Modifications to harmonic current bunching of electron beams from RF cavities due to radial boundary conditions

Harmonic current bunching of an electron beam after it passes through a standing-wave RF cavity is well known, with analytic solutions in both the transversely infinite space-charge regime and the ballistic regime. The Wallander formula is often used between these regimes for finite-sized beams, with pretty good accuracy. These formulas describe simple space-charge modes that are launched by the RF cavity, and assume a single space-charge reduction factor. However, upon close examination these formulas do not provide the complete story. Additional space-charge modes are introduced by the radial boundary conditions, and the actual total harmonic current is modified somewhat from the simple formulas, and includes beating effects between the different modes, even in the small-signal, or linear, regime. This beating leads to mode mixing and eventual damping, which also affects the long-term propagation of longitudinal space-charge waves in accelerator structures.