accelerator physicspill box cavity • needs longitudinal electric field for acceleration •...

Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020

Accelerator PhysicsFundamentals of RF Cavities

Suba De Silva, S. A. Bogacz, G. A. Krafft,

and B. Dhital

Old Dominion University / Jefferson Lab

Lecture 10


• RF Cavities

– Cavity Basics

– RF Properties

– TM Type Cavities

• Types of Cavities

– Accelerating Cavities

– Low β cavities

– Deflecting/Crabbing Cavities

• Limitations in SRF Cavities

• Losses in RF Cavities

• For normal conducting cavities

• For superconducting cavities

Outline

2


• H. Padamsee, J. Knobloch, T. Hays “ RF Superconductivity

for Accelerators”, John Wiley & Sons, Inc; ISBN 0-471-

15432-6

• Proceedings of the Workshops on RF Superconductivity

1981–2015 – (ww.jacow.org)

• CERN Accelerator School – 1955 – 2016

(https://cds.cern.ch/collection/CERN%20Yellow%20Report

s?ln=en)

Suggested Literature

3


PAS A ccelerato r Phy

RF Cavities

RF cavities made of differentmaterials, in differentshapes and sizes

1500MHz 5-cell

1300 MHz 9-cellQuarter

Wave

Cavity

Half

Wave

Cavity

Triple

Spoke

Cavity

CESR 500

MHz Cavity

(Cornell) LEP 350 MHz 4-cell Nb on Cu

4

Single

Spoke

Cavity


• Space enclosed by conducting walls that can sustain an

infinite number of resonant electromagnetic modes

• Shape is selected so that a particular mode can efficiently

transfer its energy to a charged particle

• An isolated mode can be modeled by an LRC circuit

• Lorentz force

• An accelerating cavity needs to provide an electric field

(E) longitudinal with the velocity of the particle

• Magnetic fields (H) provide deflection but no acceleration

RF Cavities

5


• Simplest form of RF resonator LC circuit

• LC circuit Pill box cavity

– Electric field is concentrated near axis

– Magnetic field is concentrated at outer cylindrical wall

RF Resonator

6


• Fields in an rf cavity are solution to the wave equation

• Subjected to boundary conditions:

– No tangential electric field

– No normal magnetic field

• Two sets of eigenmode solutions with infinite number of

modes

– TM modes Modes with longitudinal electric fields and

no transverse magnetic fields

– TE modes Modes with longitudinal magnetic fields and

no transverse electric fields

Cavity Basics

7


TM and TE Modes in a Pill Box Cavity

xmn is the nth root ofJm

x’mn is the n th root ofJ’m

8

L

2R


• TM010

– Electric field is purelylongitudinal

– Electricand magnetic fields have no angulardependence

– Frequencydependsonlyon radius, independent on length

• TM0np

– Monopolemodes that can coupleto the beamand exchange energy

• TM1np

– Dipolemodes that can deflect the beam

• TE modes

– No longitudinal E field

– Cannotcouple to the beam

– TE-typemodes can deflect thebeam

Modes in Pill Box Cavity

9


Pill Box Cavity

• Needs longitudinal electric field for acceleration

• Operated in the TM010 mode

• Hollow right cylindrical enclosureE

TM010 mode

10


TM010 Mode in a Pill Box Cavity

E

TM010 mode

R

• Frequency scales inversely with cavity radius

11


TM010 Field Profile

MagneticField

ElectricField Surface ElectricField

Surface MagneticField

• Peak surface electric

field at end plates

• Peak surface magnetic

field at outercylindrical

surface

12


Cavity RF Properties

Slide 13


• For efficient acceleration, choose a cavity geometry and a

mode where:

– Electric field is along the particle trajectory

– Magnetic field is zero along the particle trajectory

– Velocity of the electromagnetic field is matched to

particle velocity

• Accelerating voltage for charged particles

Accelerating Voltage

14


• Accelerating voltage for charged particles

• For the pill box cavity

T is the transit time factor

Accelerating Voltage (Vc)L

Enter Exit

L

2

15


• Accelerating field (gradient): Voltage gained by a particle

divided by a reference length

• For velocity of light particles:

N – no. of cells

• For less-than-velocity-of-light cavities (β < 1), there is no

universally adopted definition of the reference length

• However multi-cell elliptical cavities with β < 1

Length per cell

Accelerating Gradient (Eacc)

