Post on 03-Jan-2016
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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
Th
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Round Collimator Rectangular 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.
Introduction
ykNr
y e
'
is either small or large compared to1.
For a high energy beam passing through a symmetric collimator at a vertical distance y (y << b1) 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, re 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
z
b
1
1
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
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
Rectangular Collimator
is either small or large compared to1.
Analytical formulas for the kick factor can be found in the limits where the parameter
1
2
2 b
h
z
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
Stupakov[1] gives:
Zagorodnov[3] gives, -for a long collimator (L→∞):
-for a short collimator (L→0):
Diffractive regime
Tenenbaum[6] gives (r ≡ b1)
Rectangular Collimator
Tenenbaum[2] gives, for a short, flat collimator on the limit b1« b2:
PLACET
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
ESA Collimators
h=38 mm
38
mm
L=1000 mm
r=1/2 gap
11
22
33
66
α = 324mradr = 2 mm
α = 324mradr = 1.4 mm
α = 324mradr = 1.4 mm
α = 166mradr = 1.4 mm
α = 324mradr = 2 mm
α = 324mradr = 1.4 mm
α = 324mradr = 1.4 mm
α = 166mradr = 1.4 mm
Collimator Side view Beam view
Kick Factors for ESA Collimators
Bunch size, σz =0.5 mmColl 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-1m-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”