The transverse evolution of the envelope of an intense, unbunched ion beam in a linear transport channel can be modeled for the approximation of linear self-fields by the Kapchinskij-Vladimirskij (KV) envelope equations. Here we employ the KV envelope equations to analyze the linear stability properties of so-called mismatch perturbations about the matched (i.e., periodic) beam envelope in continuous focusing, periodic solenoidal, and periodic quadrupole transport lattices for a coasting beam. The formulation is analyzed and explicit self-consistent KV distributions are derived for an elliptical beam envelope in a periodic solenoidal transport channel. This derivation extends previous work to identify emittance measures and Larmor-frame transformations to allow application of standard form envelope equations to solenoidal focusing channels. Perturbed envelope equations are derived that include driving sources of mismatch excitation resulting from focusing errors, particle loss, and beam emittance growth. These equations are solved analytically for continuous focusing and demonstrate a factor of 2 increase in maximum mismatch excursions resulting from sudden driving perturbations relative to adiabatic driving perturbations. Numerical and analytical studies are carried out to explore properties of normal mode envelope oscillations without driving excitations in periodic solenoidal and quadrupole focusing lattices. Previous work on this topic by Struckmeier and Reiser [Part. Accel. 14, 227 (1984)] is extended and clarified. Regions of parametric instability are mapped, new classes of envelope instabilities are found, parametric sensitivities are explored, general limits and mode invariants are derived, and analytically accessible limits are checked. Important, and previously unexplored, launching conditions are described for pure envelope modes in periodic quadrupole focusing channels.

A nonrelativistic warm-fluid model is employed in the electrostatic approximation to investigate the equilibrium and stability properties of an unbunched, continuously focused intense ion beam. A closed macroscopic model is obtained by truncating the hierarchy of moment equations by the assumption of negligible heat flow. Equations describing self-consistent fluid equilibria are derived and elucidated with examples corresponding to thermal equilibrium, the Kapchinskij–Vladimirskij (KV) equilibrium, and the waterbag equilibrium. Linearized fluid equations are derived that describe the evolution of small-amplitude perturbations about an arbitrary equilibrium. Electrostatic stability properties are analyzed in detail for a cold beam with step-function density profile, and then for axisymmetric flute perturbations with ∂/∂θ=0 and ∂/∂z=0 about a warm-fluid KV beam equilibrium. The radial eigenfunction describing axisymmetric flute perturbations about the KV equilibrium is found to be identical to the eigenfunction derived in a full kinetic treatment. However, in contrast to the kinetic treatment, the warm-fluid model predicts stable oscillations. None of the instabilities that are present in a kinetic description are obtained in the fluid model. A careful comparison of the mode oscillation frequencies associated with the fluid and kinetic models is made in order to delineate which stability features of a KV beam are model-dependent and which may have general applicability.

Self-consistent Vlasov-Poisson simulations of beams with high space-charge intensity often require specification of initial phase-space distributions that reflect properties of a beam that is well adapted to the transport channel-both in terms of low-order rms (envelope) properties as well as the higher-order phasespace structure. Here, we first review broad classes of kinetic distributions commonly in use as initial Vlasov distributions in simulations of unbunched or weakly bunched beams with intense space-charge fields including the following: the Kapchinskij-Vladimirskij (KV) equilibrium, continuous-focusing equilibria with specific detailed examples, and various nonequilibrium distributions, such as the semi-Gaussian distribution and distributions formed from specified functions of linear-field Courant-Snyder invariants. Important practical details necessary to specify these distributions in terms of standard accelerator inputs are presented in a unified format. Building on this presentation, a new class of approximate initial kinetic distributions are constructed using transformations that preserve linear focusing, single-particle Courant-Snyder invariants to map initial continuous-focusing equilibrium distributions to a form more appropriate for noncontinuous focusing channels. Self-consistent particle-in-cell simulations are employed to show that the approximate initial distributions generated in this manner are better adapted to the focusing channels for beams with high space-charge intensity. This improved capability enables simulations that more precisely probe intrinsic stability properties and machine performance.

Linear and nonlinear resonant instabilities of charged-particle beams traveling in periodic quadrupole focusing channels are studied experimentally with a compact non-neutral plasma trap. The present experiments are based on the idea that the collective motion of a beam in an accelerator is physically similar to that of a one-component plasma in a trap. A linear Paul trap system named ''S-POD'' (simulator for particle orbit dynamics) was developed to explore a variety of space-charge-induced phenomena. To emulate lattice-dependent effects, periodic perturbations are applied to quadrupole electrodes, which gives rise to additional resonance stop bands that shift depending on the plasma density. It is confirmed that an mth-order resonance takes place when the corresponding tune of an mth-order collective mode m is close to a half integer.

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