RMS Envelope Equations with Space Charge

Envelope equations for a continuous beam with uniform charge density and elliptical cross-section were first derived by Kapchinsky and Vladimirsky (K-V). In fact, the K-V equations are not restricted to uniformly charged beams, but are equally valid for any charge distribution with elliptical symmetry, provided the beam boundary and emittance are defined by rms (root-meansquare) values. This results because (i) the second moments of any particle distribution depend only on the linear part of the force (determined by least squares method), while (ii) this linear part of the force in turn depends only on the second moments of the distribution. This is also true in practice for three-dimensional bunched beams with ellipsoidal symmetry, and allows the formulation of envelope equations that include the effect of space charge on bunch length and energy spread. The utility of this rms approach was first demonstrated by Lapostolle for stationary distributions. Subsequently, Gluckstern proved that the rms version of the K-V equations remain valid for all continuous beams with axial symmetry. In this report these results are extended to continuous beams with elliptical symmetry as well as to bunched beams with ellipsoidal form, and also to one-dimensional motion.