Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 1 Accelerator Physics...

J. J. Bisognano

Topic One: Acceleration

1

UW Spring 2008

Accelerator Physics

Accelerator PhysicsTopic I

Acceleration

Joseph Bisognano

Synchrotron Radiation Center

University of Wisconsin

J. J. Bisognano


2

UW Spring 2008

Accelerator Physics

Relativity

vFdt

dE

dt

pdF

mcE

vmp

cv

cv

;

1;

1 2

2

2

2

2

J. J. Bisognano


3

UW Spring 2008

Accelerator Physics

Maxwell’s Equations

0

0

0

)(

tJ

Jt

EB

t

BE

B

E

BvEqF

ooo

o

J. J. Bisognano


4

UW Spring 2008

Accelerator Physics

Vector Identity Games

dvHE

tdaHE

t

BH

t

DE

EHHEHE

EHEDLet

o

V

o

A

oo

)22

()(

)()(

)()()(

;

22

Poynting Vector Electromagnetic Energy

J. J. Bisognano


5

UW Spring 2008

Accelerator Physics

Propagation in Conductors

t

H

t

EH

t

E

t

EE

t

E

t

EE

haveweJt

EB

t

BE

From

assumeEJDensityCurrent

2

22

2

22

2

20)(

0

0,

J. J. Bisognano


6

UW Spring 2008

Accelerator Physics

Free Space Propagation

matchtdonvelocitiesandonacceleratiforwardNo

wavestransverseEkEE

cdk

dvc

kvk

withsolveseEELet

EquationWavet

EE

SpaceFree

ozoz

groupphase

kztjo

'

00

;1

;1

0

)(

2

22

J. J. Bisognano


7

UW Spring 2008

Accelerator Physics

Conductive Propagation

RoIR

o

o

Cu

kdepthskinkkk

jkjk

metermhoHz

numbersTypical

Ejz

E

tjwith

t

E

t

E

z

E

/12;2

)()(

11085.8

/108.5;10

)1(

)exp(

222

12

710

22

2

2

2

2

2

J. J. Bisognano


8

UW Spring 2008

Accelerator Physics

Boundary Conditions

2.

4

;

0&

0,

0)(0

2tan

tan4/

tan

surface

j

outsurfaceout

conductor

outcondout

Rresistsurface

Hda

dPlosspower

HeE

ConductorGood

BnKBn

BfieldsACFor

EnEEninsideE

conductorsPerfect

J. J. Bisognano


9

UW Spring 2008

Accelerator Physics

AC Resistance

!!!SHORT!GETTING ARE BUNCHES BUT

sec103;)(

;)(

)(

1132

0

00

gigahertzmanytoreallooks

copperforgjgm

en

EvenjeEjgm

ev

slowmotionelectronassume

dampingg

Eevmgdt

vdm

JacksonseeModelDrude

tj

J. J. Bisognano


10

UW Spring 2008

Accelerator Physics

Cylindrical Waveguides• Assume a cylindrical system with axis z

• For the electric field we have

• And likewise for the magnetic field

))(exp()ˆˆˆ( 000 zktjkEjEiEE gzyx

))(exp()ˆˆˆ( 000 zktjkHjHiHH gzyx

J. J. Bisognano


11

UW Spring 2008

Accelerator Physics

Solving for Etangential

ozoxoy

oyoxgoz

oxoygoz

ozoxoy

oyoxgoz

oxoygoz

ozgoyox

ozgoyox

Ejy

H

x

H

EjHjkx

H

EjHjky

Ht

DH

Hjy

E

x

E

HjEjkx

E

HjEjky

Et

BE

Hjky

H

x

H

B

Ejky

E

x

E

E

0

0

J. J. Bisognano


12

UW Spring 2008

Accelerator Physics

• Maxwell’s equations then imply (k=/c)

