SRF Course Topics at Erice
(My lectures in red)
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H. Padamsee
Superconductivity, and RF – Larbalastier, Ciovati
General comments on SC cavity design choices for accelerators
Basics of SRF cavities
– Structure Types
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Basic RF Cavity Design Principles/ Figures of Merit
– Gradient, Losses, Q, Shunt Impedance, Peak Fields…
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SC/NC comparison for CW application
Design Aspects for Multicells
Higher Order Modes and their importance in cavity design
Mechanical Aspects of Cavity Design
Couplers/Tuners/…
Cavity Performance Aspects/Cavity Technology- Antoine
– Multipacting, Breakdown (Quench), Field Emission, Q-Slope
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Fundamental critical fields/Ultimate gradient possibilities - Antoine
Cavity Fabrication /Preparation - Singer
Cavity Testing - Reschke
Wide Range of Applications
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H. Padamsee
Overall Approach
Mostly Conceptual with pictures
Will go fast through many slides, refer to text books
And tons of reviews
Some quantitative aspects – references
Draw examples from some accelerator applications
References:
Extensive Literature +
2 Text Books (1998 and 2010)
Lots of Review Papers
SRF Workshop Proceedings (1980, 83, 85….2001)
(including Tutorials) on Jacow CERN website
RAST articles (very recent)
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H. Padamsee
Saturday, 27 April
17:30
Presented
by
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General Accelerator Requirements That Drive
SC Cavity Design Choices, v/c ~ 1, v/c <1
H. Padamsee
Voltage needed
Storage Rings
CESR-III: 7 MV, KEK-B HER: 14 MV, LEP-II: 3 GV
Proton Linac: 1 GV SNS, ESS
Linac-Based FEL or ERL : 500 MeV - 5 GeV
Linear Collider: 500 - 1000 GV
Duty Factor (RF on time x Repetition Rate)
Storage Rings: CW
Linac-Based FEL or ERL CW
Proton Linacs: < 10%
Linear Collider: 0.01 - 1%
Beam Current, Ave. Beam Power, Beam loss allowed
Storage Rings: amp, MW
Linac-Based FEL or ERL 50 mA - 100 mA
Proton Linacs: 10 - 100 mA, 1- 10 MW
Linear Collider: few ma, 10 MW
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H. Padamsee
Low Velocity Accelerators
• Transition energies for v< c accelerators
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H. Padamsee
Cavity Design Choices
• Main Choices
– Particle velocity, beta = v/c
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RF Frequency
Operating Gradient
Operating Temperature
Number of Cells
Cell Shapes
Beam Aperture
Type of Structure, QWR, HWR, Spoike (low velocity)
Optimizations Involve Many Trade-offs
Best Cavity/Accelerator Performance for Least Risk
Minimize Capital + Operating Cost
Discuss parameters/dependencies
– But not the trade-offs
– Which are particular to each accelerator design
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Example Optimizations
The criteria/requirements differ depending on application:
SRF accelerator
type
Requirements
RF parameters
Cavity design
Pulsed linacs
High-gradient operation
Epk/Eacc
Hpk/Eacc
Iris and equator shape,
smaller aperture
CW linacs and
ERLs
Low cryogenic loss
(dynamic), good fill
factor
G(R/Q)
# of cells
Cell shape, smaller
aperture, larger number
of cells per cavity
(R/Q)QL of HOMs
Larger aperture, fewer
number of cells per
cavity, cavity shape
Storage rings,
ERLs
High beam current
H. Padamsee
Ideal Cavity
• Pill-box shape
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H. Padamsee
Medium and High Velocity Structures b = v/c = 0.5 -> 1
/2
RF Power In
Beam Induced Power Out
Basic Principle, v/c = 1
Multi-Cell Cavity
b/2
Single Cell
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Squeezed Cells for v/c = 0.5
Structure Examples
Structures for Particles at v < c (SNS)
H. Padamsee
For protons at 1 ~ GeV
1300 MHz Structures for Accelerating Particles at v ~ c
TESLAshape
(DESY,
TTF)
Low-Loss shape (Jlab, KEK…)
Re-entrant shape (Cornell)
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Low Velocity Structures, b = v/c = 0.01 -> 0.2
H. Padamsee
Niobium
Basic Principle
Inter-Digital
Quarter Wave
Split -Ring
Half-Wave
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Spoke
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Range of Velocity and Frequency
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Basics for Superconducting Cavities
Covered by Ciovati
10 cm
E
E
Gap = d
Vc = One Million
Volts
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H. Padamsee
RF accelerator cavities
Fields and Currents
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H. Padamsee
Figures of Merit
Accelerating Voltage/Field
(v = c Particles)
d
Enter
Exit
• Accelerating voltage then is:
T is Transit time factor = 2/p
• Accelerating field is:
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H. Padamsee
Figures of Merit for SC Cavity
Covered by Ciovati
• Accelerating Field and Q: Eacc, Q
• Stored Energy, Geometry Factor
• Peak Electric and Magnetic Field Ratios
– Epk/Eacc, Hpk/Eacc
• Shunt Impedance, Geometric Shunt
Impedance: Ra, Ra/Q
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Real Single Cell Cavities
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KEK-B Cavity
Electric field high at iris
Magnetic field high
at equator
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Importance of Figures of Merit
H. Padamsee
Peak Fields
• For Eacc  important parameter is Epk/Eacc,
– Typically 2 - 2.6
• Make as small as possible, to avoid problems
with field emission - more later.