2L

N

Vacc

accEL

2L

16

[MV/m]


• Energy density in electromagnetic field:

• Because of the sinusoidal time dependence and the 90º

phase shift, the energy oscillates back and forth between

the electric and magnetic field

• Total energy content in the cavity:

Stored Energy (U)

17


• Surface current results in power dissipation proportional to

the surface resistance (Rs)

• Power dissipation per unit area

• Total power dissipation in the cavity walls

Power Dissipation (Pdiss)

18


• For normal conductors

– per unit length

– per unit area

• For superconductors

– per unit length

– per unit area

Power Dissipation (Pdiss)

19


Quality Factor (Q0)

• Measures cavity performance as to how lossy cavity

material is for given stored energy

• For normal conducting cavities ~ 104

• For superconducting cavities ~ 1010

20


• Geometrical factor [Ω]

– Product of the qualify factor (Q0) and the surface

resistance (Rs)

– Independent of size and material

– Depends only on shape of cavity and electromagnetic

mode

– For superconducting elliptical cavities

Geometrical Factor (G)

21


• Shunt impedance (Rsh) [Ω]

• Maximize shunt impedance to get maximum acceleration

• Note: Sometimes the shunt impedance is defined as or

quoted as impedance per unit length (Ω/m)

• R/Q [Ω]: Measures of how much of acceleration for a

given power dissipation

Shunt Impedance (Rsh) and R/Q

22


• Optimization parameter:

• R/Q and RshRs

– Independent of size (frequency) and material

– Depends on mode geometry

– Proportional to no. of cells

• In practice for elliptical cavities

– R/Q ~ 100 Ω per cell

– RshRs ~ 33,000 Ω2 per cell

RshRs and R/Q

R R Rsh QR

RGsh s s

Q Q

23



Energy content

Geometrical factor

Power dissipation

2

2 2 2

0 0 1 01J (x )LR2

U E

2

0 1 012 J (x )(R L)RRs

P E

x01

2 (R L)

LG

01

1 01

x 2.40483

J (x ) 0.51915

24



Energy Gain

Gradient

Shunt impedance

Slide 25


Accelerating Cavities

Slide 26


• Beam tubes reduce the electric field on axis

– Gradient decreases

– Peak fields increase

– R/Q decreases

Real Cavities

27


Pill Box to Elliptical Cavities

Slide 28


Single Cell Cavities

Electric field high at iris

Magnetic field high at equator

• Important parameters: Ep/Eacc and Bp/Eacc

• Must minimize the ratios as smaller as possible

29


Single Cell Cavities

Cornell SC 500 MHz

270 Ω

88 Ω/cell

2.5

52 Oe/MV/m

30

1 Oe = 1 G = 0.1 mT


• What is the purpose of the cavity?

• What EM parameters should be optimized to meet the

design specs?

• Beam aperture

• Peak surface field ratios – Ep/Eacc, Bp/Eacc

• Shunt impedance – RshRs

• Higher Order Mode (HOM) extraction

Cell Shape Design

The “perfect” shape does not exist, it all

depends on your application

31


• “High Gradient” shape: lowest Ep/Eacc

• “Low Loss” shape: lowest cryogenic losses RshRs = G(R/Q)

Example: CEBAF Upgrade

32


TM-Cavity Design

• The field emission is not a hard limit in the performance of

sc cavities if the surface preparation is done in the right way

• Unlikely this, magnetic flux on the wall limits performance

of a sc cavity (Q0 decreases or/and quench). Hard limit ~180

mT for Nb

Bpeak / Eacc should be low

1. Cavities may operate at

higher gradients.

2. Cavities may operate at

lower cryogenic load.

V 2 G (R / Q)

Pdiss Rs

acc

33


New Shapes for ILC

TTF

1992

LL

2002/2004

RE

2002

f = 1300 MHz

r iris [mm] 35 30 33

kcc [%] 1.9 1.52 1.8

Epeak/Eacc - 1.98 2.36 2.21

Bpeak/Eacc [mT/(MV/m)] 4.15 3.61 3.76

R/Q [Ω] 113.8 133.7 126.8

G [Ω] 271 284 277

R/Q*G [Ω*Ω] 30840 37970 35123

Slide 34


RF Simulation Codes for Cavity Design

• 2D is fast and allowsto define geometry ofa cylindrical symmetric body(inner

and end-cells)ofthe cavity.

