Introduction to RF linear accelerators - AU · Introduction to RF linear accelerators Lars Hjorth Præstegaard, Ph.D. Technical manager Medical physicist Aarhus University Hospital

Introduction toRF linear accelerators

Lars Hjorth Præstegaard, Ph.D.Technical managerMedical physicist

Aarhus University Hospital

Aarhus University Hospital, Århus Sygehus

Outline

• Multi-cell accelerating structures

• Traveling wave acceleration

• Standing wave acceleration

• Transit time factor

• Phase stability for TW acceleration

• RFQ linear accelerator


RF linear acceleratorLinear accelerator (Linac):Acceleration along a linear path

RF linear accelerator:Linac using sinusoidally varying electromagnetic fields

Purpose:Transfer of energy from the RF wave to the particle beam

We will not consider:Induction linacs (electric field by use of Faraday's law)DC linear accelerators (limited to a few tens of MeV)

Multi-cellaccelerating structures


Electromagnetic wave in free spaceIn free space:

Transverse electromagnetic wave (TEM wave).

⇒

⇒ No electric field in the direction of propagation: Ineffective acceleration.

Both electric and magnetic field perpendicular to direction of propagation (z axis)

ω: Angular frequency (2πf).λ: Wave lengthk : Wave number (k=2π/λ).c: Speed of light

Dispersion relation for TEM wave:

ω(k)

⇒ Phase velocity = ω/k = cGroup velocity = dω/dk = c

= 2πf = 2πc/λ= kc

(links ω and λ)

E(z,t)=E0*ej(ωt-kz)


Electromagnetic wave in a waveguide

Wave propagation in the +z direction

Longitudinal electric field for propagation of a wave in a waveguide? z

z


Maxwell's equations (no sources) :

Wave equation

( ) ( )

( ) 01

0

2

2

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−Δ=⎟⎠⎞

⎜⎝⎛ ×∇

∂∂

−Δ=

⎟⎠⎞

⎜⎝⎛ ×∇

∂∂

−Δ+⋅∇∇−=⎟⎠⎞

⎜⎝⎛

∂∂

+×∇×∇−=

EBE

BEEBE

tct

tt

0=⋅∇ E

tBE∂∂

−=×∇

0=⋅∇ B

tc ∂∂

=×∇EB 2

1

012

2

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−Δ Btc

Wave equation for magnetic field:

⇒ Wave equation for electric field:

0=⋅∇ E

tBE∂∂

−=×∇

tc ∂∂

=×∇EB 2

1


Wave equation for waveguideTM wave (Bz=0):

0,012

2

2 ==⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−ΔsurfacezE

tcE

0,012

2

2 =∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−Δsurface

z

nB

tcB

Solution exist:Not suited for particle acceleration(exceptions)

TE wave (Ez=0):

Solution exist:Suited for particle acceleration


Waves in a rectangular waveguide

( ) ( ) ( ) ( )kztjz eyYxXtzyxE −= ω,,,

Ansatz for rectangular cross section of waveguide:

trigonometricfunctions

Wave propagationin the +z direction

(physical quantity: real component)

0=surfacezE

ω: Angular frequencyk: Wave number (k=2π/λ)λ: Wavelength

Figure: Color indicates Ez


Waves in a circular waveguide

( ) ( ) ( ) ( )kztjz emrRtzyxE −= ωθcos,,,

0,01112

2

22

2

22

2

2

2

==⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+∂∂

+∂∂

+∂∂

surfacezz EEtcrrrrz θ

Ansatz for circular waveguide:

Wave equation for TM wavein cylindrical coordinates:

a

z

m: Number of azimuthal oscillations (integer)



⇒ ( ) 0,01 22

2

22

2

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

∂∂

+∂∂

surfaceRrRk

crm

rrrωWave eq. + ansatz

Kr2

Solution for Kr>0:

( ) ( )rKJArR rm=

Boundary condition:( ) ( ) 0== aKJAaR rm

n'th root of Jm

mnnmr xaK =,⇒

axK mnmn

r =⇒

1st

2nd

aKr

405.2 :mode TM 01,01 =

3rd



Ez=0 at surface

( ) ( ) ( )kztjrz erKJAtzyxE −= ω

01,0,,,

TM01 mode:


Dispersion relation for circular wave guide

22,2

2

kKc nmr +=ω

Dispersion relation:

Phase velocity and group velocity:

12

2, +==

kK

ck

nmrph

ωυ12

2, +

=∂∂

=

kK

ck

nmrg

ωυ

generatorfrequency

No wave propagation for ω< ωc,01

> c < c

νph=c (free space)01,01, rc cK=ωCut-off frequency for TM01 mode:

wave in +z dir.wave in -z dir.