)(

)(

)(

)(

00220

00220

00220

00220

x

E

y

Hk

kk

jH

y

E

x

Hk

kk

jH

x

H

y

Ek

kk

jE

y

H

x

Ek

kk

jE

zzg

gy

zzg

gx

zzg

gy

zzg

gx

J. J. Bisognano


13

UW Spring 2008

Accelerator Physics

• All this implies that E0z and B0z tell it all with their

equations

• For simple waveguides, there are separate solutions with one or other zero (TM or TE)

• For complicated geometries (periodic structures, dielectric boundaries), can be hybrid modes

zgzz

zgzz

Hkky

H

x

H

Ekky

E

x

E

022

20

2

20

2

022

20

2

20

2

)(

)(

J. J. Bisognano


14

UW Spring 2008

Accelerator Physics

TE Rectangular Waveguide Mode

b

a

x

y

sderivativepartialoncondxBC

jx

Hi

y

H

kk

jE

surfacetonormalEH

ozoz

g

oz

)(

;0

22

J. J. Bisognano


15

UW Spring 2008

Accelerator Physics

a TE mode Example

c

bn

k

ckc

b

nk

dk

dv

but

c

bn

k

kcv

b

nkk

ifworks

zktjyb

nAE

g

gg

ggroup

phase

g

gx

21

21

))(())((

)(

)(

)(exp()sin(

22

22

22

222

J. J. Bisognano


16

UW Spring 2008

Accelerator Physics

Circular Waveguide TEm,nModes•

EquationsBessel

Rr

nkk

In

g

'

0)(r

RR

satisfying dependencer

an leaving ,ncos/sin ,dependence the

outfactor can egeometry w lcylindrica the

2

222

J. J. Bisognano


17

UW Spring 2008

Accelerator Physics

Circular Waveguide TEm,nModes•

)cos()cos()(

)cos()sin()(

)cos()cos()(

0

)sin()cos()()(

)sin()sin()()(

,

,

,

,

,

2

zktmrkJAH

zktmrkJAH

zktmrkJkAH

E

zktmrkJkAE

zktmrkJAE

gcnm

mk

kz

gcnm

mr

m

gcnm

mcr

z

gcnm

mc

gcnm

mr

mr

g

c

g

g

J. J. Bisognano


18

UW Spring 2008

Accelerator Physics

Circular Waveguide TMm,nModes•

0

)sin()cos()(

)sin()sin()(

)cos()cos()()(

)cos()sin()()(

)cos()cos()()(

,

,

,

,

,

2

z

gcnm

mc

gcnm

mr

mr

gcnm

mk

kz

gcnm

mr

m

gcnm

mcr

H

zktmrkJkAH

zktmrkJAH

zktmrkJAE

zktmrkJAE

zktmrkJkAE

g

c

g

g

J. J. Bisognano


19

UW Spring 2008

Accelerator Physics

Circular Waveguide Modes•

cvelocityphase

modesTETMforJJofazeron

nmmodeforcutoffk

kkk

wavelengthguide

where

mmth

cnm

cg

g

)()(/)(

,,

22

J. J. Bisognano


20

UW Spring 2008

Accelerator Physics

Cavities

:condition Resonant

pdzpyxBTE For

pdzpyxETM For

kzBkzA is dependencez General

. waveguidelcylindrica of length a on faces conducting

puttingby made be can (cavity) resonator A

z

z

,...3,2,1);/sin(),(:

,...2,1,0);/cos(),(:

cossin

)J(J of zero n is ;k

guides,circular

)(

mmth

,,nm,

c

22.,

nmnm

dp

cmnp

xwhereR

cx

for

k

d

J. J. Bisognano


21

UW Spring 2008

Accelerator Physics

Cavity Perturbations

change) tdoesn' mode :(adiabatic

invariant" remainsquency energy/fre stored

average of ratio lly,adiabatica changed

is systemgoscillatin an of statea If"

Theorem EhrenfestBoltzmann

Now following C.C. Johnson, Field and Wave Dynamics

J. J. Bisognano


22

UW Spring 2008

Accelerator Physics

Cavity Energy and Frequency

?tangential is B and normal, is E where

perturbed, is wallthe whenhappens What

tconstan E Perturbed ./

++++

- - - -

Attracts

I

-IB

E

Repels

J. J. Bisognano


23

UW Spring 2008

Accelerator Physics

Energy Change of Wall Movement

ts)measuremen pull (bead profiles

field measuring and tuning :Import

dvHEVHE

ff

UU

ff

U/f of invariance fromSlFU

HEnF

precisely, More

V

density

41

density

)()(

;