• 3D is much more time consumingbut necessary for modelingof full equipped

cavity with FPC and HOM couplersand if needed to model fabrication errors.

Also couplingstrength for FPC and dampingof HOMs can be modeled only3D.

2 2We need numerical methods:( ) A 0

The solutionto 2D (or 3D) Helmholtzequation can be analyticallyfound onlyfor

very few geometries (pillbox, sphericalresonatorsor rectangular resonator).

Approximating operator

(Finite Difference Methods)Approximating function

(Finite Element Methods)

35


• Free, 2D finite-difference code to design cylindrically

symmetric structures (monopole modes only)

• Use symmetry planes to reduce number of mesh points

SUPERFISH

File di SuperFish Generato da BuildCav F = 1472.6276 MHz

12 14 16

C:\LANLV7\HALFCEBSC.AF 11-27-2006 16:58:22

0

1

2

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4

5

6

7

8

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12

0

1

2

3

4

5

12

11

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9

8

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6

0 2 4 6 8 10

File di SuperFish Generato da BuildCav F = 1472.6276 MHz

2.8 2.9 3 3.1 3.2

C:\LANLV7\HALFCEBSC.AF 11-27-2006 16:58:22

7.30

7.40

7.50

7.60

7.70

7.80

7.90

8.00

8.10

7.30

7.40

7.50

7.60

7.70

7.80

7.90

8.00

8.10

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

http://laacg1.lanl.gov/laacg/serv

ices/download_sf.phtml

36

http://laacg1.lanl.gov/laacg/serv


• Expensive, 3D finite-element code, used to design complex

RF structure

• Runs on PC

• Perfect boundary approximation

CST Microwave Studio

https://www.3ds.com/products-services/simulia/products/cst-studio-suite/

Hexahedral mesh

37


• SLAC, 3D code, high-order Parallel Finite

Element (PFE) method

• Runs on Linux

• Tetrahedral conformal mesh

• High order finite elements (basis order p

=1 – 6)

• Separate software for user interface

(Cubit/Trelis), and visualization and

post processing (ParaView)

Omega3P – ACE3P (SLAC Code Suite)

38


Cell Shape Parametrization• Full parametric model of thecavity in terms

of 7 meaningful geometrical parameters:

Ellipseratio at theequator (R=B/A) ruled by mechanics

Ellipseratio at the iris (r=b/a)Epeak

Side wall inclination( ) and position (d)Epeak vs. Bpeak tradeoffand couplingkcc

Cavity iris radiusR iris

couplingkcc

Half-cell Length L/2=/4

Cavity radiusD

used for frequencytuning

• Behavior of all EM and mechanical properties has been found as a function ofthe above parameters L/2

39


Reducing Peak Surface Fields

Add “electric volume” at the

iris to reduce Epeak

• “Rule of thumb” for Optimizing Peak Surface Fields

Add “magnetic

volume” at the equator

to reduce Bpeak

40


Multi-Cell Cavities

Single-cell is attractive from the RF-point of

view:

• Easier to manage HOM damping

• No field flatness problem

• Input coupler transfersless power

• Easy for cleaningand preparation

• But it is expensive to base even a small

linear accelerator on the single cell. We do

it only for very high beam current

machines.

A multi-cell structure is less expensive and offers

higher real-estategradient but:

• Field flatness (stored energy)in cells

becomes sensitive to frequencyerrors of

individualcells

• Other problemsarise:HOMtrapping…

41


Cell to Cell Coupling

+ +

+ -

Symmetryplane for

the H field

Symmetryplane for

the E field

which is an additional

solution

ωo

ωπ

0

0

kcc

2

The normalized

difference between

these frequencies is a

measure of the energy

flow via the coupling

region

Slide 42


• Cost of accelerators are lower (less auxiliaries: He vessels,

tuners, fundamental power couplers, control electronics)

• Higher real-estate gradient (better fill factor)

• Field flatness vs. N (N – no. of cells)

• HOM trapping vs. N

• Power capability of fundamental power couplers vs. N

• Chemical treatment and final preparation become more

complicated

• The worst performing cell limits whole multi-cell structure

Pros and Cons of Multi-Cell Cavities

43


Multi-Cell Cavities

Mode frequencies:

ω0 – accelerating mode frequency

Voltages in cells:

Cb Ck / 2k

Ck

C

44


Cell Excitations in Pass-Band Modes9 Cell, Mode 1

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 2

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

9 Cell, Mode 3

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 4

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 5

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 6

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

9 Cell, Mode 7

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 8

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 9

-1

- 0.8

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

45


Couplingof TM010 modes of the individual cells via the iris (primarily electric

field) causes them to split into a passband ofclosely spaced modes equal in

number to the numberof cells

Pass-Band Modes Frequencies

0 1 2 3 4 5 6 7 8 9

9-cell cavity

1/2

0 (1 4k )

46

ω0


• Geometrical differences between cells causes a mixing of

the eigenmodes

• Sensitivity to mechanical deformation depends on mode

spacing

Field Flatness in Multi-Cell Cavities

47


Low Beta Accelerating Structures –

TEM Class Cavities

Slide 48


• Two main types of structure geometries

– TEM class (QW, HW, Spoke)

– TM class (Elliptical)

Classification of Structures

49


• Low β cavities: Cavities that accelerateparticleswith β < 1 efficiently

• Increased needsfor reduced-beta (β < 1) SRF cavity especially in CW

machines or high duty pulsed machine (duty > 10%)

• Reduced betaElliptical multi-cell SRF cavity

– For CW, prototypingby several R&D groups have demonstrated

as lowas β=0.47

– For pulsed,SNS β=0.61,0.81 cavities & ESS

• Ellipticalcavity has intrinsicproblemas β goes down

– Mechanicalproblem,multipacting, low rf efficiency

Low β Cavities

50


Applications of Low β Cavities

High Current Medium/Low Current

CW

Pulsed

Acceleratordriven systems

• Waste transmutation

• Energy production

Beam: p, H-, d

Production of radioactive ions

Nuclear Structure

Beam; p to U

Pulsed spallation sources

Beam: p, H-

51


Technical Issues and Challenges

High Current Medium/Low Current

CW

Pulsed

• Beam losses (~ 1 W/m)

• Activation

• High cw rf power

• Higher ordermodes

• Cryogenics losses


• Activation

• High cw rf power


• Cryogenics losses


• Activation


• High peak rf power

• Dynamic Lorentz detuning

52


Design Considerations

• Cavities with high acceptance

• Development of high cw

power couplers

• Extraction of HOM power

• Cavities with high shunt

impedance

Medium/Low Current

• Cavities with low sensitivity to vibration

• Development of microphonicscompensation

• Cavities with high shunt impedance

• Cavities with large velocity acceptance (few cells)

• Cavities with large beam acceptance (low frequency, small frequency transitions)

Note:

Large beam acceptance

• Large aperture (transverse acceptance)

• Low frequency (longitudinal acceptance)

• Cavities with high acceptance

• Development of high peak

power couplers

• Extraction of HOM power

• Development of active

compensation of dynamic

Lorentz detuning

High Current

CW

Pulsed

Slide 53


• Resonant transmission lines

– λ/4

Quarter wave

Split ring

Twin quarter wave

Lollipop

– λ/2

Coaxial half wave

Spoke

H-type

• TM type

– Elliptical

– Reentrant

• Other

– Alvarez

– Slotted iris

Basic Geometries

Slide 54


• Many different shapes and sizes

Low β Cavities

ANL cavities for RIA

SpokeQW HW

Half-wave cavity

RFQ

Slide 55


• The transit time factor is theratio of theacceleration voltage to the

(non‐physical)voltage a particlewith infinite velocity would see

• Energy gain (W):

• Assuming constantvelocity

• Transit time factor:

Transit Time Factor

56


Velocity Acceptance

• Velocity acceptance

• Lower thevelocity of the

particleor cavity β

o Faster the velocity of the

particlewill change

o Narrower the velocity range of

a particularcavity

o Smaller the numberof cavities

of that β

o More important:Particle

achieve design velocity

57

Velocity acceptance for sinusoidal field profile


QWR A capactively loaded λ/4 transmission line

Maximum voltage builds up on the open tip and

maximum current at the plate connecting the center

conductor

Beam tube is placed near the end of the tip to

produce a high voltage double gap acceleration

geometry

Quarter Wave Resonator (QWR)

