NEWS ABOUT COLLIMATOR IMPEDANCE

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NEWS ABOUT COLLIMATOR IMPEDANCE FOR PHASE 1
2
Elias Métral

• Some more info about the transverse impedance of
  the LHC
• 1st part of the answer to the question (from R.
  Assmann) about the EM fields near the collimator
  (For SPS MD in 2004) ? Deduced from Zotter2005
  formalism (circular geometry classical
  thick-wall regime)
• Preliminary results for a ceramic collimator or a
  collimator with slots or rods

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ZOTTER2005S THEORY FOR 1 GRAPHITE COLLIMATOR
3) High-frequency regime ? Not relevant for LHC!
1) Inductive-bypass regime
2) Classical thick-wall regime
Interesting frequency range for LHC ? From few
kHz to few GHz
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SIMPLEST FORMULA FOR THE LHC COLLIMATOR
TRANSVERSE IMPEDANCE (round case) (1/2)
Valid for any relatively good conductor with ?r
real and ?r 1
with
Modified Bessel function
?
?
There are Yokoyas factors to go from round to
flat (?2 / 12 and ?2 / 24 )
Classical thick-wall regime
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SIMPLEST FORMULA FOR THE LHC COLLIMATOR
TRANSVERSE IMPEDANCE (round case) (2/2)
The maximum of the real part is reached when
It is a broad maximum
?
?
This scaling was also found analytically using
the approximated model of L. Vos (as said in the
LHC Design Report, p. 100)
N.A.
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ZOOM (between 8 kHz and 20 MHz) OF THE LHC
TRANSVERSE IMPEDANCE AT TOP ENERGY (AFTER THE
SQUEEZE)

• The value of the real part of the impedance at 8
  kHz (1st unstable betatron line) is 141 M?/m
• The value of the real part of the impedance at 20
  MHz (frequency limit of the transverse damper) is
  55 M?/m
• The ratio between the two values is only 2.6
  (it would have been 50 in the case of the
  classical resistive-wall theory!)
• And, as the gain of the power amplifier rolls off
  rapidly when approaching 20 MHz, there might be
  some problems there

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SOURCE CHARGE DENSITY USED FOR THE COMPUTATIONS

• A macro-particle of charge is
  assumed to move along the pipe (in the s -
  direction) with an offset in the
  direction and with velocity

A cos m? beam generates EM fields in cos m? and
sin m? ? Different multipoles are decoupled
(consequence of the axial symmetry)
? The charge density can be written
where is the mth multipole
moment

• The cylindrical coordinate system
  is used

• Numerical values are given for the 2004 SPS
  experiment

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REMINDER Perfectly Conducting wall (1/3)
m 0
m 1
Used to compute the longitudinal impedance
Used to compute the transverse impedance
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REMINDER Perfectly Conducting wall (2/3)
Force on a particle with charge q
m 0
m 1
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REMINDER Perfectly Conducting wall (3/3)
m 0
m 1
For L 2 ? R
For L 2 ? R
Behind the bunch
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CASE OF A RESISTIVE OBJECT (1/4)
m 0
m 1
? ? ?
? z gt 0 is ahead of the beam and z lt 0 is behind
the beam
Different from Chao page 54 (see next slide)!
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CASE OF A RESISTIVE OBJECT (2/4)
I have (at the moment) for m 1
Chao finds the same result for r b and ? 0
?r 1 for Chao
Chao has (r 2 b2) instead of r 2
Chao finds a result 2 times bigger
Chao has (r 2 b2) instead of r2
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CASE OF A RESISTIVE OBJECT (3/4)
m 0
m 1
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CASE OF A RESISTIVE OBJECT (4/4)
m 0
m 1
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PRELIMINARY ANALYTICAL COMPUTATIONS OF THE
TRANSVERSE IMPEDANCE OF A CERAMIC COLLIMATOR (1/2)
? Scan in resistivity ? from 10-6 to 1020 ?m and
Many thanks to Fritz and Alexej for many
interesting discussions!
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PRELIMINARY ANALYTICAL COMPUTATIONS OF THE
TRANSVERSE IMPEDANCE OF A CERAMIC COLLIMATOR (2/2)
1 layer of thickness 2.5 cm and then Perfect
Conductor
1 layer of infinite thickness
1 layer of thickness 2.5 cm and then Perfect
Conductor
Lossy dielectric
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PRELIMINARY SIMULATIONS OF THE TRANSVERSE
IMPEDANCE OF A CERAMIC COLLIMATOR BY TOM KROYER
Typo ? below
Typo ? above
In conclusion Only preliminary results for the
moment!!! ? We still need to carefully check them!
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Full Cu
Slots
Rods
NEWS ABOUT COLLIMATOR WITH SLOTS RODS (1/2)
Current density at 10 kHz
Width 1 mm Depth 5 mm
2?2 mm2 Length 5 mm
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NEWS ABOUT COLLIMATOR WITH SLOTS RODS (2/2)

• Measurements were also started by Fritz, Benoit
  and Federico ? More news soon