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Additional resources for Particle Beam Dynamics and Phase Space [fundamentals]
5 Matching of a beam to a synchrotron in longitudinal phase space, to avoid longitudinal 42 emittance the by blowup. as Bibliography - • 1. S. Livingston, "The Development of High Energy Accelerators," Dover (1966). 2. C. Teng, "Accelerator Projects, World," in Physics of Particle Accelerators, AIP Conf. Proc. 184, p. 2116 (1989). 3. A. J. Syphers, "An Introduction to the Physics of Particle Accelerators, " Ibid. p 2. M. Sands, "The Physics of Electron Storage Rings _ An Introduction," SLAC-121 (1970).
The boundary is the separatrix. _ee ¢' Fig. 3. _Exercise ] Derive Eq. 28) from Eq. 27). +2 _ +21 ,," /_ ,, - -"- - - t - "";'-1- _ t.. ," / ,, , . '" _a° o ,-, o,, / -- ____ \_. , _-_ o-_J-_.. _, ,_-'"¢",,_ ,.. I -2 ! i " , _ ', "4.. ,. "1-"I I I MO_I_ BICKEr Fig. 3 Longitudinal phase space with sketches ries, fixed points, and separatrix. 38 of trajecto- For a particle inside the stable region, the limits of oscillation in W are given by I_ = 0, for which Eq. 10) gives two solutions, the stable fixed point ¢ = ¢0 and ¢ = zr- ¢0 -- ¢_, the second solution .
Exercise] We saw, using the smooth approximation, that cr __ u -2. Use this and the above relation between c_ and 7tr to show that 7tr _- v. This justifies the statement above that 7t_ is roughly equal to the horizontal betatron tune in a synchrotron. 35 As the beam approaches transition, r/ approaches zero, which means that the particles are locked into a particular phase relative to the rf wave. As a particular consequence, close to transition the particles constituting the bunch cannot follow as the rf bucket shape changes rapidly.