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

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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”