Transverse phase space in the presence of dispersion

A generalized formalism is used to determine the beam ellipse parameters at some location in a transversely uncoupled transport line from a sufficient number of beam size measurements. The authors allow for contributions to the beam sizes from dispersion which can be either transported from some initial point, or generated along the transport line. They derive expressions for the pure betatron emittance in terms of the effective emittance (calculated from the measurements of the total transverse beam sizes) both in the presence and absence of bending magnets. Necessary and sufficient conditions for determining the pure betatron emittance are given and it is shown to be the absolute minimum of the effective emittance. It is shown that dispersion not only directly enlarges the emittance, but also causes a beta -mismatch.<>