Beam profile studies for a phase-mixed ion beam focused with a one-eighth betatron wavelength focusing cell

The focusing properties of a one-eighth betatron wavelength focusing cell are used to determine to what extent a modification of the initial phase-space distribution of an ion beam can alter the number density profile of the beam at the focal plane. It is shown that the main modification is to alter the natural 1/r profile to include an off-axis peak. The relative difficulty with which the ion beam can be concentrated into this off-axis peak is then considered. Estimates of the source brightness (extraction ion diode source current density divided by the square of the microdivergence) required to deliver a given amount of beam current into a given annular region at the focal plane are derived.