1. The Compendium of formulae of kick factor. PLACET - ESA collimation simulation. Adina Toader School of Physics and Astronomy, University of Manchester & Cockcroft Institute, Daresbury Laboratory The University of Manchester

2. Round CollimatorRectangular Collimator Introduction z z Geometric wakefields are those who arise from a change in the vacuum chamber geometry. The geometric wake of a collimator can be reduced by adding a longitudinal taper to the collimator which minimizes the abruptness of the vacuum chamber transition. PLACET is useful tool for simulating rectangular aperture spoilers.

3. Introduction is either small or large compared to1. For a high energy beam passing through a symmetric collimator at a vertical distance y (y << b 1 ) from the axis, the mean centroid kick is given by: where N is the number of particles in the bunch, γ is the relativistic factor, r e is the classical electron radius, y is the bunch displacement and k is the (vertical) kick factor – transverse kick averaged over the length of the beam. Analytical formulas for the kick factor can be found in the limits where the parameter

4. Inductive regime Tenenbaum[2] gives: Zagorodnov[3] gives: Tenenbaum[6] gives for a round collimator of half-gap r and tapered angle α: Round Collimator

5. Stupakov[1] gives: Tenenbaum[2] gives, -for a long, round collimator: -for a short, round collimator: Diffractive regime - analytical formulas exits in the limit of short (L→0) and long (L→∞) collimator Tenenbaum[6] gives for a round collimator of half-gap r and tapered angle α: Round Collimator

6. Rectangular Collimator is either small or large compared to1. Analytical formulas for the kick factor can be found in the limits where the parameter

7. Inductive regime Tenenbaum[2] gives: Zagorodnov[3] gives: Tenenbaum[6] gives for a rectangular collimator of half-gap r and tapered angle α: Rectangular Collimator PLACET

8. Stupakov[1] gives: Zagorodnov[3] gives, -for a long collimator (L→∞): -for a short collimator (L→0): Diffractive regime Tenenbaum[6] gives (r ≡ b 1 ) Rectangular Collimator Tenenbaum[2] gives, for a short, flat collimator on the limit b 1 « b 2 : PLACET

9. Stupakov[1] gives: Tenenbaum[2] gives, Intermediate regime Tenenbaum[6] gives: Rectangular Collimator Zagorodnov[3] gives: with A=1 for a long collimator (L→∞) and A=1/2 for a short collimator (L→0). PLACET

10. ESA Collimators h=38 mm 38 mm L=1000 mm  r=1/2 gap 1 1 2 2 3 3 6 6 α = 324mrad r = 2 mm α = 324mrad r = 1.4 mm α = 324mrad r = 1.4 mm α = 166mrad r = 1.4 mm α = 324mrad r = 2 mm α = 324mrad r = 1.4 mm α = 324mrad r = 1.4 mm α = 166mrad r = 1.4 mm Collimator Side view Beam view

11. Kick Factors for ESA Collimators Bunch size, σ z =0.5 mm Coll Kick Factors (V/pC/mm) PLACET Analytic Prediction * Measured * 1 2.47 2.27 1.4±0.1 (1.0) 2 5.04 4.63 1.4±0.1 (1.3) 3 5.76 5.25 4.4±0.1 (1.5) 5 5.04 4.59 3.7±0.1 (7.9) 6 5.04 4.65 0.9±0.1 (0.9) Coll α(mrad) r (mm) LT (mm) LF(mm) σ(Ω -1 m -1 ) material 1 324 2 50.62 0 5.88e7 OFE Cu 2 324 1.4 52.40 0 5.88e7 OFE Cu 3 324 1.4 52.40 1000 5.88e7 OFE Cu 6 166 1.4 105.5 0 5.88e7 OFE Cu * PAC07 S. Molloy et al.”Measurements of the transverse wakefields due to varying collimator characteristics”