The Compendium of formulae of kick factor.

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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
The University of Manchester
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Introduction
z
z
Round Collimator
Rectangular Collimator

• 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.

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Introduction
For a high energy beam passing through a
symmetric collimator at a vertical distance y (y
ltlt b1) from the axis, the mean centroid kick is
given by
where N is the number of particles in the bunch,
? 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
is either small or large compared to1.
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Round Collimator
Inductive regime
Tenenbaum2 gives
Zagorodnov3 gives
Tenenbaum6 gives for a round collimator of
half-gap r and tapered angle a
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Round Collimator
Diffractive regime
- analytical formulas exits in the limit of short
(L?0) and long (L?8) collimator
Stupakov1 gives

• Tenenbaum2 gives,
• for a long, round collimator
• -for a short, round collimator

Tenenbaum6 gives for a round collimator of
half-gap r and tapered angle a
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Rectangular Collimator
Analytical formulas for the kick factor can be
found in the limits where the parameter
is either small or large compared to1.
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Rectangular Collimator
Inductive regime
Tenenbaum2 gives
Zagorodnov3 gives
Tenenbaum6 gives for a rectangular collimator
of half-gap r and tapered angle a
PLACET
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Rectangular Collimator
Diffractive regime
Stupakov1 gives
Tenenbaum2 gives, for a short, flat collimator
on the limit b1 b2

• Zagorodnov3 gives,
• for a long collimator (L?8)
• for a short collimator (L?0)

Tenenbaum6 gives (r b1)
PLACET
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Rectangular Collimator
Intermediate regime
Stupakov1 gives
Tenenbaum2 gives,
Zagorodnov3 gives
with A1 for a long collimator (L?8) and A1/2
for a short collimator (L?0).
Tenenbaum6 gives
PLACET
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ESA Collimators
Collimator Side view
Beam view
a 324mrad r 2 mm a 324mrad r 1.4
mm a 324mrad r 1.4 mm a 166mrad r
1.4 mm
1
a
r1/2 gap
2
3
6
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Kick Factors for ESA Collimators
Bunch size, sz 0.5 mm Coll Kick Factors
(V/pC/mm) PLACET Analytic
Prediction Measured 1
2.47 2.27
1.40.1 (1.0) 2 5.04
4.63
1.40.1 (1.3) 3 5.76
5.25
4.40.1 (1.5) 5 5.04
4.59 3.70.1
(7.9) 6 5.04
4.65 0.90.1 (0.9)

Coll a(mrad) r (mm) LT (mm)
LF(mm) s(O-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
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