Stopband


Dispersion relation for circular wave guide

Dispersion relation:


Acceleration in circular wave guidePhase velocity = ω/k > c⇒

⇒ Poor efficiency of acceleration in circular wave guide!

Wave desynchronizeswith the particle


Disk-loaded waveguide

Disk-loaded waveguide with period L:

⇒ Reflections from disks

How to slow down phase velocityof electromagnetic waveEz(x,y,z,t)=R(r)*ej(ωt-kz) in waveguide?

Positive interference of reflections: k*2L≈n*2π, n=1,2,3,.. (k=n*π/L).⇒ Large perturbation of dispersion curve for wave guide.⇒ νph, νg decreases.

cavity


Disk-loaded waveguide: DispersionDispersion relation for disk-loaded waveguide (electric coupling):

wave in +z dir.wave in -z dir.

⇒ Disks ⇒ Slowing down of phase/group velocity⇒ Frequency exist for which νph= c (acc. of electrons)

Passband:Large for large cell-to-cell coupling


Disk-loaded waveguide: Dispersion

Higher order modes (HOMs)

magneticcoupling


Disk-loaded waveguide: Dispersion

possible excitation of higherorder modes (HOMs) whic are synchronousmagnetic

couplingbetween

cells


Disk-loaded waveguide: Floquet theoremAnsatz for Ez in loss-free disk-loaded circular waveguide:

( ) ( ) ( )zktjz ezrFtzrE 0,,, −= ω

TM01: Cylindrical symmetry, 1st radial node at waveguide wallF(r,z): Periodic function with period L

Floquet theorem:For a given mode of an infinite periodic structure, the field is multiplied by a constant exp(-γ) (γ=-α+j*k0), when moving from one period to the next.

Stop band:exp(-γ) is real and less than 1 (exponential falloff)

Pass band:exp(-γ) is complex.Ez fulfills the Floquet theorem: Ez(r,z+L,t)=Ez(r,z,t)exp(-jk0L)

TM01 wave:


Disk-loaded waveguide: Space harmonics

( ) ( ) ( )∑∞

−∞=

−=n

zLnjn erazrF π2,

Fourier expansion of F(r,z):

n: Integeran:Fourier coefficient

⇒Wave eq. + ansatz

( ) ( ) 01 22

220 =⎥

⎦

⎤⎢⎣

⎡++∑

∞

−∞=

+−

nnr

zLnkjtj raKdrd

rdrdee πω

2

0

22 2

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛=

Lnk

cKr

πω Solution: an=An*J0(Kr*r)

TM01 solution for Kr,n2>0 (νp>c):

( ) ( ) ( )( )∑∞

−∞=

−=n

nrnz zktjrKJAtzrE ωexp,, 0

Lnkknπ2

0 +=

Sum of infinite number of

traveling waves:Space harmonics


Disk-loaded waveguide: Space harmonicsTM01 solution for Kr,n

2<0 (νp<c):

( ) ( ) ( )( )∑∞

−∞=

−=n

nrnz zktjrkIAtzrE ωexp,, 0

dnkknπ2

0 +=

Sum of infinite number of

traveling waves:Space harmonics

22rr

Kk −=


Disk-loaded waveguide: Space harmonics

• n=0: Principal wave (largest amplitude)• Shift of dispersion curve for n=0 wave with nπ/L, n=..., -2, -1, 1, 2,...• Phase velocity for n'th space harmonics:

Lnkknph π

ωωυ20 +

==

Properties of space harmonics:

Arbitrary low phase velocity for large n

Lnkknπ2

0 +=

generatorfrequency

Traveling wave (TW)acceleration


TW accelerationTraveling wave (TW) acceleration:

Disk-loaded waveguide (slowed-down wave)Input of electromagnetic wave at first cellAbsorption of microwave in load after the last cellWave synchronous with the beamInjection of particle beam along axis of disk-loaded waveguide

microwave load

electron gun


TW acceleration: Constant impedance

Resistive losses in walls + Energy transfer to the beam⇒ Reduction of microwave power along disk-loaded waveguide:


TW acceleration

Only waves synchronous with the beam affect the beam

generatorfrequency

n=0: Large wave amplitude (65-80 %)Wave synchronous with the beam

High acceleration efficiency:


TW acceleration: Power dissipation in wallsDefinition of shunt impedance per unit length:

ω∝−

=dzdP

EZ zs

20

Ez0: Axial electric field of 0th space harmonic synchronous with the beam.

dP/dz: RF power dissipated in the walls per unit length

Amplitude of accelerating field for a given dissipated RF power.

