The Energy-Dependent Betatron Phase Advance in the Blowout Regime–Comparison of Two Methods for Estimation

For small excursions of an electron within a continuous focusing ion channel, the phase advance of the socalled betatron oscillations of that electron–equivalent to the number of oscillations per unit length of that electron about the central restoring force of the ion’s space-charge force in the blowout regime–are independent of the (negligible) amplitude of its radial excursion. Here, we compare this one-dimensional (z-dependence only) model of the betatron phase advance with a two-dimensional (x, z) model that includes the finite transverse excursion of the electrons. For experimentally realized sizes of an ensemble of electrons focused into a continuously-focusing ion channel, the 2-D model predicted phase advance (or number of oscillations) is surprisingly close to that of the 1-D model where the radial excursion is ignored.