Nonlinear δ F simulation studies of high-intensity ion beam propagation in a periodic focusing field

This paper makes use of the nonlinear Vlasov–Poisson equations to describe the propagation of an intense, non-neutral ion beam through a periodic focusing solenoidal field with coupling coefficient ${\kappa }_z{\mathrm{(s+S)=\kappa }}_z\mathrm{(s)}$ in the thin-beam approximation ${\mathrm{(r}}_b\mathrm{\ll S).}$ The nonlinear $\mathrm{\delta F}$ formalism is developed for numerical simulation applications by dividing the total distribution function $F_b$ into a zero-order part ${\mathrm{(F}}_b^0)$ that propagates through the average focusing field ${\mathrm{\kappa ̄}}_z=\mathrm{const},$ plus a perturbation ${\mathrm{(\delta F}}_b)$ which evolves nonlinearly in the zero-order and perturbed field configurations. To illustrate the application of the technique to axisymmetric, matched-beam propagation, nonlinear $\mathrm{\delta F}$ -simulation results are presented for the case where $F_b^0$ corresponds to a thermal equilibrium distribution, and the oscillatory component of the coupling coefficient, ${\mathrm{\delta \kappa }}_z{\mathrm{(s)=\kappa }}_z{\mathrm{(s)-\kappa ̄}}_z,$ turns on adiabatically over many periods S of the focusing lattice. For adiabatic turn-on of ${\mathrm{\delta \kappa }}_z\mathrm{(s)}$ over 20–100 lattice periods, the amplitude of the mismatch oscillation is reduced by more than one order of magnitude compared to the case where the field oscillation is turned on suddenly. Quiescent, matched-beam propagation at high beam intensities is demonstrated over several hundred lattice periods.