rms Envelope Equations in the Presence of Space Charge and Dispersion

The matching of charged particle beams into transport lines is usually done by using the rms envelope equations. The usual rms envelope equations, however, do not apply in the presence of bending magnets and a longitudinal momentum spread. A new set of equations is needed that simultaneously describes the rms envelopes of the beam and the dispersion function. A derivation is outlined in this paper. The new equations will make it possible to achieve proper matching of the rms envelopes and dispersion in the regime of highly space charge dominated beams.