RF Cavity Design Erk Jensen CERN BE/RF CERN Accelerator School Accelerator Physics (Intermediate level) Darmstadt 009 CAS Darmstadt '09 RF Cavity Design 1
Overview DC versus RF Basic equations: Lorentz & Maxwell, RF breakdown Some theory: from waveguide to pillbox rectangular waveguide, waveguide dispersion, standing waves waveguide resonators, round waveguides, Pillbox cavity Accelerating gap Induction cell, ferrite cavity, drift tube linac, transit time factor Characterizing a cavity resonance frequency, shunt impedance, beam loading, loss factor, RF to beam efficiency, transverse effects, Panofsky Wenzel, higher order modes, PS 80 MHz cavity (magnetic coupling) More examples of cavities PEP II, LEP cavities, PS 40 MHz cavity (electric coupling), RF Power sources Many gaps Why? Example: side coupled linac, LIBO Travelling wave structures Brillouin diagram, iris loaded structure, waveguide coupling Superconducting Accelerating Structures RFQ s CAS Darmstadt '09 RF Cavity Design
DC VERSUS RF CAS Darmstadt '09 RF Cavity Design 3
DC accelerator potential W q E d s q DC versus RF RF accelerator CAS Darmstadt '09 RF Cavity Design 4
Lorentz force A charged particle moving with velocity through an electromagnetic field experiences a force d d p t q E v B v p v m W mc The energy of the particle is pc mc W kin mc 1 Change of W due to the this force (work done) ; differentiate: WdW c pdp qc p dw qv Edt E v B dt qc p E dt Note: no work is done by the magnetic field. CAS Darmstadt '09 RF Cavity Design 5
Maxwell s equations (in vacuum) why not DC? 1) t 0 1 B c t E t E J 0 B 0 E B 0 c E 0 E DC ( ): which is solved by Limit: If you want to gain 1 MeV, you need a potential of 1 MV! 0 ) Circular machine: DC acceleration impossible since E ds 0 With time varying fields: E t B E ds B t da CAS Darmstadt '09 RF Cavity Design 6
Maxwell s equation in vacuum (contd.) CAS Darmstadt '09 RF Cavity Design 7 0 1 E t c E E E E 0 1 E t c E vector identity: curl of 3 rd and of 1 st equation: t with 4 th equation : i.e. Laplace in 4 dimensions 0 0 0 0 1 E B t E B E t c B
Another reason for RF: breakdown limit surface field, in vacuum,cu surface, room temperature Wang & Loew, SLAC-PUB-7684, 1997 Approximate limit for CLIC parameters (1 GHz, 140 ns, breakdown rate: 10-7 m -1 ): 60 MV/m Kilpatrick 1957, 4.67 f E c e 4.5 f in GHz, E c in MV/m E c CAS Darmstadt '09 RF Cavity Design 8
FROM WAVEGUIDE TO PILLBOX CAS Darmstadt '09 RF Cavity Design 9
Homogeneous plane wave E u B u k r y x cos t k r cos t k r cos z sin c x x Wave vector : the direction of is the direction of propagation, the length of k is the phase shift per unit length. behaves like a vector. k k k k c k c c E y φ z k z c c 1 CAS Darmstadt '09 RF Cavity Design 10
Wave length, phase velocity The components of are related to the wavelength in the direction of that component as etc., to the phase velocity as. CAS Darmstadt '09 RF Cavity Design 11 c k 1 c z c k c k c z z k k c k c c k z z z f k v, z x E y
Superposition of homogeneous plane waves E y x z + = Metallic walls may be inserted where without perturbing the fields. Note the standing wave in x direction! This way one gets a hollow rectangular waveguide E y 0 CAS Darmstadt '09 RF Cavity Design 1