58


Quarter Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β

Vp – voltage on the center conductor with outer conductor at ground

Peak Magnetic Field

Geometrical Factor Energy Content

59


Quarter Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β


Shunt Impedance

Power Dissipation (Ignore losses in the shorting

end plate)

R/Q

60


Half–Wave Resonator

61

2b

2a

L

E

B


Half Wave Resonator (HWR)HWR A λ/2 transmission line

Equivalent to 2 QWR facing each other and connected

Magnetic field loop around the inner conductor with peak

fields at the shorted ends

Beam tube is placed at the center of the inner conductor to

coincide with the maximum voltage

Same accelerating voltage is obtained at about 2 times larger

power in QWR (PHWR ~ 2PQWR)

62


Half Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β


Geometrical Factor

Peak Magnetic Field

Energy Content

63


Half Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β


Shunt Impedance

Power Dissipation (Ignore losses in the shorting end plates)

R/Q

64


There have been extensive efforts for design optimization especiallyto reduce

the ratios of

Ep/Eacc and Bp/Eacc

• ControllingA/B (Ep/Eacc)and C/D (Bp/Eacc) Shapeoptimization

• Flat contactingsurface at spoke base will also help in minimization of Bp/Eacc

• For thesecavities:

– Calculationsagree wellEp/Eacc~3,Bp/Eacc ~(7~8)mT/(MV/m)

– Though it is tricky to obtainprecise surface field information from the

3D simulation

Spoke Resonators

325 MHz, β=0.17 (FNAL)

65


• Ep/Ea ~ 3.3, independent of β

• Bp/Ea ~ 8 mT/(MV/m), independent of β

Spoke Resonators

Surface electric field Surface magnetic field

Lines: Elliptical Squares:Spoke

66


• QRs ~ 200 β [Ω]

• Rsh/Q ~ 205 [Ω] independent of

Spoke Resonators


67


•

• U/E2 ~ 200 β2

Rsh Rs ~ 40000 β [Ω2]

[mJ]

Spoke Resonators

Rsh Rs U/E2


0

5000

10000

15000

20000

25000

30000

35000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Beta

Rs

hR

sp

er

ce

ll

2

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Beta

U/E

2p

er

ce

ll(m

J)

@1

MV

/m,

500

MH

z

68


Features of Spoke Cavities

• Small Size

– About half of TM cavity of same

frequency

• Allows low frequency at reasonable

size

– Possibility of 4.2 K operation

– High longitudinal acceptance

• Fewer number of cells

– Wider velocity acceptance

350 MHz, =

0.45

69


Deflecting / Crabbing Cavities

70


Tail of the bunch

deflected down

• Both deflectingand crabbingresonantcavities are required to generate a

transverse momentum

• Can be produced by eitheror by both transverse electric (E t) and magnetic

(Bt) fields

• Lorentzforce:

• Types of designs:

– TM-typedesignsMain contributionfromBt

– TE-typedesignsMain contribution fromE t

– TEM-typedesigns Contributionfromboth E t and B t

Deflecting/Crabbing ConceptComplete bunch is deflected

Head of the bunch

deflected up

71


Deflecting Cavities

f0 = 1497 MHz

fd = 499 MHz

• RF frequency(fd) = 499 MHz

• Beam energy (E) = 11.023GeV

• Deflecting angle (θ)

• Transverse voltage (Vt)

θ

E0

Beam direction

eVT

• Completebunch is deflected with the

transverse kick applied at the centerof the

bunch

72


• Luminosity with no crabbing

system:

• Luminosity with crabbing

system:

– For head-on collision

• Transverse voltage

Crabbing Cavities

crossing angle

Crabbing

Cavity

crossing angle

Head-on collision

73


• For particlesmoving virtually at v=c, the integrated transverse force

(kick) can be determined from the transverse variation of the integrated

longitudinal force

• Accordingto the theorem:

