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H. Padamsee SRF Course Topics at Erice

(My lectures in red)

• Superconductivity, and RF – Larbalastier, Ciovati• General comments on SC cavity design choices for accelerators• Basics of SRF cavities

– Structure Types• Basic RF Cavity Design Principles/ Figures of Merit

– Gradient, Losses, Q, Shunt Impedance, Peak Fields…• 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• Fundamental critical fields/Ultimate gradient possibilities - Antoine• Cavity Fabrication /Preparation - Singer• Cavity Testing - Reschke• Wide Range of Applications

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Mostly Conceptual with picturesWill go fast through many slides, refer to text booksAnd tons of reviews

Some quantitative aspects – referencesDraw examples from some accelerator applications

References: Extensive Literature +2 Text Books (1998 and 2010)Lots of Review PapersSRF Workshop Proceedings (1980, 83, 85….2001)(including Tutorials) on Jacow CERN websiteRAST articles (very recent)

Overall Approach

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Presented by

Saturday, 27 April

17:30

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H. PadamseeGeneral Accelerator Requirements That Drive

SC Cavity Design Choices, v/c ~ 1, v/c <1

Voltage needed

Storage Rings

CESR-III: 7 MV, KEK-B HER: 14 MV, LEP-II: 3 GV

Duty Factor (RF on time x Repetition Rate)Storage Rings: CW

Proton Linacs: < 10%

Linear Collider: 0.01 - 1%

Linear Collider: 500 - 1000 GV

Beam Current, Ave. Beam Power, Beam loss allowedStorage Rings: amp, MW

Proton Linacs: 10 - 100 mA, 1- 10 MW

Proton Linac: 1 GV SNS, ESS

Linear Collider: few ma, 10 MW

Linac-Based FEL or ERL : 500 MeV - 5 GeV

Linac-Based FEL or ERL CW

Linac-Based FEL or ERL 50 mA - 100 mA

Low Velocity Accelerators

• Transition energies for v< c accelerators

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Cavity Design Choices

• Main Choices – Particle velocity, beta = v/c – 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

Example Optimizations

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The criteria/requirements differ depending on application:

SRF accelerator type

Requirements RF parameters Cavity design

Pulsed linacs High-gradient operationEpk/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

Storage rings, ERLs

High beam current (R/Q)QL of HOMsLarger aperture, fewer

number of cells per cavity, cavity shape

Storage rings, ERL injectors

High beam power Pcouper

Larger aperture, fewer number of cells per

cavity (single-cell cavities for SR)

Ideal Cavity

• Pill-box shape

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Medium and High Velocity Structures = v/c = 0.5 -> 1

Single Cell

RF Power InBeam Induced Power Out

Multi-Cell Cavity

Squeezed Cells for v/c = 0.5

Basic Principle, v/c = 1

/2

/2

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1300 MHz Structures for Accelerating Particles at v ~ c

Structures for Particles at v < c (SNS)For protons at 1 ~ GeV

TESLA-shape

(DESY, TTF)

Low-Loss shape (Jlab, KEK…)

Re-entrant shape (Cornell)

Structure Examples

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Low Velocity Structures, = v/c = 0.01 -> 0.2

Basic Principle

Quarter Wave

Half-Wave SpokeSplit -Ring

Inter-Digital

Niobium

Range of Velocity and Frequency

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18Basics for Superconducting Cavities

Covered by Ciovati

10 cm

E

Vc = One Million

Volts

E

Gap = d

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RF accelerator cavitiesFields and Currents

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• Accelerating voltage then is:

• Accelerating field is:

Figures of MeritAccelerating Voltage/Field

(v = c Particles)

d

Enter Exit

T is Transit time factor = 2/

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Figures of Merit for SC CavityCovered 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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H. PadamseeReal Single Cell Cavities

KEK-B Cavity

Electric field high at iris

Magnetic field high at equator

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• 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.

Importance of Figures of MeritPeak Fields

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Hpk

Peak fields for low beta cavities are higher

Typical

Epk/Eacc = 4 - 6

Hpk/Eacc = 60 - 200 Oe/MV/m

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Figures of Merit Dissipated Power, Stored Energy, Cavity Quality (Q)

•Stored energy is:

•Dissipation in the cavity wall given by surface integral:

UTrf Pc

= 2 •Define Quality (Q) as

which is ~ 2 number of cycles it takes to dissipate the energy stored in the cavity Easy way to measure Q

• Qnc ≈ 104, Qsc ≈ 1010

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Galileo, 1600 AD

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Figures of MeritShunt Impedance (Ra)

•Ra/Q only depends on the cavity geometry Cavity design

• Shunt impedance (Ra) determines how much acceleration one gets for a given dissipation (analogous to Ohm’s Law)

Another important figure of merit is

To maximize acceleration, must maximize shunt impedance.

