Simultaneous correction of chromaticity and orbit dispersion in a strong focussing machine

Two quantities play a central role in that part of nonlinear optics which deals with the effects of a finite momentum spread in a particle beam: the orbit position whose derivative with respect to momentum is called 'orbit dispersion' and the betatron wave number whose derivative WRT momentum is the 'chromaticity'. The orbit dispersion varies with the azimuth and is essentially horizontal in a machine with a horizontal symmetry plane; parasitic radial fields induce a vertical component. The chromaticity is a scalar quantity related to the integral of the focusing strength, over one turn of the machine; it is defined for the horizontal and vertical planes. After recalling the general motion equation in the horizontal plane, the orbit dispersion and the chromaticity will be treated. Closed formulae are given for lumped elements and special emphasis is put on dipoles with a high deflection angle. (8 refs).