TRANSVERSE RESISTIVE-WALL

1
TRANSVERSE RESISTIVE-WALL IMPEDANCE FROM
ZOTTER2005S THEORY
Elias Métral (25 5 min, 25 slides)
Low-frequency regime

• Introduction and motivation
• Numerical applications for a
• LHC collimator (vs. Burov-Lebedev2002 and
  Bane1991)
• SPS MKE kicker (vs. Burov-Lebedev and 2-wire
  measurements)
• Review of Zotters theory (? For a circular beam
  pipe)
• Any number of layers
• Any beam velocity
• Any frequency ? Unification of 3 regimes (BL,
  thick-wall and Bane)
• Any s (conductivity), e (permittivity) and µ
  (permeability)
• Conclusion and work in progress

High-frequency regime
2
INTRODUCTION AND MOTIVATION (1/2)
THE MOTIVATION THE LHC GRAPHITE COLLIMATORS

• First unstable betatron line

• Skin depth for graphite (? 10 µOm)

• Collimator thickness

?
? One could think that the classical thick-wall
formula would be about right
3
INTRODUCTION AND MOTIVATION (2/2)

• In fact it is not ? The resistive impedance is
  2 orders of magnitude lower at 8 kHz !

? A new physical regime was revealed by the LHC
collimators
Usual regime
New regime
4
N.A. FOR A LHC COLLIMATOR (1/3)
COMPARISON ZOTTER2005-BUROVLEBEDEV2002
Classical thick-wall
BLs results (real and imag. parts) in black
dots without and lines with copper coating
5
N.A. FOR A LHC COLLIMATOR (2/3)
GLOBAL PLOT FROM ZOTTER2005
Low beam velocity case (e.g. PSB
, )
Same as Bane1991
Negative
AC conductivity
6
N.A. FOR A LHC COLLIMATOR (3/3)
7
N.A. FOR A SPS MKE KICKER
8
REVIEW OF ZOTTERS THEORY (1/17)

• 1) Maxwell equations
• In the frequency domain, all the field quantities
  are taken to be proportional to
• Combining the conduction and displacement current
  terms yields

with
9
REVIEW OF ZOTTERS THEORY (2/17)

• 2) Scalar Helmholtz equations for the
  longitudinal field components
• Using ,
  one obtains (using the circular cylindrical
  coordinates r, ?, z)

• The homogeneous equation can be solved by
  separation of variables

10
REVIEW OF ZOTTERS THEORY (3/17)
m is called the azimuthal mode number (m1 for
pure dipole oscillations)
?
and
k is called the wave number
Reinserting the time dependence ( ), the
axial motion is seen to be a wave proportional to
, with phase velocity which may
in general differ from the beam velocity
R (r) is given by
with
Radial propagation constant
The solutions of this differential equation are
the modified Bessel functions
and
11
REVIEW OF ZOTTERS THEORY (4/17)

• Conclusion for the homogeneous scalar Helmholtz
  equations
• For pure dipole oscillations excited by a
  horizontal cosine modulation propagating along
  the particle beam, one can write the solutions
  for Hz and Ez as
• Sine and cosine are interchanged for a purely
  vertical excitation (see source fields)
• Only the solutions of the homogeneous Helmholtz
  equations are needed since all the regions
  considered are source free except the one
  containing the beam where the source terms have
  been determined separately by Gluckstern (CERN
  yellow report 2000-011) ? See slide 14

C1,2 and D1,2 are constants to be determined
12
REVIEW OF ZOTTERS THEORY (5/17)

• 3) Transverse field components deduced from the
  longitudinal ones using Maxwell equations (in a
  source-free region)

13
REVIEW OF ZOTTERS THEORY (6/17)

• 4) Source of the fields Ring-beam distribution ?
  Infinitesimally short, annular beam of charge
  and radius traveling with
  velocity along the axis

• Charge density in the frequency domain

where is the horizontal
dipole moment
14
REVIEW OF ZOTTERS THEORY (7/17)

• Longitudinal source terms (from Gluckstern) ?
  Valid for , i.e. in the vacuum
  between the beam and the pipe region (1)

with
and will be determined by the
boundary conditions at b and d
15
REVIEW OF ZOTTERS THEORY (8/17)

• 5) The total (i.e. resistive-wall space charge)
  horizontal impedance

?
with L the length of the resistive pipe and
16
REVIEW OF ZOTTERS THEORY (9/17)

• The space-charge impedance is obtained with a
  perfect conductor at r b, i.e. when
  and , with

?

• If

?
and
?
17
REVIEW OF ZOTTERS THEORY (10/17)

• The resistive-wall impedance is obtained by
  subtracting the space-charge impedance from the
  total impedance

?
Only remains to be determined (by field
matching)
18
REVIEW OF ZOTTERS THEORY (11/17)

• 6) Field matching
• At the interfaces of 2 layers (r constant) all
  field strength components have to be matched,
  i.e. in the absence of surface charges and
  currents the tangential field strengths
  and have to be continuous
• Then the radial components of the displacement
  and of the induction are also
  continuous, i.e. matching of the radial
  components is redundant
• At a Perfect Conductor (PC)
  ?
• At a Perfect Magnet (PM)
  ?
• At r ? Infinity ? Only is
  permitted as diverges

19
REVIEW OF ZOTTERS THEORY (12/17)

• General form of the field strengths in region (p)

(Ep, Gp, ap and ?p) are constants to be determined

20
REVIEW OF ZOTTERS THEORY (13/17)

• 7) General 1-layer formula

a2 and ?2 are determined by the boundary
conditions at the outer chamber wall r d
21
REVIEW OF ZOTTERS THEORY (14/17)
d ? ?, or PC or PM
(2) Layer 1

• d ? ?

• Perfect Conductor (PC)

(1) Vacuum
Beam
a
b

• Perfect Magnet (PM)

d
22
REVIEW OF ZOTTERS THEORY (15/17)

• 8) General 2-layer formula

where the parameters (E2, a2) are 2 parameters
out of 4 (a2, ?2, E2 and G2) which have to be
found by solving the matching equations at each
layer boundary. The 4 unknowns are given by the
following system of 4 linear equations
23
REVIEW OF ZOTTERS THEORY (16/17)
1,2,3 refer to the vacuum (between the beam and
the first layer), the first and second layer
respectively
24
REVIEW OF ZOTTERS THEORY (17/17)
(3) Layer 2
e ? ?, or PC or PM
(2) Layer 1
(1) Vacuum

• e ? ?

Beam
a
b
d
e
25
CONCLUSION AND WORK IN PROGRESS

• Zotter2005s formula has been compared to other
  approaches from Burov-Lebedev2002, Tsutsui2003
  (theory and HFSS simulations) and Vos2003 (see
  Ref. 6 below) ? Similar results obtained in the
  new (Burov-Lebedev2002) low-frequency regime
• Work in Progress
• Multi-bunch or long bunch ? Wave velocity ?
  Beam velocity
• Finite length of the resistive beam pipe

REFERENCES
1 B. Zotter, CERN-AB-2005-043 (2005) 2 R.L.
Gluckstern, CERN yellow report 2000-011
(2000) 3 E. Métral, CERN-AB-2005-084
(2005) 4 A. Burov and V. Lebedev, EPAC02
(2002) 5 K. L.F. Bane, SLAC/AP-87 (1991) 6
F. Caspers et al., EPAC04 (2004) 7 H.
Tsutsui, LHC-PROJECT-NOTE-318 (2003) 8 L. Vos,
CERN-AB-2003-005-ABP (2003) and
CERN-AB-2003-093-ABP (2003)