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Evaluation - Analytic Expressions

1.5 GHz pillbox cavity, R = 7.7 cm, d = 10 cm

Low Beta Elliptical Cavities

• A progression of compressed elliptical cavity shapes at the same rf frequency but for decreasing values

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Current and Voltage for QWRReviews/Tutorials by Delayen and Facco

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Current and Voltage for Half-Wave Resonator

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(a) (b) (c) (d)

Spoke is Half-Wave Resonator

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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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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 . The transit time factor falls off steeply with on either side of the optimum

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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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Real Cavities - Codes

• Adding beam tubes reduces Ra/Q by about x2 =>

• 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.

June 22, 2011 44

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 45

2D code example: SLANS/CLANS

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.

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More on 3D EM Codes

• 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 un- structured 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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H. PadamseeSample Result from Codes

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Elliptical cavity (TM-class) Half-wave cavity (TEM-class)

equator

E and H Fields for Multi - Spoke

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H. PadamseeDesign 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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270 ohmΩ88 ohm/cell 2.552 Oe/MV/m

Cornell SC 500 MHz

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• Example: Assume we make this cavity out of copper

• Want to operate CW at 500 MHz and

• 1 MV (3 MV/m)

• Rs = 6 mohm

Q = 45,000Ra = 4 Mohm

Pdiss = 250 kWThis 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.

Copper Cavity ExampleCW and Low Gradient

R/Q = 89 Ohm

Rs = √ ( f 0 )

f = RF frequency r = DC resistivity

= permeability of free space

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* DF

Want high Vacc

Depends only on geometry. Maximize this for copper cavities

Determined by the material being used

If dissipation is too large,must reduce duty factor

Minimizing Losses

Pdiss =Vacc

2

Ra

Vacc2

Ra/Q * Q=* DF

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H. PadamseeOptimizing CW Copper CavitiesHigh Current Application

• 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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Superconducting Cavity

• 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 10-

100 times less than that of Cu cavities.

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H. PadamseeA challenge of the SC option is cryogenics

Refrigerator efficiencies are low

And one has to add other heat contributions from conduction, radiation, helium distribution. Carnot efficiency of frigand technical efficiency of frig machinery

Carnot = 4.5300 - 4.5

= 0.015

technical = 0.20total = 0.003 = 1/333

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H. PadamseeSRF Requirements & Limitations

• Cryogenic system.

Hi Tech:Ultra-clean preparation and

assembly required

Max Eacc = 50 MV/mCovered by Ciovati

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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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(Some) Further SC Advantages

• Freedom to adapt design better to the accelerator requirements allows, for example, the beam-tube 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., B-factories)

– Reduce the amount of beam scraping less activation in, e.g., proton machines (important for, e.g., SNS, Neutrino factory)

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Higher-Order-Mode Excitation

time

The bunched beam excites higher-order-modes (HOMs) in the cavity.

bunch

bunch

bunch

Slide 64

Hig

her

Ord

er

Mo

de a

nd

Be

am

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HOMs

• 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 n 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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0 20 40 60 80 1000

20

40

60

80

100

HOM frequency [GHz]

% o

f H

OM

po

wer

loss

ab

ove

fH

OM

Matthias Liepe03/08/02

Monopole Higher-Order-Mode Single Bunch Power Excitation per 9-Cell Cavity

U N I V E R S I T YCORNELL

HOM damping 2

bunch

Ptotal = 160 W

P(f<3.5 GHz) = 30 WP(f>3.5 GHz) = 130 W P(f>5 GHz) = 115 WP(f>10 GHz) = 83 WP(f>20 GHz) = 51 WP(f>40 GHz) = 23 WP(f>80 GHz) = 5 W

bunch = 0.6 mm qbunch = 77 pC

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• 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 is bunch displacement off-axis, a is the cavity aperture (radius), n the angular HOM frequency of mode n. Rd/Q0 is the dipole mode impedance,

• Each dipole mode has two polarizations.

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H. PadamseeHigher Order Modes

Monopole HOM above cut off

Dipole HOM above cut off

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H. PadamseeWakefields 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

• Higher frequency, structures– 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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H. PadamseeSummary of Design Trade-OffsWhat RF Frequency?

Frequency, low beta

• Gap = /2

• So lower beta, prefers lower frequency

• Structure dimensions increase with low freq

• BCS losses decrease with low freq

Range of Velocity and Frequency

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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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• Iris diameter• Increasing the iris diameter increases cell-to-

cell 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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• 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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• 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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