)(ˆ

2

0

2

0

2

0

2

0

2

0

2

0

J. J. Bisognano


24

UW Spring 2008

Accelerator Physics

Bead Pull

J. Byrd

J. J. Bisognano


25

UW Spring 2008

Accelerator Physics

Lorentz Theorem

• Let and be two distinct solutions to Maxwell’s equations, but at the same frequency

• Start with the expression

),( HE

),( HE

)()(

)()(

)()[(

HEEH

HEEH

HEHE

J. J. Bisognano


26

UW Spring 2008

Accelerator Physics

Vector Arithmetic

ncancelatio

scalarbuttensornotif

EEjEEj

HHjHHj

etcHjE

With

abba

abba

aa

,,

0)()(

.,

��

��

��

�

J. J. Bisognano


27

UW Spring 2008

Accelerator Physics

• Using curl relations for non-tensor one can show that expression is zero

• So, in particular, for waveguide junctions with an isotropic medium we have

S1

S2

S3

0)]()[(321

daHEHESSS

J. J. Bisognano


28

UW Spring 2008

Accelerator Physics

Scattering Matrix• Consider a multiport device

• Discussion follows Altman

S1

S2

Sp

J. J. Bisognano


29

UW Spring 2008

Accelerator Physics

S-matrix

• Let ap amplitude of incident electric field normalize so that ap2 = 2(incident power) and bp2 = 2(scattered power)

)( ppp

ppp

baconstH

baE

J. J. Bisognano


30

UW Spring 2008

Accelerator Physics

Two-Port Junction

• Port X Port Y

a1

b1

a2

b2

J. J. Bisognano


31

UW Spring 2008

Accelerator Physics

Implication of Lorentz Theorem

•

YPortaSaHaSaE

XPortaSHaSE

YPortaSHaSE

XPortaSaHaSaE

Define

aSaSb

aSaSb

22222222

212212

121121

11111111

2221212

2121111

J. J. Bisognano


32

UW Spring 2008

Accelerator Physics

Lorentz/cont.

• Lorentz theorem implies

• or

TSSorSS

SSSS

SSSS

2112

22212221

11121211

0)1()1(

)1())(1(

J. J. Bisognano


33

UW Spring 2008

Accelerator Physics

Unitarity of S-matrix• Dissipated power P is given by

• For a lossless junction and arbitrary this implies

))1(,(

),(),(

),(),(

)( **

aSSa

aSaSaa

bbaa

bbaaP mmm

mm

a

unitaryeiSS .,.1

J. J. Bisognano


34

UW Spring 2008

Accelerator Physics

Symmetrical Two-Port Junction•

2

2

2

2

2

2211

2212

1211

1

1

1

1

1

i

iS

i

iS

sssymmetry

ss

ssS

J. J. Bisognano


35

UW Spring 2008

Accelerator Physics

Powering a Cavity

•

b1

a1

b2

a2

))2(exp(22 iba L

J. J. Bisognano


36

UW Spring 2008

Accelerator Physics

Power Flow

•

2

1

)2(

2

2

1

)2(

2

2

1

2

1

2

2

1

1

1

1

b

b

ebai

ebia

a

a

i

i

i

i

L

L

J. J. Bisognano


37

UW Spring 2008

Accelerator Physics

Power Flow/cont.