Rectangular waveguide Fundamental (TE 10 or H 10 ) mode in a standard rectangular waveguide. E.g. forward wave -y power flow electric field power flow: 1 Re cross section E H * d A x z power flow magnetic field z x CAS Darmstadt '09 RF Cavity Design 13
Waveguide dispersion k z Im g e.g.: TE 10 wave in rectangular waveguide: slow wave free space, ω/c j c j Z0 a cutoff a general cylindrical waveguide: c fast wave j Z 0 k c j for TE, Z 0 for TM j In a hollow waveguide: phase velocity > c, group velocity < c CAS Darmstadt '09 RF Cavity Design 14
Waveguide dispersion (continued: Higher Order Modes) TE 0 TE 01 TE 10 free space, ω/c c TE 10 k z Im CAS Darmstadt '09 RF Cavity Design 15
General waveguide equations: Transverse wave equation (membrane equation): boundary condition: longitudinal wave equations (transmission line equations): TE (or H) modes T c T c 0 TM (or E) modes n T 0 T 0 du d z d I d z z z Z 0 0 I U Z z 0 z 0 propagation constant: characteristic impedance: ortho normal eigenvectors: transverse fields: longitudinal field: H z c c Z 0 u j e z T E H I TU z j c j 1 c U z zu z e e Z 0 j CAS Darmstadt '09 RF Cavity Design 16 E z e c c T z T I j
CAS Darmstadt '09 RF Cavity Design 17 TM (E) modes: TE (H) modes: y b n x a m na mb ab T n m H mn cos cos 1 ) ( y b n x a m na mb ab T E mn sin sin ) ( 0 0 1 i for i for i m m a T mn m mn mn m m E mn cos sin J J 1 ) ( m m a m T mn m mn m mn m H mn sin cos J J ) ( TE (H) modes: TM (E) modes: where Ø = a a b b n a m c c a c c mn
Standing wave resonator Same as above, but two counter running waves of identical amplitude. electric field no net power flow: 1 Re cross section E H * d A 0 magnetic field (90 out of phase) CAS Darmstadt '09 RF Cavity Design 18
Round waveguide parameters used in calculation: f = 1.43, 1.09, 1.13 f c, a: radius E B TE 11 : fundamental mode TM 01 : axial electric field TE 01 : lowest losses! f c 87.85 GHz a / mm f c 114.74 GHz a / mm 334.74 GHz a / mm CAS Darmstadt '09 RF Cavity Design 19 f c
Pillbox cavity TM 010 mode (only 1/8 shown) electric field magnetic field CAS Darmstadt '09 RF Cavity Design 0
Pillbox cavity field (w/o beam tube) CAS Darmstadt '09 RF Cavity Design 1 a a T 01 1 01 01 0 J J 1,... 01.40483 a a a B a a a a E z 01 1 01 1 0 01 1 01 0 01 0 J J 1 J J 1 j 1 a c pillbox 01 0 a h a h Q R pillbox ) ( sin J 4 01 01 1 3 01 h a a Q pillbox 1 01 The only non vanishing field components : h Ø a 377 0 0 for later:
ACCELERATING GAP CAS Darmstadt '09 RF Cavity Design
Accelerating gap We want a voltage across the gap! It cannot be DC, since we want the beam tube on ground potential. Use Eds The shield imposes a db da dt upper limit of the voltage pulse duration or equivalently a lower limit to the usable frequency. gap voltage The limit can be extended with a material which acts as open circuit! Materials typically used: ferrites (depending on f-range) magnetic alloys (MA) like Metglas, Finemet, Vitrovac resonantly driven with RF (ferrite loaded cavities) or with pulses (induction cell) CAS Darmstadt '09 RF Cavity Design 3
Linear induction accelerator E ds B da t compare: transformer, secondary = beam Acc. voltage during B ramp. CAS Darmstadt '09 RF Cavity Design 4
Ferrite cavity PS Booster, 98 0.6 1.8 MHz, < 10 kv gap NiZn ferrites CAS Darmstadt '09 RF Cavity Design 5
Gap of PS cavity (prototype) CAS Darmstadt '09 RF Cavity Design 6
Drift Tube Linac (DTL) how it works For slow particles! E.g. protons @ few MeV The drift tube lengths can easily be adapted. electric field CAS Darmstadt '09 RF Cavity Design 7
