# Tutorial on Linear Colliders by F. Zimmermann

By F. Zimmermann

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Jackson, T. Nicol, “Design of a High Duty Cycle, Asymmetric Emittance RF Photoinjector for Linear Collider Applications,”, Proc. C. (1993). E. E. Carlsten, “Space-Charge Induced Emittance Growth in the Transport of High-Brightness Electron Beams,” IEEE PAC Washington (1987). [96] R. Brinkmann, Ya. Derbenev, K. Fl¨ottmann, “A Flat Beam Electron Source for Linear Colliders,” TESLA 99-09 (1999). [97] R. Brinkmann, Y. Derbenev, K. Fl¨ottmann, “A Low Emittance, Flat-Beam Electron Source for Linear Colliders,” Proc.

3944 (1965). [2] U. Amaldi, “A Possible Scheme to Obtain e+ e− Collisions at Energies of Hundreds of GeV,” Physics Letters 61B no. 3 (1976). [3] R. , “SLAC Linear Collider Design Handbook,” Stanford Linear Accelerator Center (1984). [4] J. L. Bane T. L. Spence, C. Adolphsen, “Observation and Control of Emittance Growth in the SLC Linac,” 14th Int. Conference on High Energy Accelerators, Tsukuba, Japan, Part. Accel. 30, 97 (1990). [5] R. W. Assmann, “Beam dynamics in SLC,” 17th IEEE Particle Accelerator Conference (PAC 97), Vancouver, Canada, and SLAC-PUB-7576 (1997).

2π)3/2 αC Eσδ2 (136) The term Z/n refers to the longitudinal impedance, with Z denoting the impedance at nω0 (ω0 is the angular revolution frequency), and n the number of revolution harmonics. Assuming a constant number and size of strong inductive impedance sources, such as rf cavities, the longitudinal impedance decreases inversely with the bending radius [12], Z/n ∝ 1/ρ. On the other hand, αC decreases more strongly, as αC ∝ 1/ρ2 (since D/ρ ∝ 1/ρ2 ). Recalling the assumed energy scaling E ∝ 1/ρ1/3 , we can stay below the threshold, if the current per bunch decreases as 1/ρ2/3 .