cornell universityliepe/webpage/docs/p4456l19.pdfenergy density for te mode 2,2-1-0.5 0 0.5 1-1-0.5...

Matthias Liepe, P4456/7656, Spring 2010, Cornell University Slide 1

5. RF Systems and Particle Acceleration5.1 Waveguides

5.1.3 Cylindrical Waveguides5.2 Accelerating RF Cavities

5.2.1 Introduction5.2.2 Traveling wave cavity: disk loaded

waveguide5.2.3 Standing wave cavities5.2.4 Higher-Order-Modes5.2.5 The pillbox cavity5.2.6 SRF primer


Mode for particle acceleration: TM01 )cos()()(00 tzkJExE zrr

zz ω−=

)sin()(')(101 tzkJkrExE zrr

zzr ω−−=

0)( =xE ϕ

0)( =xBr

)sin()(')(12 01 tzkJrExB zrr

cz ωωϕ −−=


ϕinRS

nzz erJBxBxE nm )()(,0)( 0==

R)(

)(

2222

222

zczzki

zzzki

EBkB

BEkE

zc

zc

×∇−∇=

×∇+∇=

⊥⊥−⊥

⊥⊥−⊥

ωω

ωω

( )( )

( )( )

)cos()(

)cos()(

)sin()('

0

)sin()('

)cos()(

0

10

12

10

2

10

2

10

12

2

2

tzknrJBB

tzknrJRnkBBikB

tzknrJRkBBikB

E

tzknrJRBBiE

tzknrJRnBBiE

zRS

nz

zRS

nrR

SzzrSR

z

zRS

nSzzrSR

zr

z

zRS

nSzrSR

zRS

nrR

SzrSR

r

nm

nm

nmnm

nm

nmnm

nm

nmnm

nm

nmnm

ωϕ

ωϕ

ωϕ

ωϕωω

ωϕωω

ϕϕ

ϕ

ϕ

−+=

−+−=∂=

−+−=∂=

=

−+=∂−=

−+−=∂=


0 1 2 3 4 5 6

-1

-0.5

0

0.5

1

E

B

Bx

z


0)(,)()( 0 == xBerJExE zin

RZ

nznm

ϕ

R)(

)(

2222

222

zczzki

zzzki

EBkB

BEkE

zc

zc

×∇−∇=

×∇+∇=

⊥⊥−⊥

⊥⊥−⊥

ωω

ωω

0

)sin()('

)cos()(

)cos()(

)cos()(

)sin()('

2

2

0

10

1

0

10

1

0

=

−+−=∂=

−+=∂−=

−+=

−+−=∂=

−+−=∂=

z

zRZ

nczrZR

zRZ

nrR

ZczrZR

r

zRZ

nz

zRZ

nrR

ZzzrZRk

zRZ

nzzrZRk

r

B

tzknrJEEiB

tzknrJnEEiB

tzknrJEE

tzknrJnkEEiE

tzknrJkEEiE

nm

nm

nm

nmnm

nm

nm

nmnm

z

nm

nm

z

ωϕ

ωϕ

ωϕ

ωϕ

ωϕ

ωωϕ

ωϕ

ω

ϕϕ


0 1 2 3 4 5 6

-1

-0.5

0

0.5

1

E

B

Bx

z


0 1 2 3 4 5 6

-1

-0.5

0

0.5

1

TES

TMS

TESx

z


Energy density for TE mode 2,2

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

010002000

3000

4000

-1

-0.5

0

0.5

Energy density for TM mode 2,2

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

020004000

6000

8000

-1

-0.5

0

0.5

Energy density for TM mode 2,2

-1-0.5

00.5

1 0

2

4

6

02000400060008000

-1-0.5

00.5

TEUTMU TMU

x x

xy y z

x




• Use high DC voltage to accelerate particles

• No work done by magnetic fields

Cockroft and Walton's electrostatic accelerator (1932)

Protons were accelerated and slammed into lithium atoms

producing helium and energy.

DC Accelerators:

Use time-varying fields!


Taiwan Light Source cryomodule

RF Cavities

• The main purpose of using RF cavities in accelerators is to provide energy gain to charged-particle beams

• The highest achievable gradient, however, is not always optimal for an accelerator. There are other factors (both machine-dependent and technology-dependent) that determine operating gradient of RF cavities and influence the cavity design, such as accelerator cost optimization, maximum power through an input coupler, necessity to extract HOM power, etc.

• In many cases requirements are competing.


