Image-charge effects on the envelope dynamics of an unbunched intense charged-particle beam

The root-mean-squared (rms) envelope equations are derived and analyzed for an unbunched intense charged-particle beam in an alternating-gradient focusing field and a cylindrical conducting pipe. All higher-order image-charge effects from the cylindrical pipe are expressed in terms of so-called multiple moment factors in the rms beam envelope equations, and the multiple moment factors are evaluated. Numerical results show that for vacuum phase advance σ_ν < 90°, the image-charge effects on the matched and slightly mismatched beam envelopes are negligibly small, at all orders, for all beams with arbitrary beam density profiles (including hollow density profiles) as well as for arbitrary small apertures (including beams with large aspect ratios). However, the main unstable region for the envelope evolution with image-charge effects, which occurs for 90° < σ_ν < 270°, depending on the value of the normalized beam intensity SK / ε, is found to be narrower than its counterpart without image-charge effects.