TW power:

dzdPwQ

−= ω

wP gν=

Q factor:

w: Stored energy per unit lengthνg: Group velocity

α(z): field attenuation per unit length

Attenuation of electric field:

( ) ( ) ( )zEzdz

zdEz

z0

0 α−=


TW acceleration: Power dissipation in walls

Definition of Zs, Q, and P ⇒ ( ) ( )20 zEZ

QzP z

s

g

ων

=

( )gQ

zνωα

2=

Attenuation of TW power:

( ) ( ) ( ) ( ) ( )zPzdz

zdEzEZ

Qdz

zdPz

s

g αων

22 0 −==

Definition of Q and P ⇒


( ) ( )

( )l

elPZq

lelqEdzeEqW

l

s

l

z

lz

z

α

αα

αα

−

−−

−=

−==Δ ∫102

100 00

0

TW acceleration: Constant impedanceConstant impedance accelerating structure:Uniform cell geometryQ, Zs, νg and α do not depend on z

( )( ) z

z

z

eE

zEα−

=

00

0

Energy gain for synchronous particle at wave crest:

l: Length of acc. structure

Maximum of ΔW (αl=1.26):

( ) lPZqW s 0903.0max =Δ

Ez0(l) = 0.28 , P(l) = 0.08 (goes into the load)

Importance of shunt impedance

Tuning of αl by changing νg (disk aperture)

Low αl: High power loss in loadHigh αl: High power dissipation to walls


TW acceleration: Constant gradientConstant gradient accelerating structure:Ez0 does not depend on zStructure geometry depend on z (for example change of iris diameter)νg and α depends on z (sensitive to iris geometry)Q and Zs=Ez0

2/(-dP/dz) do not depend on z (approximately correct)

⇒ dP(z)/dz = constant ⇒ ( ) ( ) ( ) ( ) zl

PlPPzP 00 −+=

( )( )( )

( )

( )∫∫ −=llP

P

dzzzPzdP

00

2 α

Use of eqn. for attenuation of TW power:

⇒( ) ( ) ( )zPz

dzzdP α2−=

⇒ ( ) ( ) ( )∫== −l

dzzePlP0

2 ,0 αττ

⇒ ⇒ ( ) ( ) ( )τ210 −−−= el

Pdz

zdP( ) ( ) ( )⎟⎠⎞

⎜⎝⎛ −−= − τ2110 e

lzPzP


TW acceleration: Constant gradient

Energy gain for synchronous particle at wave crest:

( ) ( ) ( ) ( )τ20

00 1000 −−===Δ ∫ elPZqlqEdzEqW sz

l

z

Tuning of τ by changing νg (disk aperture)

( ) ( ) ( ) ( )τ220 10 −−−=−= e

lPZ

dzzdPZzE s

sz

That is

Maximum of ΔW (τ infinite):

( ) lPZqW s 0max =Δ Importance of shunt impedance


TW acceleration: X-band TW structure


TW acceleration: Stanford linear accelerator

Constant gradient:

Energy: 50 GeV electrons (3 km linac with 932 linac sections)

Uniform power dissipationLower peak surface electric field

Structure geometry: Iris tapering from 1.3 cm to 1.0 cmCavity radius tapering: 4.2 cm to 4.1 cm

Compromise between high energy gain and short filling time Choice of τ: 0.57


TW acceleration: Choice of frequency

Low frequency:Large mechanical tolerancesLarge beam apertureWake fields ∝ ω2 (long.)/ω3 (transv.)Stored energy per unit length (∝ω-2)

High frequency:Efficient acceleration (Zs∝ω½)Higher threshold for breakdown

LEP cavities: 350 MHZLEP cavities: 350 MHZGradient: 6 MV/mGradient: 6 MV/m

CLIC cavities: 30 GHZCLIC cavities: 30 GHZGradient: 150 MV/mGradient: 150 MV/m


TW acceleration: Choice of mode

2π/3 mode (TW favorite):• High shunt impedance.• Reasonable mode separation.• Shorter settling time than the π mode.

