Abstract
We use the linearized Vlasov-Poisson equations to study the response of a Kapchinskij-Vladimirskij beam to magnetic multipole errors in a circular lattice. This work extends the calculation of Gluckstern [Proceedings of the Linac Conference, 1970 (Fermilab, Batavia, IL, 1970), p. 811] to the case of nonideal periodic lattices. The smooth approximation is assumed. We determine the resonance conditions as well as the amplitude of the excited collective modes as a function of the error size outside the stopbands. We find that the frequencies associated with lattice resonances are a subset of the beam natural eigenfrequencies. The result is used to study the motion of test particles crossing the boundary of the beam core. Close to resonance the model predicts the emergence of a halo if sufficiently large gradient errors are present. Application is made to the University of Maryland Electron Ring.
Highlights
In a circular accelerator or storage ring a major limitation to the machine luminosity is caused by the presence of magnetic field errors in the beam optics
If there is no coupling between the motion in the horizontal and vertical planes the resonance conditions in terms of the tune in either plane are given by n0 mn, where m and n are integers
An interesting question is the extent to which the results of our resonance analysis can be applied to an alternate gradient (AG) lattice
Summary
In a circular accelerator or storage ring a major limitation to the machine luminosity is caused by the presence of magnetic field errors in the beam optics. Our goal here is to further clarify our understanding of the collective behavior of a beam by placing the emphasis on deriving the resonance conditions and on determining the dependence of the collective modes on the size of the periodic lattice errors that drive them This will allow us to outline the proper framework to study the motion of test particles and correct some earlier works (see e.g., [10]) that overlooked the space charge perturbation associated with the collective modes. The calculation takes into account the forces due to both the gradient errors and the space charge perturbation excited by those errors We show how these particles under certain conditions can generate a halo surrounding the beam core. III and IV, while in Appendix C we illustrate the mechanism of cancellation between the lattice error and space charge perturbation forces mentioned above using the alternate method of the envelope equations
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