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Understanding Damping Rings and FODO Lattices: Betatron Phase Advance and Twiss Functions

This lecture exercise focuses on the fundamental principles of damping rings and the FODO lattice structure, illustrated through a practical example. It covers key parameters such as betatron phase advance and the Twiss functions (β, α) at the center of the focusing (QF) and defocusing (QD) quadrupoles. Using the thin lens approximation, we derive the matrix representation of a FODO cell and explore the behavior of the various functions as a function of phase advance, particularly at 180º. This analysis is crucial for understanding beam dynamics in particle accelerators.

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Understanding Damping Rings and FODO Lattices: Betatron Phase Advance and Twiss Functions

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  1. Lecture 5 Damping Ring basicsExercise solution

  2. QF QD QF FODO Latticean example  12 FODO cells Qx = 3.15, Qz = 3.15 x FODO cell x= y= 94.5º

  3. Thin Lens FODO Cell In thin lens approximation and representing dipoles as drift spaces the matrix of a FODO cell can be written as Comparing this with the matrix of a periodic structure evaluate the betatron phase advanceand the Twiss functions  and  at the center of QF (optionally QD) Optional: Observe the behavior of  and D as a function of ; what happens at 180º?

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