• Equally important is Hpk/Eacc, to maintain SC
– Typically 40 - 50 Oe/MV/m or 4 – 5 mT/MV/m
• Hpk/Eacc can lead to premature quench
problems (thermal breakdown).
• Ratios increase significantly
– when beam tubes are added to the cavity
– or when aperture is made larger.
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H. Padamsee
Peak fields for low beta cavities are higher
Typical
Epk/Eacc = 4 - 6
Hpk/Eacc = 60 - 200 Oe/MV/m
Hpk
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Figures of Merit
Dissipated Power, Stored Energy, Cavity Quality (Q)
•Dissipation in the cavity wall given by
surface integral:
•Stored energy is:
•Define Quality (Q) as
U
= 2p
Trf Pc
which is ~ 2p number of cycles it takes to dissipate the
energy stored in the cavity  Easy way to measure Q
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Qnc ≈ 104,
Qsc ≈ 1010
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H. Padamsee
Galileo, 1600 AD
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H. Padamsee
Figures of Merit
Shunt Impedance (Ra)
• Shunt impedance (Ra) determines how
much acceleration one gets for a given
dissipation (analogous to Ohm’s Law)
 To maximize acceleration, must maximize shunt impedance.
Another important figure of merit is
•Ra/Q only depends on the
cavity geometry  Cavity design
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H. Padamsee
Evaluation - Analytic Expressions
1.5 GHz pillbox cavity, R = 7.7 cm, d = 10 cm
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H. Padamsee
Low Beta Elliptical Cavities
• A progression of
compressed elliptical
cavity shapes at the same
rf frequency but for
decreasing b values
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Current and Voltage for QWR
Reviews/Tutorials by Delayen and Facco
H. Padamsee
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H. Padamsee
Current and Voltage for Half-Wave Resonator
(a)
(b)
(c)
(d
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H. Padamsee
Spoke is Half-Wave Resonator
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H. Padamsee
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One-gap (pill-box) transit time factor for low velocity, using two different
aperture values 15 mm and 30 mm. Acceleration takes place efficiently
above β ~2g/λ and it is maximum at β=1. Here g is the gap and b is the
aperture.
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H. Padamsee
Transit Time for Standard QWR
• 2-gap transit time factor for the π-mode compared to the 1-gap.
Acceleration is maximum for a particular value of b. The transit
time factor falls off steeply with b on either side of the optimum
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H. Padamsee
Transit Time Factor for Low-Velocity
• Normalized transit time factor vs. normalized velocity
β/β0, for cavities with different number of equal gaps
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H. Padamsee
Real Cavities - Codes
• Adding beam tubes reduces Ra/Q by about x2 =>
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for Cu cavities use a small beam hole.
• Peak fields also increase.
– Can be a problem for high gradient cavities
• Analytic calculations are no longer possible
– especially if cavity is shaped is to optimize peak fields.
•  Use numerical codes.
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RF design tools
 Design of elliptical cavities is performed in two steps: 2D
and 3D.
 2D codes (Superfish, SLANS/CLANS, …)
 faster and allow to design geometry of the
cylindrically-symmetric body of the cavity.
 3D codes (MAFIA, Microwave Studio, HFSS, Omega-3P,
GdfidL, …)
 necessary to complete the design by modeling the
cavity equipped with fundamental power couplers,
HOM loop couplers, calculating coupling strength,
etc.
Peak surface fields
2D code example: SLANS/CLANS
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 SuperLANS (or SLANS) is a computer program
designed to calculate the monopole modes of
RF cavities using a finite element method of
calculation and a mesh with quadrilateral
biquadratic elements.