π mode:• Long time fill time of accelerating structure (small group velocity).• Sensitive to frequency errors• Small separation of neighbor modes (only useful for a small number of

cavities.

π/2 mode:• Low shunt impedance per unit length.• Large separation to neighbor modes (large νg).• Insensitive to geometrical errors (cancellation of 1st

order perturbations).

Standing wave (SW)acceleration


SW acceleration

Full reflection of traveling waves at structure ends

Periodically-loaded waveguide with reduced apertures at ends:

⇒ Standing waves

Different to TW structure (αl≈1:sequential filling of all cavities)Low field attenuation (αl<<1)


SW acceleration: Modes

⇒ Family of N normal modes (q=0, 1, 2,..., N-1)N allowed values of wave vector k: k=πq/(N-1)=phase advance per cavityHighly resonant structure

Array of N cavities behaves as N coupled harm. oscillators:

oscillator 1 oscillator 2 oscillator 3

coupling coupling

Dispersion curve:


0 1 2 3 4 5 6 7 8 9 10

q=0q=1q=2q=3q=10

SW acceleration: Energy gain

• N+1 modes characterized by the number of half field oscillations q • Only N+1 allowed values of k.

( )tN

qnEEz ωπ coscos0 ⎟⎠⎞

⎜⎝⎛=

Axial electric field for N+1 cavities:E0: Amplitude of electric fieldn: Cavity number (0,1,2,...,N)q: Mode number (0,1,2,...,N)πq/N: Phase advance per cell


SW acceleration: Energy gainVelocity of particle synchronous with mode q:

qNL

NLqks πω

πωωυ ===

Energy gain for particle at cavity n=0 at t=0):

( )⎪⎩

⎪⎨⎧

=+

<<+

=⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+=

⎟⎠⎞

⎜⎝⎛==Δ

∑

∑∑

=

==

NqLEN

NqLEN

NqnLEe

LN

qnEeLEeW

N

n

N

n

N

nn

,0,1

0,2

22cos12

cos

0

0

0

0

0

2

00

π

π

backward wave

0 and π modes: Energy gain doubled due to backward wave(same shunt impedance as that of TW acc.)

NqnnLt sn

ωπυ

==


SW acceleration: Energy gainSW acceleration with π mode:

Particle in everyother cavity


SW acceleration: Energy gain0- and π-modes: Large energy gain

Large group velocityInsensitive to geometrical errorsSmall energy gain

π/2-mode:

π/2 mode

bi-periodic π/2 mode

coupling cavity(magnetic coupling)

Biperiodic π/2-mode SW accelerating structure:All advantages for 0, π/2, π-modes


SW acceleration: Medical linacVarian 600c biperiodic π/2-mode SW structure:

microwaves incoupling cavity

Normal cavity


SW acceleration: Drift tube linac (DTL)

SW acceleratorTM010 mode of tank (q=0)Reverse field shielded by drift tube QP inside drift tubeLength of drift tubes increase with particle velocity

Transit time factor


Transit time factorAccelerating cavity:

( ) ( ) ( )∑∞

−∞=

−=n

nnrnz zjkrkIAzrE exp, ,0

dnkknπ2

0 +=

TM01 solution in disk-loaded structure (νp<c: Kr,n2<0):

2222 ⎟

⎠⎞

⎜⎝⎛+=−=

ckKk nrr

ωwhere


Transit time factorReplacement of sum with integral:

( ) ( ) ( )2

220 ,cos, ⎟

⎠⎞

⎜⎝⎛+== ∫

∞

∞− ckkdkkzrkIAzrE rrkz

ω

Inversion of Fourier integral:

( ) ( ) ( )2

220 ,cos,

21

⎟⎠⎞

⎜⎝⎛+== ∫

∞

∞− ckkdkkzzrErkIA rzrk

ωπ

Assume constant electrical field in the gap for r=a:

( ) 22,, gzgEzaEz ≤≤−=

⇒( )

( )akIkgkgEgA

rk

0

12

2sin2π

=

⇒ ( ) ( ) ( )( ) ( )∫

∞

∞−

= dkkzakIrkI

kgkgEgzrE

r

rz cos

22sin

2,

0

0

π


Transit time factorEnergy gain in gap:

( ) ( ) ( )

( )

( ) ( )( ) ( )ϕ

ϕνωϕ

νω

ϕω

cos2

2sin

sinsincoscos,

cos,,,

0

0

2

2

2

2

2

2

akIrkI

kgkgqEg

dzzzzrEq

dztzrEqdztzrEqW

r

r

L

L ppz

L

Lz

L

Lzgap

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=

+==Δ

∫

∫∫

−

−−

ϕ: RF phase (rel. to wave crest) when the particle passes the center of the gap (t=0)z= νpt: Particle positionνp: Particle velocityk=ω/νp: Wave vector for synchronous waves

(only synchronous waves contribute to acceleration)


Transit time factor

( ) ( )( ) ( ) ( )rkqEgT

akIrkI

kgkgqEgW

r

rgap ,cos

22sin

0

0 ==Δ ϕ

( )2

2sinkg

kg

Reduced field on axis due to opening in drift tube(the field at the drift tube bore radius is larger).

Energy gain in gap:

Reduction of the optimum energy gain qEg*cos(ϕ):

( )( )akI

rkIr

r

0

0

Change of the field during passage of the particle(small for small g and fast particle)

Transit time factor:

( ) ( ) ( )( )akI

rkIkg

kgrkTr

r

0

0

22sin, =

Phase stability for TWacceleration


Phase stability for TW electron accelerators

Accelerator model:Drift spaces (no field) separated by thin gaps where forces are applied as impulses.

Energy gain from gap n-1 to gap n:

( ) ( )( )

( )ϕ

πϕπγ

sin

2cos2cos

20

20

mcLeE

mcLeE

n

−=

−+=Δ

γ: Relativistic gamma factor = W/(mc2)E0: Average longitudinal electric field on axis in cellL: Cell lengthm: Electron massϕ: Phase relative to synchronous phase = π/2 (optimum phase for injection)

nn-1



( )ϕγγ sin20

mceE

dLd

dzd n −=≈

Energy gain per unit length:

RF phase change from gap n-1 to gap n:

( )nwavenn tt ωωϕ −=Δ ,

tn: Time for particle to travel from gap n-1 to gap n

RF phase change per unit length:

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=⎟

⎟⎠

⎞⎜⎜⎝

⎛−−=≈

1111

2γγ

βω

ββωϕϕ

phph

n

ccdLd

dzd

β: Relativistic beta factor = νp/c

(1)

(2)


Phase stability for TW electron acceleratorsCombining eqn. (1) and (2) + some manipulation:

( )( )2

0max 11coscos phph

ph

mceE

βββγβ

ωϕϕ −−−=−

Case νph<c: ( ) 21 mcW −= γ

Curve in phase space for each choice of ϕmax

Inefficient acceleration:1. High energy electron (fast) overtakes the TW wave (non synchronous)⇒ 2. Energy loss.⇒ 3. The eletron becomes synchronous.⇒ 1. etc


Phase stability for TW electron acceleratorsCase νph=c (βph=c):

( ) 21 mcW −= γ

The phase only decreases (open curves in phase space)

Optimum acceleration: Electron phase decreases asymptotically to ϕ=ϕmax

Electrons injected with inappropriate phase (red) are lost (bunching)

( )βγωϕϕ −=− 1coscos0

max eEmc

⇒

optimum asymptoticphase ϕmax=-π/2

011<⎟

⎟⎠

⎞⎜⎜⎝

⎛−−=

ββωϕ

phcdzd

Optimum acc.: Inject the beam on this curve

Where



Minimum injection energy (π mode):

2

2

2

20

2

2

20

22

2

20

22

massrest electron cellin gain energy 11,

11

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

+−

=+

−=

ππχ

χχ

ω

ωβmc

LeE

mEemc

mEemc

Acceleration at RF crest (ϕmax=-π/2) for injection at ϕ=0:

( ))1(

10 β

βω+−

=emcE

( ) ( )( )β

βωβγωϕϕ+−

=−==−1111coscos

00max eE

mceEmc

⇒ Required field:

Large field required for small initial energy

Radio frequency quadrupole (RFQ)linear accelerator


RF quadrupole (RFQ) linac

RFQ linac:Bunching of beamFocusing of beamAcceleration of beam

No beamloss!

