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Since the 1960's, many e+e~ colliders have been built. Experience in the operation of high energy colliders has led to an understanding of beam dynamics problems such as beam-beam interactions, nonlinear resonances, collective (coherent) beam instability, wakefield and impedance, intrabeam scattering, etc. Some e+e~ colliders now in operation are CESR at Cornell, SLC and PEP at SLAC, PETRA and DORIS at DESY, VEPP's at Novosibirsk, TRISTAN at KEK, and LEP at CERN. The drive to reach higher energy provided the incentive for the high power klystron.

Rev. 32, 492 (1911); I. Langmuir, Phys. Rev. 32, 450 (1913). T. Forrester, Large Ion Beams (Wiley, New York, 1988). EXERCISE 33 1 position vector as R = rr + ss. Let VQ(S) be the electric potential on the axis of symmetry. Show that the electric potential V(r, s) and the electric field E = Err + Ess are y(4) VW V(r,s)=V0(s)-^rr*-^rri , , 1/(3) + ---, T/(5) where VJj correspond to nth-derivative of Vb with respect to s. The equation of motion for a non-relativistic particle in the electric field is mR = eE, where the overdot represents the time derivative.

Single-cell cavities routinely reach 30 MV/m and beyond. O. E. Edlefsen, Science, 72, 376 (1930). g. M. McMillan, Early Days in the Lawrence Laboratory (1931-1940), in New directions in physics, eds. N. M. , New York, 1987). S. Livingston, Particle Accelerators: A Brief History, (Harvard, 1969). 12 CHAPTER 1. INTRODUCTION reproduced below: One evening early in 1929 as I was glancing over current periodicals in the University library, I came across an article in a German electrical engineering journal by Wideroe on the multiple acceleration of positive ions.