 SLANS has the ability to calculate the mode
frequency, quality factor, stored energy, transit
time factor, effective impedance, max electric
and magnetic field, acceleration, acceleration
rate, average field on the axis, force lines for a
given mode, and surface fields.
 Later versions, SLANS2 and CLANS2,
calculate azimuthally asymmetric modes, and
CLANS and CLANS2 can include into
geometry lossy materials.
H. Padamsee
More on 3D EM Codes
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Under the SciDAC collaboration, (Scientific Discovery through
Advanced Computing), SLAC has developed parallel processing finite
element electromagnetic codes to obtain gains in accuracy, problem
size and solution speed by harnessing the computing power and
exploiting the huge memory of the latest supercomputers, e. g. the
IBM/SP (Seaborg) at NERSC and the Cray/X1E (Phoenix) at NLCF.
The suite of electromagnetic codes available are based on unstructured grids for high accuracy, and use parallel processing to
enable large-scale simulation.
The new modeling capability supports meshing, solvers, refinement,
optimization and visualization.
The code suite to date includes the eigensolver Omega3P, the S-matrix
solver S3P, the time-domain solver T3P, and the particle tracking code
Track3P.
Direct simulations of the entire cavity with input and HOM couplers
have been carried out.
TEM-class drift-tube loaded cavities and Spoke Resonators have been
designed using modern 3D simulation codes such as MAFIA,
Microwave Studio, SOPRANO
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Sample Result from Codes
H. Padamsee
equator
Elliptical cavity (TM-class)
Half-wave cavity (TEM-class)
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E and H Fields for Multi - Spoke
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Design Cavity Shape Consequences
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Influence of Aperture
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Aperture
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Equator Radius
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Iris Radius
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Summary of Aperture Effects
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H. Padamsee
Cornell SC 500 MHz
270 ohmΩ
88 ohm/cell
2.5
52 Oe/MV/m
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Copper Cavity Example
CW and Low Gradient
Rs = √ (p f m0 r)
• Example: Assume we
make this cavity out of
copper
• Want to operate CW at
500 MHz and
• 1 MV (3 MV/m)
R/Q = 89 Ohm
H. Padamsee
f = RF frequency
r = DC resistivity
m0 = permeability of free space
• Rs = 6 mohm
Q = 45,000
Ra = 4 Mohm
Pdiss = 250 kW
This would result in a overheating of copper cell. Water-cooled
copper cavities at this frequency can dissipate about 40 kW.
(CW) copper cavity design is primarily driven by the
requirement that RF losses must be kept small.
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H. Padamsee
Minimizing Losses
If dissipation is too large,
must reduce duty factor
Want high Vacc
Pdiss =
Vacc
Ra
2
* DF =
Vacc2
Ra/Q * Q
* DF
Depends only on geometry.
 Maximize this for copper cavities
Determined by the materia
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being used
Optimizing CW Copper Cavities
High Current Application
H. Padamsee
• Use small beam tubes
• Use reentrant design to reduce
surface magnetic currents.
•  Ra/Q = 265 Ohm
•  Pdiss = 80 kW @ 3 MV/m
• Still have to reduce voltage to
0.7 MV.
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H. Padamsee
Superconducting Cavity
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Recalculate Pdiss with SC Nb
at 4.2 K, 1 MV, and 500
MHz.
Q = 2 x 109 (Rs 15 n)
 Ra= 5.3 x 1011
 Pdiss= 1.9 W!
 Pac= 660 W = AC power
(Frig. efficiency = 1/350)
 Include cryostat losses,
transfer lines, etc.
 Pacincreases, but is still 10100 times less than that of
Cu cavities.
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A challenge of the SC option is cryogenics
H. Padamsee
Refrigerator efficiencies are low
And one has to add other heat contributions from
conduction, radiation, helium distribution.
Carnot efficiency of frig
and technical efficiency of frig machinery
Carnot =
4.5
300 - 4.5
= 0.015
technical
= 0.20
total = 0.003 = 1/333
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SRF Requirements & Limitations
H. Padamsee
• Cryogenic system.
Hi Tech:
Ultra-clean preparation and
assembly required
Max Eacc = 50 MV/m
Covered by Ciovati
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H. Padamsee
SC Advantages
• Power consumption is much less  operating cost
savings, better conversion of ac power to beam
power.
• CW operation at higher gradient possible  Less
klystron power required  capital cost saving
• Need fewer cavities for CW operation  Less beam
disruption
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H. Padamsee
(Some) Further SC Advantages
• Freedom to adapt design better to
the accelerator requirements
allows, for example, the beamtube size to be increased:
– Reduces the interaction of the beam
with the cavity (scales as size3) 
The beam quality is better preserved
(important for, e.g., FELs).
