te cyl cornel university

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 zrrzz =

)sin()(')(101 tzkJkrExE zrrzzr =

0)( =xE 0)( =xBr

)sin()(')(12 01 tzkJrExB zrrcz =


inRS

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

R)(

)(

2222

222

zczzki

zzzki

EBkB

BEkE

zc

zc

=

+=

( )( )

( )( )

)cos()()cos()(

)sin()('0

)sin()(')cos()(

0

10

12

10

2

10

2

10

12

2

2

tzknrJBBtzknrJRnkBBikB

tzknrJRkBBikBE

tzknrJRBBiE

tzknrJRnBBiE

zRS

nz

zRS

nrR

SzzrSR

z

zRS

nSzzrSR

zr

z

zRS

nSzrSR

zRS

nrR

SzrSR

r

nm

nm

nmnm

nmnmnm

nmnmnm

nm

nmnm

+=

+==

+==

=

+==

+==


0 1 2 3 4 5 6-1

-0.5

0

0.5

1

E

B

Bx

z


0)(,)()( 0 == xBerJExE zinRZnz nm

R)(

)(

2222

222

zczzki

zzzki

EBkB

BEkE

zc

zc

=

+=

0)sin()('

)cos()()cos()(

)cos()()sin()('

2

2

0

10

10

10

1

0

=

+==

+==

+=

+==

+==

z

zRZ

nczrZR

zRZ

nrR

ZczrZR

r

zRZ

nz

zRZ

nrR

ZzzrZRk

zRZ

nzzrZRk

r

BtzknrJEEiB

tzknrJnEEiBtzknrJEE

tzknrJnkEEiEtzknrJkEEiE

nmnm

nmnmnm

nm

nmnmnm

z

nmnmz


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

00.5

1 -1

-0.5

0

0.5

1

01000200030004000

-1-0.5

00.5

Energy density for TM mode 2,2

-1-0.5

00.5

1 -1

-0.5

0

0.5

1

02000400060008000

-1-0.5

00.5

Energy density for TM mode 2,2

-1-0.5

00.5

1 0

2

4

6

02000400060008000

-1-0.5

00.5

TEU TMU 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 (19 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

gradient (820 MV/m)

Relaxed HOM damping requirements

Low average RF power (513 kW)

CEBAF cavities

ELBE cryomodule


SRF cavities Moderate

gradient (1520 MV/m)

Strong HOM damping (Q = 102104)

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

operation

koperation


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

p=integer

d

pdz=

pdkz2

=


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.

p=2

p=3

p=4

ccsh PVR

2=


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

=

TE TM

TM010

0,0 == HE nn

0122 =

H

Etc

00010

10

00

,405.2

405.2

405.2

==

=

=

Rc

eRrJEiH

eRrJEE

ti

tiz


Higher Frequency (Order) Standing Wave Modes

TMmnp (TEmnp),m, n, p Ez (Hz) , r, z

TM010


T

T = 2/ Eacc Eacc = Vc/d d = /2

( )

== dzezEV czizc 0,0

TdE

cdcd

dEdzeEVd

czic =

== 00

0

00

0

2

2sin0

dEVdTttT centerexittransit 00222 ====

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 dvdvU2

02

0 21

21 EH

( )2221 HE += u


Quality factor

2 ~104 ~1010

( )0

0000

00012losspower average

energy stored

====

cc PU

TPUQ

=

Ss

VdsR

dvQ 2

2000 H

H

-1000 -500 0 500 10000

1

cavit

y fie

ld [a

rb. u

nits

]

Frequency 1.3 GHz [Hz]

Bandwidth

(1) The RF system has a resonant frequency

(2) The resonance curve has a characteristic width

0

Q20 =

Geometry factor

G

G = 257Ohm

sRGQ =0

=

S

Vds

dvG 2

200

H

H


Shunt impedance and R/Q

Pc

R/Q = 196 Ohm

ccsh PVR 2

2=

ccsh PVR

2=

caccsh P

Er4

=2

UV

QR csh

0

2

0 =


GR/Q

)/()/)(()/( 02

00

2

00

22

QRGRV

RQRQRV

QRQV

RVP

shsc

sshsc

shc

shcc

=

=

==


270 88 /cell

2.552 Oe/(MV/m)

Cornell SC 500 MHz

Hpk Epk R/Q

Pillbox vs. real life cavity


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