Drift tube linac practical implementations CAS Darmstadt '09 RF Cavity Design 8
Transit time factor If the gap is small, the voltage is small. E z dz If the gap large, the RF field varies notably while the particle passes. Define the accelerating voltage Transit time factor E z e E j z c z dz dz V gap E z e j z c dz Example pillbox: transit time factor vs. h sin 01 h h 01 a a h/ CAS Darmstadt '09 RF Cavity Design 9
CHARACTERIZING A CAVITY CAS Darmstadt '09 RF Cavity Design 30
Cavity resonator equivalent circuit Simplification: single mode I G V gap I B Generator P Beam R/β C L R L=R/(Q 0 ) β: coupling factor Cavity C=Q/(R 0 ) R: Shunt impedance L C : R upon Q CAS Darmstadt '09 RF Cavity Design 31
Resonance CAS Darmstadt '09 RF Cavity Design 3
Reentrant cavity Nose cones increase transit time factor, round outer shape minimizes losses. nose cone Example: KEK photon factory 500 MHz R probably as good as it gets this cavity optimized pillbox R/Q: 111 Ω 107.5 Ω Q: 4470 41630 R: 4.9 MΩ 4.47 MΩ CAS Darmstadt '09 RF Cavity Design 33
k loss R Q V 4 W 0 gap 1 C Loss factor V (induced) Impedance seen by the beam I B Beam Energy deposited by a single charge q: Voltage induced by a single charge q: V k gap loss q k loss 1 0-1 q R/ C Cavity 0 e 0 5 10 15 0 t f 0 L R CAS Darmstadt '09 RF Cavity Design 34 Q L L=R/(Q 0 ) C=Q/(R 0 ) t
Summary: relations V gap, W, P loss gap voltage k R Q loss V gap W 0 R 0 Q V gap 4 W R shunt V gap P loss Energy stored inside the cavity 0 Q W P loss Power lost in the cavity walls CAS Darmstadt '09 RF Cavity Design 35
Beam loading RF to beam efficiency The beam current loads the generator, in the equivalent circuit this appears as a resistance in parallel to the shunt impedance. If the generator is matched to the unloaded cavity, beam loading will cause the accelerating voltage to decrease. The power absorbed by the beam is gap the power loss P. V R For high efficiency, beam loading shall be high. B The RF to beam efficiency is. 1 Re CAS Darmstadt '09 RF Cavity Design 36 1 1 V R gap I B V * gap I B I I G
Resonance frequency Transit time factor Characterizing cavities field varies while particle is traversing the gap Shunt impedance gap voltage power relation V R gap 0 E z e E j z c 1 L C Circuit definition z dz dz shunt P loss V Linac definition gap R shunt P loss Q factor 0 W QP loss R/Q independent of losses only geometry! R Q Vgap W 0 L C R Q V gap W 0 loss factor k loss R 0 Q V 4 W CAS Darmstadt '09 RF Cavity Design 37 gap k loss R 0 4 Q V gap 4 W
Example Pillbox: 0 R Q pillbox pillbox 01 c a a Q pillbox 1 3 01 a h 01 4 J 1 sin ( h a 01 01 h ) a 01.4048 Cu 0 0 5.810 377 7 S/m CAS Darmstadt '09 RF Cavity Design 38
Higher order modes external dampers... R 1, Q 1, 1 R, Q, R 3, Q 3, 3... n 1 n n 3 I B CAS Darmstadt '09 RF Cavity Design 39
Higher order modes (measured spectrum) without dampers with dampers CAS Darmstadt '09 RF Cavity Design 40
Pillbox: Dipole mode (TM 110 ) (only 1/8 shown) electric field (@ 0º) magnetic field (@ 90º) CAS Darmstadt '09 RF Cavity Design 41
Panofsky Wenzel theorem For particles moving virtually at v=c, the integrated transverse force (kick) can be determined from the transverse variation of the integrated longitudinal force! j F F c Pure TE modes: No net transverse force! Transverse modes are characterized by the transverse impedance in -domain the transverse loss factor (kick factor) in t-domain! W.K.H. Panofsky, W.A. Wenzel: Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields, RSI 7, 1957] CAS Darmstadt '09 RF Cavity Design 4