• NC or SC• Relatively low

gradient (1…9 MV/m)

• Strong HOM damping (Q ~ 102)

• High average RF power (hundreds of kW)

CESR cavities

KEK cavity

PEP II Cavity


• NC or SC• High gradients• Moderate HOM

damping reqs. • High peak RF power

ILC: 21,000 cavities!ILC / XFEL cavities


• SRF cavities• Moderate to low


• Relaxed HOM damping requirements

• Low average RF power (5…13 kW)

CEBAF cavities

ELBE cryomodule


• SRF cavities• Moderate


• Strong HOM damping (Q = 102…104)

• Low average RF power (few kW)

Cornell ERL cavities

BNL ERL cavity



Standing wave cavity. Traveling wave cavity (wave guide).


• Cylindrical Waveguide: TM01 has longitudinal electric field and could in principle be used for particle acceleration

• But: phase velocity of wave > c > speed of particle-> no average energy transfer to beam !

• Solution: Disc Loaded Waveguide• Iris shaped plates at constant separation in waveguide lower phase

velocity• Iris size is chosen to make the phase velocity equal the particle

velocity


Vgroup = Vparticle

ωoperation

koperation


• Irises form periodic structure in waveguide-> Irises reflect part of wave-> Interference-> For loss free propagation: need disk spacing d

p=integer

d

pd zλ=

pdkz

π2=

π


Selection of integer “p”:

Long initial settling or filling time,not good for pulsed operation with very short pulses.

Small shunt impedance per length (shunt impedance determines how much acceleration a particle can get for a given power dissipation in a cavity).

Common compromise.

λ

p=2

p=3

p=4

c

csh P

VR2

=


• Time dependent electromagnetic field inside metal box

• Energy oscillates between electric and magnetic field!




500MHz Cavity of DORIS:GHz4967.01.23 )(

010 == Mfcmr

l The frequency is increased and tuned bya tuning plunger.

l An inductive coupling loop excites themagnetic field at the equator of the cavity.


LC1

=ω

TE TM

TM010

0ˆ,0ˆ =⋅=× HE nn

01 22 =

∂

∂−∇

HE

tc

0

0010

10

00

,405.2

405.2

405.2

εµ

ηω

ηω

ϕ

ω

==

−=

=

Rc

eRrJEiH

eRrJEE

ti

tiz


Higher Frequency (Order) Standing Wave Modes

TMmnp (TEmnp),m, n, p Ez (Hz) ϕ, r, z

TM010


T

T = 2/π Eacc Eacc = Vc/d

d = βλ/2

( )∞

∞−

== dzezEV czizc

βωρ 0,0

TdE

cdcd

dEdzeEVd

czic ⋅=

== 00

0

00

0

2

2sin

0

βω

βω

βω

dEVdTttT centerexittransit 00 222 π

βλ ===−=ωωωω t

E z


• Surface currents (∝ H) result in dissipation proportional to the surfaceresistance (Rs):

• Dissipation in the cavity wall given bysurface integral:

Energy density in electromagnetic field:

Because of the sinusoidal time dependence and 90° phase shift, the energy oscillates back and forth between the electric and magnetic field. The stored energy in a cavity is given by

==VV

dvdvU 20

20 2

121 EH εµ

( )2221 HE ⋅+⋅= µεu


Quality factor•

2π ~104 ~1010

( )0

000

0

000

12losspower average

energy storedωω

τωπωω

Δ====

⋅≡

cc PU

TPUQ

=

Ss

V

dsR

dvQ 2

200

0H

Hµω

-1000 -500 0 500 10000

1

cavi

ty fi

eld

[arb

. uni

ts]

Frequency – 1.3 GHz [Hz]

Bandwidth

(1) The RF system has a resonant frequency

(2) The resonance curve has a characteristic width

0ω

Q20ωω =Δ

Geometry factor

G

G = 257Ohm

sRGQ =0

=

S

V

ds

dvG 2

200

H

Hµω


Shunt impedance and R/Q

Pc

R/Q = 196 Ohm

c

csh P

VR2

2=

c

csh P

VR2

=

c

accsh P

Er4

=2

UV

QR csh

0

2

0 ω=


G⋅R/Q

)/()/)(()/( 0

2

00

2

00

22

QRGRV

RQRQRV

QRQV

RVP

sh

sc

sshs

c

sh

c

sh

cc ⋅

⋅=

⋅=

⋅==


270 88 /cell

2.552 Oe/(MV/m)

Cornell SC 500 MHz

Hpk Epk R/Q

Pillbox vs. “real life” cavity


cornell universityliepe/webpage/docs/p4456l19.pdfenergy density for te mode 2,2-1-0.5 0 0.5 1-1-0.5...

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