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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
Aarhus University Hospital, Århus Sygehus
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
Aarhus University Hospital, Århus Sygehus
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)
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Electromagnetic wave in a waveguide
Wave propagation in the +z direction
Longitudinal electric field for propagation of a wave in a waveguide? z
z
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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
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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
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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
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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)
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Waves in a circular waveguide
⇒ ( ) 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
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Waves in a circular waveguide
Ez=0 at surface
( ) ( ) ( )kztjrz erKJAtzyxE −= ω
01,0,,,
TM01 mode:
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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
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Dispersion relation for circular wave guide
Dispersion relation:
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Acceleration in circular wave guidePhase velocity = ω/k > c⇒
⇒ Poor efficiency of acceleration in circular wave guide!
Wave desynchronizeswith the particle
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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
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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
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Disk-loaded waveguide: Dispersion
Higher order modes (HOMs)
magneticcoupling
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Disk-loaded waveguide: Dispersion
possible excitation of higherorder modes (HOMs) whic are synchronousmagnetic
couplingbetween
cells
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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:
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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
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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 −=
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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
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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
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TW acceleration: Constant impedance
Resistive losses in walls + Energy transfer to the beam⇒ Reduction of microwave power along disk-loaded waveguide:
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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:
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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−=
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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 ⇒
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( ) ( )
( )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
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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
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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
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TW acceleration: X-band TW structure
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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
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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
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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
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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)
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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:
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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
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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
==
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SW acceleration: Energy gainSW acceleration with π mode:
Particle in everyother cavity
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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
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SW acceleration: Medical linacVarian 600c biperiodic π/2-mode SW structure:
microwaves incoupling cavity
Normal cavity
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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
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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
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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
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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)
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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
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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
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Phase stability for TW electron accelerators
( )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)
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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
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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
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Phase stability for TW electron accelerators
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
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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
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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
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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:
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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
Aarhus University Hospital, Århus Sygehus
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
Aarhus University Hospital, Århus Sygehus
RF quadrupole (RFQ) linacElectric field components:
( ) ( ) ( )( )
( )
( ) ( )kzkrIkAVzUE
rVAUr
E
kzkrIkArAVrUE
z
r
sin2
2sin1
cos2cos22
010
01
11001
=∂∂
−=
⋅=∂∂
−=
+⋅−=∂∂
−=
θθ
θ
θ
Acceleration
RF breakdown
Aarhus University Hospital, Århus Sygehus
RF breakdown
RF breakdown⇒ damage to the cavity surface
RF breakdown provides an upper limit for the accelerating gradient
Aarhus University Hospital, Århus Sygehus
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
)
Aarhus University Hospital, Århus Sygehus
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