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Lecture 5 Damping Ring basics Exercise solution

This lecture exercise explores the fundamentals of Damping Rings through the lens of the FODO cell design, applying the thin lens approximation and representing dipoles as drift spaces. By formulating and comparing the matrix of a FODO cell with that of a periodic structure, we evaluate the betatron phase advance, φ, and the Twiss parameters, β₆ and α₆, at the center of QF (and optionally QD). Observations reveal that at φ = 180°, the FODO cell transitions to unstable behavior with βF approaching infinity and βD approaching zero.

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Lecture 5 Damping Ring basics Exercise solution

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  1. Lecture 5 Damping Ring basicsExercise solution

  2. 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?

  3. Solution or F = D = 0

  4. F (blue) and D (magenta) vs phase advance for FODO cell L=1.5 m For =180º F and D0 The motion is unstable

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