•

)(22

)11

)11

1(

2

1)2(2

)2(2

2

1)2(2

)2(22

1

resonancenatpeakb

ae

eb

ae

eb

i

i

i

i

L

L

L

L

J. J. Bisognano


38

UW Spring 2008

Accelerator Physics

Optimization

• With no beam, best circumstance is ; I.e., no reflected power

01 b

2

1

111

2

2

2

22

L

couplingcriticale

ee

L

L

L

J. J. Bisognano


39

UW Spring 2008

Accelerator Physics

At Resonance

•

fieldpowerConstbThen

Let

ab

sosmallis

poweraRecall

ae

L

L

L

1

2

2/

,

11

2

2

122

1

12

J. J. Bisognano


40

UW Spring 2008

Accelerator Physics

Shunt Impedance

• Consider a cavity with a longitudinal electric field along the particle trajectory

• Following P. Wilson

z1z2

J. J. Bisognano


41

UW Spring 2008

Accelerator Physics

Shunt Impedance/cont

)exp()(

)sin()(

)cos()(

)(

}){()(

2

1

2

1

2

1

00

2

1

max

)/(

0

zc

izEdzV

zc

zEdzS

zc

zEdzC

iSCeezEdzV

icrelativistttczezEE

z

z

z

z

z

z

ticztiz

z

ti

z

J. J. Bisognano


42

UW Spring 2008

Accelerator Physics

Shunt Impedance/cont.

• Define

• where P is the power dissipated in the wall (the term)

• From the analysis of the coupling “”

• where is the generator power

edanceImpShuntP

VR

2

max0

Le

gPRV 012

gP

J. J. Bisognano


43

UW Spring 2008

Accelerator Physics

Beam Loading• When a point charge is accelerated by a cavity, the loss

of cavity field energy can be described by a charge induced field partially canceling the existing field

• By superposition, when a point charge crosses an empty cavity, a beam induced voltage appears

• To fully describe acceleration, we need to include this voltage

• Consider a cavity with an excitation V and a stored energy

• What is ?

2VW

J. J. Bisognano


44

UW Spring 2008

Accelerator Physics

Beam Loading/cont.

• Let a charge pass through the cavity inducing and experiencing on itself.

• Assume a relative phase

• Let charge be bend around for another pass after a phase delay of

bV

bfV

J. J. Bisognano


45

UW Spring 2008

Accelerator Physics

Beam Loading/cont.

• Ve

V2

V1V1 +V2

J. J. Bisognano


46

UW Spring 2008

Accelerator Physics

Beam Loading/cont.• With negligible loss

• But particle loses

• Since is arbitrary, and

22cos2 bVW

sinsincoscos2)cos1(2

)cos([

2

bbbb

bee

qVqVqfVV

WU

qVqVqVU

21

2 fand

qVb

J. J. Bisognano


47

UW Spring 2008

Accelerator Physics

Beam Loading/cont.

• Note: we have same constant (R/Q) determining both required power and charge-cavity coupling

)2

(

)/(

1/

)/(

0

2

00

22

QR

qV and

RQ finally, and

Q of indep. (R/Q) ;losses wall

losses coupling

QQwhere

QRV

W

QQRV

RV

PandPW

QNow

b

J. J. Bisognano


48

UW Spring 2008

Accelerator Physics

Beam Induced Voltage• Consider a sequence of particles at Tf 1

20

RIqeqV

summing

etqV

excites particle each resonance, on

eIeqffntqtI

TQR

nTQ

QR

total

tQ

QR

0,

tin

n

tin

22

22

22 )cos(

)/()(0

J. J. Bisognano


49

UW Spring 2008

Accelerator Physics

Summary of Beam Loading

• References: Microwave Circuits (Altman); HE Electron Linacs (Wilson, 1981

Fermilab Summer School)

LL

L

La

abgag

QiQ

RZ

QQ

RR

IRVPRV

2tan;;21

1;

1

1;

1

2

0

0

0

J. J. Bisognano


50

UW Spring 2008

Accelerator Physics

Vector Addition of RF Voltages

Vb

Vc

Vb

Vbr

Vg

Vgr

J. J. Bisognano


51

UW Spring 2008

Accelerator Physics

Vector Algebra

sincos)sin(coscos

cos)cos(coscos

by crest -off Beam

cos;cos

1;

