147 Superconducting Cavities - Cornell University

[email protected] Class Phys 488/688 Cornell University 03/14/2008

CHESS & LEPPCHESS & LEPPCHESS & LEPPCHESS & LEPPCHESS & LEPPCHESS & LEPPCHESS & LEPPCHESS & LEPP

147

Normal conducting cavitiesNormal conducting cavitiesNormal conducting cavitiesNormal conducting cavities• Significant wall losses.Significant wall losses.Significant wall losses.Significant wall losses.• Cannot operate continuously with Cannot operate continuously with Cannot operate continuously with Cannot operate continuously with

appreciable fields.appreciable fields.appreciable fields.appreciable fields.• Energy recovery was therefore not Energy recovery was therefore not Energy recovery was therefore not Energy recovery was therefore not

possible.possible.possible.possible.

• Very low wall losses.Very low wall losses.Very low wall losses.Very low wall losses.• Therefore continuous operation Therefore continuous operation Therefore continuous operation Therefore continuous operation

is possible. is possible. is possible. is possible.

• Energy recovery becomes Energy recovery becomes Energy recovery becomes Energy recovery becomes possible.possible.possible.possible.

Q = 1010

E = 20MV/m

A bell with this Q

would ring for a year.

Superconducting CavitiesSuperconducting CavitiesSuperconducting CavitiesSuperconducting Cavities



148

The filed in many cells can be excited by a single power source and a single inputcoupler in order to have the voltage of several cavities available.

Without the walls: Long single cavity with too large wave velocity.

Thick walls: shield the particles from regions with decelerating phase.k

vω=ph

zk

ω

phvc

MV12.2kW125

100.18R

MHz5006

s

→Ω=

=resfExample: PETRA Cavity

MulticellMulticellMulticellMulticell standingstandingstandingstanding----wave cavitieswave cavitieswave cavitieswave cavities



149

The iris size is chosen to let the phase velocity equal the particle velocity.

Long initial settling or filling time,not good for pulsed operation.

Small shunt impedance per length.

Common compromise.

λ

Modes in WaveguidesModes in WaveguidesModes in WaveguidesModes in Waveguides



150

The iris size is chosen to let the phase velocity equal the particle velocity.

Standing wave cavity. Traveling wave cavity (wave guide).modekLoss free propagation: dnk π2=

d

Disc Loaded WaveguidesDisc Loaded WaveguidesDisc Loaded WaveguidesDisc Loaded Waveguides



151

(1) Linearization:

Transport maps of cavitiesTransport maps of cavitiesTransport maps of cavitiesTransport maps of cavities

0),,0(),,( 21 =⋅∇⇒∂−= EtzEtzrE zzr

r

rr

(2) Equation of motion:

]'''[]))1(([

])([)('

002111

2

21

02

2

0

200

0

appraEE

aEEaFFa

a

pzztcv

vdzdr

vpq

zztcv

zr

vpq

zxvp

ppx

+−≈+∂−−−=

+∂+∂−=−=

=

EBtzEtzrB tcztr

c

rrr∂=×∇⇒∂= 22

12

11 ),,0(),,(ϕ

( )22'

4

'

2

'''211

2'

)()'''(''

'

3

2

pp

p

p

p

ppp

p

pp

urpaappru

prpau

pru

−=−+++−≈

+=

=Focusing !Focusing !Focusing !Focusing !

p denotes pp denotes pp denotes pp denotes p0000 for simplicityfor simplicityfor simplicityfor simplicity

High energy approximation gggg2<<1:



152

Transport maps of cavitiesTransport maps of cavitiesTransport maps of cavitiesTransport maps of cavities

(3) Average focusing over one period with relatively little energy change:

24/ ','' 2

2

puup

=∆−≈ ∆

(4) Continuous energy change:

',' pp =ΩΩ≈

( )2

2

22

2

4

/1 ''pdp

d uuu Ω∆Ω

−≈≈

( )

( )

2

2

2222

2

2

2

2

2

2

2

3

2

32

2

2

2

''''))ln(()(

)(

111

4

1/1

4

/

4

11

ppppdpd

pppdpd

dpd

pppdpd

dpd

dpd

rppr

rrrr

rrrprpr

ε

ε

ηηηηη −=+=⇒−=

−=−≈+

−≈−+=

−Ω∆

Ω∆



153

Transport maps of traveling wave cavitiesTransport maps of traveling wave cavitiesTransport maps of traveling wave cavitiesTransport maps of traveling wave cavities

−

=

−

=

−=−=

i

i

pp

pp

pp

pp

pp

pp

pp

a

r

a

r

B

A

pp

pp

a

r

BA

i

ii

ii

'1'

'

2

0

01

))ln(cos())ln(sin(

))ln(sin())ln(cos(

0

01

))ln(cos())ln(sin(

))ln(sin())ln(cos(

0

01

)sin()cos()(,''

εεεεε

ε

εεεε

ε

εξεξξηηεη

( ) 1)(cos',)cos('

