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Chromatic Coupling and Beam-Beam Effects

Chromatic Coupling and Beam-Beam Effects. Yunhai Cai October 26, 2006 SLAC PEP-II MAC meeting. Definition of Coupling Parameters. Given one-turn matrix M, we can decouple it with a symplectic transformation:. where u 1 and u 2 can be parameterized as if no coupling case and w is a

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Chromatic Coupling and Beam-Beam Effects

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  1. Chromatic Coupling and Beam-Beam Effects Yunhai Cai October 26, 2006 SLAC PEP-II MAC meeting

  2. Definition of Coupling Parameters Given one-turn matrix M, we can decouple it with a symplectic transformation: where u1 and u2 can be parameterized as if no coupling case and w is a symplectic matrix: There are ten independent parameters. Bar notes symplectic conjugate. g2=1-det(w).

  3. Covariant Matrix and Eigen Emittances Given coupled lattice parameters and eigen emittance, we can calculate the S-Matrix: where s1 and s2 are defined as the case of no coupling: This formula is derived based on the fact that eigen emittances are Invariance around the ring.

  4. Covariant Matrix and Beam Profile for a Gaussian Beam Useful matrix elements at any given point in the ring: Tiled beam distribution: e1 and e2 are eigen emittances and they are invariance like tunes in the ring. For a typical electron ring e1>>e2. Y is tilt angle, a and b are the beam s’s of major and minor axis respectively.

  5. Luminosity Degradation due to Coupling at the IP w11=0.012 w12=0.003 (m) w21=1.0 (m-1) w22=0.15

  6. Chromatic w11 1.5E-4 at 0.1%

  7. Chromatic w12 4.0E-4 at 0.1%

  8. Chromatic w21 0.1 at 0.1%

  9. Chromatic w22 0.02 at 0.1%

  10. Conclusion • Chromatic coupling may reduce the luminosity at a few percentage level.

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