1

2

2

brgrc

brgrca

ibrb

igrg

abrgagr

VVV

VVVV

eVVeVV

IRVPRV

J. J. Bisognano


52

UW Spring 2008

Accelerator Physics

Required Generator Power

• Trig yields

}]sincos)1(

[sin

]cos)1(

{[cos

cos

1

4

)1(

2

22

2

22

c

a

c

a

a

cg

V

IR

V

IR

R

VP

J. J. Bisognano


53

UW Spring 2008

Accelerator Physics

E.g, assume

•

beam thengacceleratiin and

cavity in then dissipatioin usedpower all

1cos

1

power when Minimum

)cos(

4

)1(

sin)1(

tan

0

2

0

0

22

bca

cg

c

b

c

a

a

brcg

c

a

PPR

VP

P

P

V

IR

R

VVP

V

IR

J. J. Bisognano


54

UW Spring 2008

Accelerator Physics

Scaling of Shunt Impedance

• Consider a pillbox cavity of radius b & length L

)sin()()/(

)cos()(

100

00

tkrJZEH

tkrJEEz

J. J. Bisognano


55

UW Spring 2008

Accelerator Physics

Pillbox Cavity

• The energy stored and power loss are given by

SCfor Const R Cu; for R

pJLbZ

EbRdaH

RP

pJLEbdvEW

2

ss

A

ss

sVz

2/10

01

2

12

0

2

02

01

2

10

2

0

20

)2

(

)()(2

)(22

J. J. Bisognano


56

UW Spring 2008

Accelerator Physics

Summary of Scaling•

SCLR

Cu;LR

SCR Cu; R

LbL

WV

QR

SC Cu; Q

RRLbbLb

bPW

Q

b

ss

12/1

22/1

21

22

22/1

2

1

1)(

1

J. J. Bisognano


57

UW Spring 2008

Accelerator Physics

More Multicell Cavities

• Given a solution of a single cell cavity, one can consider the coupling of multiple cells in that mode by an expansion where

– the expansion coefficients give the strength of excitation of each cell in that mode

– coupling comes from perturbation of purely conductive boundary by holes communicating field between cells

• This is a recondite subject, with all sorts of dangers from “conditional” covergence of Fourier series; see Slater (and Gluckstern) for a complete picture of this

n

a

n

a

n

a

n

a HhH and EeE

J. J. Bisognano


58

UW Spring 2008

Accelerator Physics

Periodic Structures•

J. J. Bisognano


59

UW Spring 2008

Accelerator Physics

Expansion Equations

encieseigenfrequ separateget just weand zero, is RHS the

cavity, in current no and cells between holes no For

HEndadtd

EJdvkekhdtd

HEndaEJdvdtd

ekedtd

have we,conditionsboundary conductive pure For

n

aAV

n

a

n

a

n

a

n

a

n

a2

2

n

aAV

n

a

n

a

n

a

n

a2

2

))(()(

))(()(

2

2

J. J. Bisognano


60

UW Spring 2008

Accelerator Physics

Holes• But if there are holes in the cavity talking to

neighboring cells, we have

• E.g., Bethe says

Tangential electric

field

)()(2

)(1

0

2/122

0

2/122

tan Hnraik

ErarE z

J. J. Bisognano


61

UW Spring 2008

Accelerator Physics

Coupled Equations

•

)cos1/((;2

)12(sin

)cos1/((;cos

0)(2

)(

2

0

2

0

11

22

0

2

Nq

eNnq

x

cells N and /2/

of detuning w/nterminatio cell" full" ForNq

eNqn

x

cells 1)(N n,terminatio cell" half" For

xxx

form take equations mode, singlea For

(q)

n

ti(q)

n

(q)

n

ti(q)

n

nnn

q

q

J. J. Bisognano


62

UW Spring 2008

Accelerator Physics

8 Cell Cavity Modes

0.80.850.90.951

1.051.11.151.2

0 2 4 6 8 10

Series1

J. J. Bisognano


63

UW Spring 2008

Accelerator Physics

• Result of Floquet Theorem that solutions of differential equations with

periodic coefficients have form of periodic function times exp(jz)• Phase velocities less than c particle acceleration possible