0,)cos()cos(

)cos()cos(

0

2

21

21

02

0

2212

00

000

2

00

2

0−=

==

=+=

+−+=

∑∑

∑

∑

∞

=

∞

=

∞

=

∞

=

ngg

nn

nnLn

nnLnz

ngpgp

zng

kztzngE

εϕϕ

αϕα

ϕωα

π

π



154

Transport maps of standing wave cavitiesTransport maps of standing wave cavitiesTransport maps of standing wave cavitiesTransport maps of standing wave cavities

−

=

i

i

pp

pp

pp

pp

pp

pp

a

r

a

ri

ii

ii

'1' 0

01

))ln(cos())ln(sin(

))ln(sin())ln(cos(

0

01

εεεεε

ε

1'

,'

)]cos(2[

][

,,,)cos()cos(

0

2

21

0

22

0

041

)()(

41

*0

0

2200

02

02

02

02

2

−=

=

=

=++=

+++=

===+=

∑∑

∑∑

∑

∑

∞

=

∞

=

∞

−∞=

∞

−∞=

−+−

∞

−∞=

−+−−++

−

∞

−∞=

nff

nn

zin

nn

zin

nn

imn

imn

n

zinzinzmnizmnin

nnLn

zinnz

n

LL

LLLL

L

fp

fp

efegegeg

eeeeg

ggmkkzttkzegE

ε

ϕ

ωϕω

ππ

ππππ

π

ϕϕ

ϕϕϕϕ

π



155

Phase space Phase space Phase space Phase space precervationprecervationprecervationprecervation in cavitiesin cavitiesin cavitiesin cavities

−

=

i

i

M

pp

pp

pp

pp

pp

pp

a

r

a

ri

ii

ii

444444444 3444444444 21'

1' 0

01

))ln(cos())ln(sin(

))ln(sin())ln(cos(

0

01

εεεεε

ε

ppiM =)det(

Average focusing over one period with relatively little energy change:

Because the determinant is not 1, the phase space volume is no longer conserved but changes by p0/p.A new propagation and definition of Twiss parameters is therefore needed:

)sin(2 01 φψβγβ +=

rrJr



156

TwissTwissTwissTwiss parameters in accelerating cavitiesparameters in accelerating cavitiesparameters in accelerating cavitiesparameters in accelerating cavities

( )

( )[ ]

++

−=

+=+

==⇒

++

+−+−=

++

−+

−+−+−=

++−≈

+++−==

=−=

+

+

+

+

)cos(

)sin(02

'

1:choice,'

)cos(

)sin(

'

)()(2

)cos(

)sin(

''''2

'2'

]')''(['

)]cos()sin([2'

,'

0

0

12

2'

0

0'

'2

'''21

0

0

'4

'32

''''

211

0'

02

2

121

2'

2

222

'

2

φψφψβ

βγα

βψ

φψφψ

ψββαβ

φψφψ

βψψβαψββαα

φψφψ

γβα

ββ

βα

β

βαβψ

β

ββψ

β

βα

βα

pp

pp

pmc

pp

pp

pp

pp

pp

pmc

pp

p

pp

ppp

pmc

p

pmc

Ja

r

K

AA

KJ

Ja

apppKra

Jra



157

Beta functions in accelerating cavitiesBeta functions in accelerating cavitiesBeta functions in accelerating cavitiesBeta functions in accelerating cavities

For systems with changing energy one uses the normalized Courant-Snyder invariant JJJJnnnn = J = J = J = J bbbbrrrr ggggr

pp

pmc

nJa

r2

'

0

0

1~~,

)sin(

)cos(02 βαα

φψφψβ

ββα +=

++

−=

( ) ( ) nmcp

mcp J

a

r

ara

r

ar=

=

+

2

~1

2~

1~1

~

~0

0

2

βαα

βββα

βαββ

αβ

&&&

Reasons:Reasons:Reasons:Reasons:(1)(1)(1)(1) JJJJ is the phase space amplitude of a particle in (x , a)(x , a)(x , a)(x , a) phase space, which is

the area in phase space (over 2p) that its coordinate would circumscribe during many turns in a ring. However, a=pa=pa=pa=pxxxx/p/p/p/p0000 is not conserved when p0 changes in a cavity. Therefore J is not conserved.

(2)(2)(2)(2) JJJJnnnn = J p= J p= J p= J p0000/mc/mc/mc/mc is therefore proportional to the corresponding area in (x , (x , (x , (x , ppppxxxx))))phase space, and is thus conserved.



158

The The The The normaizednormaizednormaizednormaized emittanceemittanceemittanceemittance

Remarks:Remarks:Remarks:Remarks:(1) The phase space area that a beam fills in (x , a) phase space shrinks during

acceleration by the factor pi/p. This area is the emittance e.(2) The phase space area that a beam fills in (x , px) phase space is conserved.

This area (divided by mc) is the normalized emittance en.

rr

nγβ

εε =

x

ax 1'=

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147 Superconducting Cavities - Cornell University