A general equation for the centroid motion of free, continuous, intense beams propagating off axis in solenoidal periodic focusing fields is derived. The centroid equation is found to be independent of the specific beam distribution and may exhibit unstable solutions. A new Vlasov equilibrium for off-axis beam propagation is also obtained. The properties of the equilibrium and the relevance of centroid motion to beam confinement are discussed. DOI: 10.1103/PhysRevLett.93.244801 PACS numbers: 41.85.Ja, 05.45.-a A fundamental understanding of the kinetic equilibrium and stability properties of certain systems is sought. These systems include periodically focused high-current, low emittance beams. The understanding we seek is crucial for a variety of applications. Among them are advanced particle accelerators and coherent radiation sources. For a long time, the Kapchinskij-Vladimirskij (KV) distribution [1] was the only Vlasov [2] equilibrium distribution known for the propagation of periodically focused intense particle beams. Equilibrium and stability analysis based on the KV beam have been critical to the development and understanding of the physics of intense beams [3][4][5][6][7][8][9][10][11]. More recently, it has been shown that the KV distribution can be generalized to allow for rigid beam rotation with respect to the Larmor frame in periodic solenoidal focusing fields [12]. Studies indicate that rotation may have an important role in particle beam stability [13].In the derivation of these Vlasov equilibria it is always assumed that the beam is perfectly aligned with the symmetry axis of the focusing field [1,6,12]. Actually, this simplifying assumption is generally used in the analysis of intense beams [8] because the axis is an equilibrium for the beam centroid, and the equilibrium is stable if smooth-beam approximations are employed where the periodic fluctuations of the focusing field are averaged out [14]. In some cases, however, we may expect the onset of a parametric resonance involving the centroid motion and the focusing field oscillations. This would destabilize the centroid motion and heavily affect the overall beam dynamics. In such conditions the averaging procedure is no longer valid and a detailed description of the centroid dynamics becomes mandatory.In this Letter, we derive from a kinetic VlasovMaxwell description a general equation for the centroid motion of free, continuous, intense beams propagating off axis in solenoidal periodic focusing fields. It is shown that the centroid obeys a Mathieu-type equation. The equation is independent of the specific beam distribution and becomes unstable whenever the oscillatory frequency of the centroid, which is related to the rms focusing field strength per lattice, is commensurable with the focusing field periodicity itself. In the particular case of a uniform beam density around the beam centroid, we show that there exists a self-consistent Vlasov equilibrium distribution for the beam dynamics. The beam envelope that determines the outer radi...

The role of the centroid dynamics in the transport of periodically focused particle beams is investigated. A Kapchinskij-Vladimirskij equilibrium distribution for an off-axis beam is derived. It is shown that centroid and envelope dynamics are uncoupled and that unstable regions for the centroid dynamics overlap with previously stable regions for the envelope dynamics alone. Multiparticle simulations validate the findings. The effects of a conducting pipe encapsulating the beam are also investigated. It is shown that the charge induced at the pipe may generate chaotic orbits which can be detrimental to the adequate functioning of the transport mechanism.