• From Slater, RMP 20,473

J. J. Bisognano


64

UW Spring 2008

Accelerator Physics

Pi Mode• Tesla Pi-mode

J. J. Bisognano


65

UW Spring 2008

Accelerator Physics

Floquet Theorem

• For example, for a disk loaded circular cylindrical structure, the TM01 is of the form

2220 ;

2

)(exp()(),,(

nnn

nnnnnz

kkd

n

where

ztjrkJatzrE

J. J. Bisognano


66

UW Spring 2008

Accelerator Physics

Field Relations for Cylindrical Systems

•

)(ˆ1ˆ

ˆ -ˆ1

))((ˆ1

)(ˆ

)((ˆ1

/;)eE(r,

as varyingfields Assume

n-ti

rAz

rrz

AA

r

AAr

rA

Define

ArA

rz

rr

A

z

A

rAz

Ar

rA

cklet

rtz

zzzt

rzr

z

z

J. J. Bisognano


67

UW Spring 2008

Accelerator Physics

Field Relations for Cylindrical Systems•

EiHik

E

HiEik

H

HE for solveCan

HHEi

EEHi

yield equations curl sMaxwell'

ztnzt

n

t

ztnzt

n

t

tt

tzztt

tzztt

)(1

)(1

,

22

22

J. J. Bisognano


68

UW Spring 2008

Accelerator Physics

Integrated Force at v=c• Let be longitudinal force seen by a particle.

Consider a trajectory z=vt, r=r0. The integrated force is then=

• Only q=/v contributes;i.e.n=/v

),,( rtzF

);,/(~

);,(~)(

);,(~

);,(~)(

);,()(

0

0

0

0

0

rvFd

rqFqvdqd

rqFedqdted

rqFeevtzdqdtdzd

rtzFvtzdtdzF

iqvtti

iqzti

J. J. Bisognano


69

UW Spring 2008

Accelerator Physics

E.g., a TM Mode

•

constriE

constriH

rkconstE

ngE

erkJconstE

nt

t

nz

in

nnz

ˆ

ˆ

)(1(

0.,.

))((

222

22

J. J. Bisognano


70

UW Spring 2008

Accelerator Physics

Force on Relativistic Particle

•

particle icrelativist-ultra for

deflection no causes mode TM pure So,

constrc

icriBvEF

cwith

constriE

constriB

HB

nr

n

nr

0)1

(

/

2

J. J. Bisognano


71

UW Spring 2008

Accelerator Physics

Panofsky Wenzel Theorem

• Pure TE mode doesn’t kick; pure TM mode, as in previous example, behaves to cancel denominator, so falls off as -2 ; hybrid modes don’t cancel denominator, so finite kick may obtain even when v=c

ztzt

n

n

ztnzt

n

ztnzt

n

ccv

nr

Eiv

Ecv

ki

F

HiEik

v

EiHik

F

synchfor k withBvEF

)1(

)(1

)(1

2

2

22

22

22

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Superconductivity

• Basic mechanism– Condensation of charge carriers into Cooper pairs,

coupled by lattice vibrations– Bandgap arises, limiting response to small

perturbations (e.g., scattering)– No DC resistance

• At temperatures above 0 K, some of the Cooper pairs are “ionized”

• But for DC, these ionized pairs are “shorted out” and bulk resistance remains zero

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RF Superconductivity• But pairs exhibit some inertia to changing

electromagnetic fields, and there are some residual AC fields (sort of a reactance)

• These residual fields can act on the ionized, normal conducting carriers and cause dissipation

• But it’s very small at microwave frequencies (getting worse as f2)

• At 1.3 GHz, copper has Rs~10 milli-ohm• At 1.3 GHz, niobium has Rs~800 nano-ohm at 4.2 K• At 1.3 GHz, niobium has Rs~15 nano-ohm at 2 K• Q’s of 1010 vs. 104

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CEBAF RF Parameters

• Superconducting 5-cell cavity in CEBAF

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CEBAF Cavity Assembly

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Cavity Specifications• Frequency 1497 MHz

