• C.E. Mitchell, A. Dragt
  UMD, College Park, Maryland

Transfer maps for magnetic elements in storage and damping rings can depend sensitively on nonlinear fringe-field and high-order-multipole effects. The inclusion of these effects requires a detailed and realistic model of the interior and fringe magnetic fields, including their high spatial derivatives. A collection of surface fitting methods has been developed for extracting this information accurately from 3-dimensional magnetic field data on a grid, as provided by various 3-dimensional finite element field codes. The virtue of surface methods is that they exactly satisfy the Maxwell equations and are relatively insensitive to numerical noise in the data. These techniques can be used to compute, in Lie-algebraic form, realistic transfer maps for the proposed ILC Damping Ring wigglers. An exactly-soluble but numerically challenging model field is used to provide a rigorous collection of performance benchmarks.

• G. Bassi
  The University of Liverpool, Liverpool
• J.A. Ellison, K.A. Heinemann
  UNM, Albuquerque, New Mexico

We consider a 2D bunch represented by \mathcal N simulation particles moving on arbitrary planar orbits. The mean field of the bunch is computed from Maxwell's equations in the lab frame with a smoothed charge/current density, using retarded potentials. The particles are tracked in beam frame, thus requiring a transformation of densities from lab to beam frame. We seek improvements in speed and practicality in two directions: (a) choice of integration variables and quadrature rules for the field calculation; and (b) finding smooth densities from scattered data. For item (a) we compare a singularity-free formula with the retarded time as integration variable, which we used previously, with a formula based on Frenet-Serret coordinates. The latter suggests good approximations in different regions of the retardation distance, for instance a multipole expansion which could save both time and storage. For item (b) we compare various ideas from mathematical statistics and numerical analysis, e.g., quasi-random vs. pseudo-random sampling, Fourier vs. kernel smoothing, etc. Implementations in a parallel code with \mathcal N up to a billion will be given, for a chicane bunch compressor.

• S. Franke, W. Ackermann, T. Weiland
  TEMF, TU Darmstadt, Darmstadt

The Vlasov equation describes the evolution of a particle density under the effects of electromagnetic fields. It is derived from the fact that the volume occupied by a given number of particles in the 6D phase space remains constant when only long-range interaction as for example Coulomb forces are relevant and other particle collisions can be neglected. Because this is the case for typical charged particle beams in accelerators, the Vlasov equation can be used to describe their evolution within the whole beam line. This equation is a partial differential equation in 6D and thus it is very expensive to solve it via classical methods. A more efficient approach consists in representing the particle distribution function by a discrete set of characteristic moments. For each moment a time evolution equation can be stated. These ordinary differential equations can then be evaluated efficiently by means of time integration methods if all considered forces and a proper initial condition are known. The beam dynamics simulation tool V-Code implemented at TEMF utilizes this approach. In this paper the numerical model, main features and designated use cases of the V-Code will be presented.

• M.M. Dehler
  PSI, Villigen

For the 250 MeV test injector, it is planned to use a 2.6 cell RF gun originally developed for high current and charge operation in the CLIC test facility CTF-2. First start-to-end simulations assuming perfect field symmetries show, that this gun should be able to generate bunches at 200 pC with an emittance of below 400 nm rad, which would be compatible with the requirements for the SwissFEL. This gun uses double side coupled RF feeds in the last cell as well as tuners in the last two cells, which give transverse multipole effects in the field and phase space distribution and may lead to a deteriorated emittance. Since the beam in the last cell is already relativistic at energies between 4 and 6.4 MeV, this effect can be computed in a clean way by looking at the distributions of the integrated beam voltage at the cavity iris and deriving any transverse kicks via the Panovsky-Wenzel theorem. Doing this approach for the various operation modi planned for the PSI injector shows an emittance dilution well below the critical thresholds.

• B. Erdelyi
  Northern Illinois University, DeKalb, Illinois
• S.L. Manikonda
  ANL, Argonne

Most charged particle beams under realistic conditions have Gaussian density distributions in phase space, or can be easily made so. However, for several practical applications, beams with uniform distributions in physical space are advantageous or even required. Liouville’s theorem and the symplectic nature of beam’s dynamic evolution pose constraints on the feasible transformational properties of the density distribution functions. Differential Algebraic methods offer an elegant way to investigate the underlying freedom involving these beam manipulations. Here, we explore the theory, necessary and sufficient conditions, and practicality of the uniformization of Gaussian beams from a rather generic point of view.

• J.M. Maus, R.A. Jameson, A. Schempp
  IAP, Frankfurt am Main

For the development of high energy and high duty cycle RFQs accurate particle dynamic simulation tools are important for optimizing designs, especially in high current applications. To describe the external fields in RFQs, the Poisson equation has to be solved taking the boundary conditions into account. In PteqHI this is now done by using a finite difference method on a grid. This method will be described and simulation results will be compared to different RFQ particle dynamic codes.