• Acceleration of low velocity beams: 0.01-0.06 times c (ions)

• Often preaccelerator for regular ion linacs (DTLs)

• Replaces often DC preaccelerators• Electric force stronger than

magnetic force for low velocities• Velocity-independent focusing

(focusing by electric field)microwave

input


RF quadrupole (RFQ) linac

microwaveinput

Focusing and acceleration in an RFQ:

Transverse component of E: FocusingLongitudinal component of E: Acceleration

ion

E field

RFQ electrodes


RF quadrupole (RFQ) linacQuasistatic approximation:Ignore induced electric field (Faraday's law)Good approximation when a,ma<<λ

0112

2

2

2

2 =∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

zUU

rrUr

rr ϑ

ω: Angular frequencyϕ: Initial phase of potential

( ) ( ) ( )φωθθ += tzrUtzrV sin,,,,,Time-dependent scalar potential for electric field :

U(r,θ,z) is a solution of Laplace's equation:


RF quadrupole (RFQ) linacGeneral solution to Laplace's equation:

( ) ( )

( ) ( ) ( )∑∑

∑

+

=∞

=

n lnln

n

nn

lkznlkrIAV

nrAVzrU

cos2cos2

2cos2

,,

2

0

20

θ

θθl+n=2p+1, p=0, 1, 2,...±V/2: Electrode potentialI2n(x): Modified Bessel function of order 2nϕ: Initial phase of potentialk=2π/LL: Period of structure modulationLowest order terms:

( ) ( ) ( ) ( )[ ]kzkrIArAVzrU cos2cos2

,, 0102

01 += θθ

electric QP potential(focusing)

acceleration

β: Relativistic beta factor of TRF: RF periodλ: Wavelength

Synchronous acceleration:L = particle motion during one RF cycle

= cβTRF = βλ


RF quadrupole (RFQ) linacBoundary condition:

( ) ( )2

2,0,0,0, VmaUaU == βλ

period=βλ

( ) ( )mkaIkaImmA

002

2

101

+−

=

⇒ ( )( ) 2010201 11a

kaIAa

A χ≡−=

Large m ⇒ Large acceleration

Small a ⇒ Large focusing⇒

Focusing

Acceleration


RF quadrupole (RFQ) linacElectric field components:

( ) ( ) ( )( )

( )

( ) ( )kzkrIkAVzUE

rVAUr

E

kzkrIkArAVrUE

z

r

sin2

2sin1

cos2cos22

010

01

11001

=∂∂

−=

⋅=∂∂

−=

+⋅−=∂∂

−=

θθ

θ

θ

Acceleration

RF breakdown


RF breakdown

RF breakdown⇒ damage to the cavity surface

RF breakdown provides an upper limit for the accelerating gradient


RF breakdownKilpatrick's criteria for RF breakdown:

( )kk EEf 5.8exp64.1 2 −= f: RF frequency (MHz)Ek: Critical field at cavity surface (MV/m)

The criteria is conservative for clean surfaces and short pulses⇒ Es≅2Ek is usually tolerated

0

20

40

60

80

100

120

140

0 5000 10000 15000 20000 25000 30000

Frequency (MHz)

E k (M

eV/m

)


LiteratureWave guides:• J. D. Jackson, Classical Electrodynamics, John Wiley & Sons.• D.J. Griffiths - Introduction to Electrodynamics, Prentice-Hall.• http://www.temf.de/Field-animations.6.0.html?&L=1

RF Linear accelerators:• E.A. Knapp et al., Coupled resonator model for standing• wave tanks. Rev. Sci. Instr., v. 38, n. 11, p. 22, 1967• E.A. Knapp et al., Standing wave accelerating structures for• high energy linacs. Rev. Sci. Instr., v. 39, n. 7, p. 31, 1968.• Introduction to RF linear accelerators, Mario Weiss, CERN Accelerator School: 5th

General accelerator physics course, Jyväskylä, Finland, 7 - 18 Sep 1992.• Principles of RF linear accelerators, Thomas P. Wangler, John Wiley & Sons,1998.• Medical Electron Accelerators, C. J. Karzmark, C. S. Numan, and E. Tanabe,

McGraw-Hill, 1993.• http://cas.web.cern.ch/cas/• http://uspas.fnal.gov/lect_note.html

Documents

Introduction to RF linear accelerators - AU · Introduction to RF linear accelerators Lars Hjorth Præstegaard, Ph.D. Technical manager Medical physicist Aarhus University Hospital