• Nominal length 0.5 meters

• Gradient >5 MeV/m

• Accel Current 200A 5 passes

• Number cells 5

• R/Q 480 ohms

• Nominal Q0 2.4·109

• Loaded QL 6.6 ·106

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What’s QL ?• Run in linac mode, essentially on crest, = 0• In storage rings, 0 for longitudinal focusing• Wall losses V2/R=(2.5 · 106volts)2 /(480·2.4·109 )

are 5.4 watts

• Power to beam: (200A 5 passes)·(2.5 · 106 volts) is 2500 watts

6

0

0

102.5)1/(

460/1

QQ

PP

L

b

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Would Copper Work• Typical Q is now 104

• Wall losses (2.5 · 106volts)2 /(480·104 ) are now 1.3 MW vs. beam power of 2500 watts

• Some optimizations could yields “2’s” of improvement• More importantly, SRF losses are at 2 K, which

requires cryogenic refrigeration.• Efficiencies are order 10-3

• So, 5 watts at 2 K is 5 kW at room temperature, but still factor of 100 to the good

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Higher Order Modes

• There are higher order modes, which can be excited by the beam

• These can generate wall losses, and fields can act on beam to generate destructive collective effects

• First question is whether wall losses are large or small compared to fundamental wall losses

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Loss Factors• When a bunch passes through a cavity, it loses

energy of

MV.yield to cycles Q/ for builds

this Q, high is lfundamenta Sincevoltgenerating

pCq bunch, CEBAF a For

pCVk mode lfundamenta For

k factor loss defines which

kqQR

qW

superbunch

bunchbunch

.1

,6.0

/2

2

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Loss Factor for HOMs• Bunch spectrum extends out to• For a typical 1 ps linac bunch,

• For for a 1.5 GHz fundamental, there are many tens of longitudinal HOMs for the beam to couple; coupling is weaker than to fundamental because of more rapid temporal and spatial variation

• From codes,

t 1

GHzfHz 100;1012

pcVk /20

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Power Estimates• For 1 mA CEBAF 5-pass beam, with 0.5

pC/superbunch, we have only10mW of loss

• But in, say FEL application, with 100 pC bunches at 5mA, we have 10 watts in the wall, more than the power dissipated by fundamental!

• So extraction of HOM power is issue for high current applications with short bunches

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Couplers and Kicks• Waveguide couplers break cylindrical symmetry

• Result is that the nominal TM01mode now has TE content and m0; hybridized

• By Panofsky-Wenzel, introduces steering and skew quad fields that require compensation

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Homework: Topic I• From Accelerator Physics, S.Y. Lee

– Reading: Chapter 3, VIII, p. 352 & onward– Problems:3.8.1 (really table 3.10), 3.8.2, 3.8.4

• The next generation CEBAF cavity can achieve 20 MV/m gradients with a 7 cell structure. a) Assuming R/Q is 7/5 higher, what would be the heat load generated per cavity if the Q is unchanged? b) How much higher a Q is necessary to main the heat load at the levels from the first generation cavity discussed in the lecture?

• Consider a single cell RF system operating at 500 MHz with an effective length of 0.3 meters, which is to be operated at a gradient of 2 MV/m with beam current of 100 mA. Assume the R/Q of the cavity is 100 ohms and that the Q from resistive wall losses is 40,000.A) Calculate the optimal coupling coefficient for powering the cavity with the 100 a mA beam passing through it.B) Calculate the power necessary with 100 mA beam to power the cavity.C) Calculate the power received by the beam and the power dissipated in wall losses.D) Describe what will happen to the power requirements and reflected power from the cavity if the beam is lost, but the feedback system attempts to maintain the 2 MV/meter gradient.

• A CEBAF problem: Use the nominal specs given in lecture• a) For a 1 mA at 5 MV/m calculate total power and reflected power on resonance and

10 degrees off resonance with bunch on crest.• b) If this beam is allowed to pass for a second time through the cavity for energy

recovery at 170 degrees off accelerating crest, simultaneously with the first pass beam on crest, calculate the total and reflected power when the cavity is tuned on resonance.

Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 1 Accelerator Physics...

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