– In a pureTE modethe contribution to thedeflection fromthe

magnetic field is completelycancelled by thecontributionfromthe

electric field

Panofsky–Wenzel Theorem

c

• Transverse momentumis related to the gradient of the longitudinal

electric field alongthebeam axis

j Ft t Fz

74


Deflecting/Crabbing CavitiesKEK TM110 crabbingcavity Superconducting4-rod cavity

BNL doublequarter-wavecavity

75


Deflecting/Crabbing Cavities

76

RF-Dipole Cavity


LHC High Luminosity Upgrade

77

SPS Cryomodule with 2 DQW crab cavities –

Demonstrated crabbing of first proton beam


Cavity Limitations

78


Cavity Performance

79


Multipacting

80

• Multipacting condition – A large amount of secondary

electrons are emitted from the cavity surface by the

incident primary electrons


• Resonant condition

– The secondaryelectronshave localized and

sustainableresonanttrajectorieswith thecavity rf

fields

– Impact energies correspondsto a secondary

emission yield (SEY) greater than one

Multipacting

δmax

EIEmax EII

Impact Energy [eV]

Sec

ond

ary

Em

issi

on

Yie

ld(δ

)

EI 150eV EII 2000eVFor Nb:

81


Field Emission• Characterized by exponential decayin Q0

• Exponential increaseof losses due to acceleration ofelectron field

emission

• With the increasingrf field the field emitters generate an electron current

leadingto excessive heatingand x-rays producedby bremsstrahlung

• It is a general difficulty in acceleratingstructures,but doesnot present an

ultimate fundamental limit to the maximum surface electric field

• Main cause ofFE is particulatecontamination

Temperature map shows line

heating along the longitude at

the location of the emitter

82


• Fowlerand Nordheim(FN) showed that when thework function barrier

at themetal surface is lowered by an applied surface electric field,

electronscan tunnelthrough

• Electrostaticpotential of the metal-vacuuminterface

• Tunnelingcurrentdensity

Field Emission

No electric field applied Electric field applied

83


• Modified Fowler Nordheimequationfor rf fields

• Field emitters

Field Emission

• FE can be prevented by propersurface

preparation and contaminationcontrol

• Possible to reduce if not completely

eliminateFE using CW RF

processing, High-power Pulsed

Processing(HPP) and/orHelium

processing

84


• Ponderomotive effects: changes in frequency caused by the

electromagnetic field (radiation pressure)

– Static Lorentz detuning (CW operation)

– Dynamic Lorentz detuning (pulsed operation)

• Microphonics: changes in frequency caused by connections

to the external world

– Vibrations

– Pressure fluctuations

• Note: The two are not completely independent. When phase

and amplitude feedbacks are active, the ponderomotive

effects can change the response to external disturbances

Ponderomotive Effects

85


Lorentz Force Detuning

-0.006

020

z [mm]40

60

P[N

/mm

^2

]

• Electromagnetic fields in a cavity exert Lorentzforces on the cavity wall. The

force per unit area (radiation pressure)is given by

92 kA/m

50 MV/m0.003

-0.003

0

E and H at Eacc =

25 MV/m in

TESLAinner-cup

• Residual deformationof the cavity shapeshifts the

resonant frequency

kL – Lorentzcoefficient

86


• Localized heating

• Thermal breakdown occurswhen theheat generated at thehot spot is

larger than that can be evacuated to the heliumbath

• Both the thermal conductivityand thesurface resistanceof Nb are

highly temperaturedependentbetween2 and 9K

Thermal Breakdown

87


Losses in RF Cavities

88


• Losses are given by Ohm’s Law where σ is the

conductivity

• In a cavity, rf magnetic field drives an oscillating current in

the cavity wall

• Following Maxwell’s equations

• Neglecting displacement current

Losses in Normal Conducting Cavities

89


• Considering the cavity as wall as a local plane surface, solve

one dimensional problem at the surface for uniform magnetic

field in y direction

• with field decaying into the conductor over the skin depth


90


• From Maxwell equation

– A small tangential electric field component decays into

the conductor

• Surface impedance

• Surface resistance is the real part of surface impedance


91


Anomalous Skin Effect

• Surface resistance becomes independent of the conductivity at low

temperatures

• As the temperature decreases, the conductivity σ increases− Skin depth decreases

− Skin depth (the distance over which fields vary) can become less then the

mean free path of the electrons (the distance they travel before being

scattered)

− Electrons do not experience a constant electric field over a mean free path

− Local relationship between field and current is not valid

Slide 92


Anomalous Skin Effect• Theory introduces a dimensionless parameter αs, which depends on the

temperature-independent product of the mean free path l and the resistivity ρ = 1/σ