– HOMs are removed more easily 
better beam stability  more current
accelerated (important for, e.g., Bfactories)
– Reduce the amount of beam scraping
 less activation in, e.g., proton
machines (important for, e.g., SNS,
Neutrino factory)
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H. Padamsee
bunch
bunch
time
bunch
The bunched beam excites higher-order-modes
(HOMs) in the cavity.
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Slide 64
Higher Order Mode and Beam
Higher-Order-Mode Excitation
H. Padamsee
HOMs
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A charged particle beam excites a wide spectrum of higher order
modes, depending on the impedances (i.e., the R/Q )of the modes.
The resulting electromagnetic field left behind by the beam is called the
wake-field,
Beam deposits power in high impedance monopole higher order
modes (HOM).
HOMs can also cause longitudinal beam instabilities and increase the
energy spread.
=> HOMs must be properly extracted and damped,
Energy lost by the passage a single bunch, charge q, is given by
where wn is the angular frequency of mode n, Ra/Q `is the geometric
shunt impedance of the monopole,
kn is also referred to as the loss factor of mode n.
The total power deposited depends on the number of bunches per
second, or the beam current.
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Monopole Higher-Order-Mode Single Bunch
Power Excitation per 9-Cell Cavity
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CORNELL
UNIVERSITY
Matthias Liepe
03/08/02
bunch
bunch = 0.6 mm
HOM
damping 2
% of HOM power loss above f
HOM
100
qbunch = 77 pC
Ptotal
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P(f<3.5 GHz) = 30 W
P(f>3.5 GHz) = 130 W
P(f>5 GHz) = 115 W
P(f>10 GHz) = 83 W
P(f>20 GHz) = 51 W
P(f>40 GHz) = 23 W
P(f>80 GHz) = 5 W
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40
20
0
0
= 160 W
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40
60
80
HOM frequency [GHz]
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H. Padamsee
• Beam excited dipole and quadrupole modes with high
impedance can deflect the beam to cause transverse
instabilities.
• Dipoles have the highest impedance. The energy lost by a
charge to the dipole mode is given by
• Where r is bunch displacement off-axis, a is the cavity aperture
(radius), wn the angular HOM frequency of mode n. Rd/Q0 is the
dipole mode impedance,
• Each dipole mode has two polarizations.
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Higher Order Modes
H. Padamsee
Dipole HOM above cut off
Monopole HOM above cut off
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Wakefields and Effects
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Summary of Aperture Effects
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Aperture and HOMs
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Higher-Order Modes Impedance
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Summary of Design Trade-Offs
What RF Frequency?
H. Padamsee
• Higher frequency, b=1structures
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Higher BCS losses per m of structure
Higher R/Q (per meter)
Higher wakefields
Overall smaller aperture for beam
More cells per meter
Above 3 GHz, Global thermal instability limit
Less material – lower cost?
– Less surface area for defects to cause field
emission and breakdown
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Frequency, low beta
• Gap = b/2
• So lower beta, prefers lower frequency
• Structure dimensions increase with low
freq
• BCS losses decrease with low freq
H. Padamsee
Range of Velocity and Frequency
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H. Padamsee
Cell Shape
• Most beta = 1 structures are based on the elliptical cavity
• Parameters to be manipulated for optimization are
• cell diameter at equator, iris diameter and shape of
transition from equator to iris.
• The cavity shape is designed for low Epk/Eacc to
minimize field emission, low Hpk/Eacc for best thermal
stability, and high cell to-cell coupling for enhancement of
field flatness.
• The tilt of the cell wall provides stiffness against
mechanical deformations and is a better geometry for acid
draining and water rinsing for surface preparation.
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H. Padamsee
• Iris diameter
• Increasing the iris diameter increases cell-tocell coupling and the peak field ratios
(Epk/Eacc, Hpk/Eacc)
• but decreases the characteristic shunt
impedance (R/Q) as well as loss factors of
the higher modes (kll and kt).
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H. Padamsee
• Increasing the surface area of the
equatorial region, lowers the peak
surface magnetic field.
• Reducing the iris aperture also lowers
the peak surface fields, but raises the
wakefields.
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H. Padamsee
• The optimum iris diameter depends on
accelerator type and operating
conditions.
• For high gradient application the peak
field ratios must be minimized (smaller
iris).
• For high current applications, the HOM
loss factors must be small (larger iris).
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