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Anderson, O. A.

Paper Title Page
THPAS003 Exact Analytic Solution of the Envelope Equations for a Matched Quadrupole-Focused Beam in the Low Space Charge Limit 3513
 
  • O. A. Anderson
    LBNL, Berkeley, California
  • L. L. LoDestro
    LLNL, Livermore, California
  
  Funding: Supported by U. S. Depatment of Energy under contract number DE-AC02-05CH11231

The Kapchinskij-Vladimirskij equations describe the evolution of the beam envelopes in a periodic system of quadrupole focusing cells and are widely used to help predict the performance of such systems. Being nonlinear, they are usually solved by numerical integration. There have been numerous papers describing approximate solutions with varying degrees of accuracy. We have found an exact solution for a matched beam in the limit of zero space charge. The model is FODO with a full occupancy, piecewise-constant focusing function. Our explicit result for the envelope a(z) is exact for phase advances up to 180 degrees and all other values except multiples of 180 degrees. The peak envelope size is minimized at 90 degrees. The higher stable bands require larger, very accurate, field strengths while producing significantly larger envelope excursions.