• αs strongly depends on mean free path l

Slide 93

• Classical expression for the surface resistivity

is valid when a αs ≤ 0.016. In the anomalous

limit, when αs ∞

• For intermediate values


• Superconductivity – Discovered in 1911 Kammerlingh-Onnes, is

a phenomenon where below a certain temperature, called the

critical temperature (Tc), some materials show a sudden drop of

the dc electrical resistance to zero

Superconductivity

Kamerlingh Onnes and van der Waals in

Leiden with the helium 'liquefactor' (1908)

94


Meissner Effect

Meissner Effect – Discovered by Meissner and Ochsenfeld in 1933

• Ability to completelyexpel an externallyappliedmagnetic field when cooled

down belowthecritical temperature(Tc)

• Superconductorbehaves as a perfect diamagnet

• Surface currentscreated at the

surface generates a magnetic field

that cancelsthe external magnetic

field inside the superconductor

• These surface currentsdo not

decay with time dueto thezero

resistance in thesuperconductor

95


• Critical magnetic field (Bc) – Field beyond at which

superconductivity is destroyed

Critical Magnetic Field

Type I – Soft Superconductors Type II – Hard Superconductors

96

Tc – critical temperature


Surface Resistance of Superconductors

• Superconductors are free of power dissipation at static fields

• In microwave fields with the time dependent magnetic field in

the penetration depth will generate an electric field

• Electric field will induce oscillations in the normal electrons,

which will lead to power dissipation

Slide 97


• Macroscopic theory of superconductivity

• Coexistence of:

– super electrons (ns)

– normal electrons (nn)

– Total density

• Only super electrons are accelerated by the

constant electric field (E)

• Super current density

• Super electrons are not affected by the normal

electrons

Two Fluid Model

98


• Using Maxwell’s equations

• Field penetration in the superconductor

• London penetration depth

Distance to which a magnetic field

penetrates into a superconductor

London Equations

99


Fundamental Lengths

• London penetration depth (λL)

– Distance over which magnetic fields decay in superconductors

• Pippard coherence length (ξ0)

– Distance over which the superconducting state decays

100

• Type II

superconductors –

λL >> ξ0

• Type I

superconductors –

ξ0 >> λL


Material Parameters

101


• Bardeen-Cooper-Schrieffer Theory (1957) – Nobel prize in 1972

• Macroscopical Microscopical representation

BCS Theory

102


• Cooper pairs – Pair of electrons formed due to electron-phonon

interaction that dominates over the repulsive Coulomb force

• Moving electron distorts the lattice and leaves behind a trail of

positive charge that attracts another electron moving in

opposite direction

• Has lower energy than the two separate electrons

• Therefore, electron pairs form bound states of lower energy

which are stable than the Fermi ground state

• Strong overlap of many Cooper pairs

results in the macroscopic phase coherence

Cooper Pairs

103


Energy Gap

• Energy gap (Δ) – gap around Fermi level between ground

state and excited state• At 0 < T < Tc not all electrons are

bounded in to Cooper pairs

• Density of unpaired electrons is given

by

Normal

conductorSuperconductor

104


• At the presence of an rf field

– Cooper pairs move without resistance Do not

dissipate power

– Due to inertial mas of Cooper pairs they cannot follow an AC

electromagnetic fields instantly and do not shield it perfectly

– Remaining residual field accelerates the normal electrons that

dissipate power

• More normal electrons The material is more lossy

– Losses decrease with temperature below Tc

• Faster the field oscillates the less perfect the shielding

– Losses increase with frequency

Losses in Superconductor

105


Surface Impedance

• Following two fluid model

• Total current density

• Surface impedance:

• Skin depth

106


BCS Surface Resistance

• Surface impedance

• Taking may parameters into

account Mattis and Bardeen

developed theory based on

BCS theory

107

A – material parameters


• Surface resistance of superconductors (Rs)

• Residual resistance (Rres) due to:

– Dielectric surface contaminants (gases, chemical residues, ..)

– Normal conducting defects, inclusions

– Surface imperfections (cracks, scratches, ..)

– Trapped flux during cool down through critical temperature

– Hydrogen absorption during chemical processing

• An approximation of RBCS for Nb:

Surface Resistance

108

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