Equilibrium and Stability of Off-Axis Periodically Focused Particle Beams

Abstract

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 In such conditions the averaging procedure is no longer valid and a detailed description of the centroid dynamics becomes mandatory.In this Letter, we derive from a kinetic VlasovMaxwell description a general equation for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...

Combined centroid-envelope dynamics of intense, magnetically focused charged beams surrounded by conducting walls

“…The present results extend the previous investigation on the coupling of envelope and centroid dynamics in the absence of surrounding walls. 6,7 In these previous works it has been possible to show formally the uncoupled nature of the combined dynamics. Here we make use of analytical estimates as well as Poincaré plots and full simulations to conclude likewise.…”

“…The conclusion is that the coupling of the centroid and envelope dynamics is absent in the present case of beams surrounded by conducting walls, similar to what happens when walls are not present. 6,7 It should be remarked that while the beam preserves axisymmetry with respect to its own center, it cannot exchange energy with the centroid motion, since under this symmetry condition the centroid is unaware of the beam size; Eq. ͑8͒.…”

“…One thing we know for sure though is that when the walls are absent, centroid and envelopes become uncoupled. 6,7 Another key information, given by Gauss' law, is that even in the presence of conducting walls, surface charges induced by circular beams perfectly centered at the symmetry axis do not act in the inner region r Ͻ r w . These two facts suggest that provided the centroid oscillations are moderate, an initially circular beam approximately preserves its circular shape at least during the initial stages of the dynamics.…”

“…6,7 Under these conditions there is no available channel along which the excess energy of a possible centroid motion could be thermalized or converted into emittance, which is ultimately associated with the rms size of the beam envelope. The fact is that one has not the needed nonlinearity to secure the coupling between the envelope and the centroid.…”

Nonlinear coupling between breathing and quadrupole-like oscillations in the transport of mismatched beams in continuous magnetic focusing fields

“…In that regard, many studies have been made since the 1980s on the linear stability of uniform and periodically focused beams, where vanishing small deviations from the matched ͑equilib-rium͒ solution were taken into account. [5][6][7][8][9][10][11] They detected the occurrence of different instability modes which compromise beam transport for certain parameters of the system. Of particular relevance for axisymmetric solenoidal focusing is the breathing mode that induces increasing-amplitude axisymmetric oscillations of the beam envelope around its matched ͑equilibrium͒ value; and the quadrupole-like mode that induces elliptic oscillations of the beam, breaking its symmetry.…”