matrix formalism developed for the analysis of transverse coupling is generalized to the horizontal-longitudinal case. A measure of residual coupling (as distinguished from the unavoidable coupling associated with dispersion) is defined. General expressions for the crabbing angle and residual coupling are given in terms of transfer matrix elements