CERN/PS 80 MHz cavity (for LHC) inductive (loop) coupling, low self inductance CAS Darmstadt '09 RF Cavity Design 43
Higher order modes Example shown: 80 MHz cavity PS for LHC. Color coded: E CAS Darmstadt '09 RF Cavity Design 44
MORE EXAMPLES OF CAVITIES CAS Darmstadt '09 RF Cavity Design 45
PS 19 MHz cavity (prototype, photo: 1966) CAS Darmstadt '09 RF Cavity Design 46
Examples of cavities PEP II cavity 476 MHz, single cell, 1 MV gap with 150 kw, strong HOM damping, LEP normal conducting Cu RF cavities, 350 MHz. 5 cell standing wave + spherical cavity for energy storage, 3 MV CAS Darmstadt '09 RF Cavity Design 47
CERN/PS 40 MHz cavity (for LHC) example for capacitive coupling coupling C cavity CAS Darmstadt '09 RF Cavity Design 48
RF POWER SOURCES CAS Darmstadt '09 RF Cavity Design 49
RF Power sources > 00 MHz: Klystrons Thales TH1801, Multi Beam Klystron (MBK), 1.3 GHz, 117 kv. Achieved: 48 db gain, 10 MW peak, 150 kw average, η = 65 % db: output power input power 4.8 10 < 1000 MHz: grid tubes pictures from http://www.thales electrondevices.com Tetrode IOT UHF Diacrode CAS Darmstadt '09 RF Cavity Design 50
RF power sources 10000 Typical ranges (commercially available) grid tubes 1000 klystrons CW/Average power [kw] 100 10 solid state (x3) IOT CCTWTs 1 Transistors 0.1 10 100 1000 f [MHz] 10000 CAS Darmstadt '09 RF Cavity Design 51
Example of a tetrode amplifier (80 MHz, CERN/PS) 400 kw, with fast RF feedback 18 Ω coaxial output (towards cavity) kv DC anode voltage feed through with λ/4 stub tetrode cooling water feed throughs coaxial input matching circuit CAS Darmstadt '09 RF Cavity Design 5
MANY GAPS CAS Darmstadt '09 RF Cavity Design 53
What do you gain with many gaps? The R/Q of a single gap cavity is limited to some 100 W. Now consider to distribute the available power to n identical cavities: each will receive P/n, thus produce an accelerating voltage of R P n. The total accelerating voltage thus increased, equivalent to a total equivalent shunt impedance of nr. P/n P/n P/n P/n P V acc n R n nrp 1 3 n CAS Darmstadt '09 RF Cavity Design 54
Standing wave multicell cavity Instead of distributing the power from the amplifier, one might as well couple the cavities, such that the power automatically distributes, or have a cavity with many gaps (e.g. drift tube linac). Coupled cavity accelerating structure (side coupled) The phase relation between gaps is important! CAS Darmstadt '09 RF Cavity Design 55
Example of Side Coupled Structure A 3 GHz Side Coupled Structure to accelerate protons out of cyclotrons from 6 MeV to 00 MeV Medical application: treatment of tumours. Prototype of Module 1 built at CERN (000) Collaboration CERN/INFN/ Tera Foundation CAS Darmstadt '09 RF Cavity Design 56
LIBO prototype This Picture made it to the title page of CERN Courier vol. 41 No. 1 (Jan./Feb. 001) CAS Darmstadt '09 RF Cavity Design 57
TRAVELLING WAVE STRUCTURES CAS Darmstadt '09 RF Cavity Design 58
Brillouin diagram Travelling wave structure L/c speed of light line, /c synchronous L CAS Darmstadt '09 RF Cavity Design 59
Iris loaded waveguide 11.4 GHz structure (NLC) 1 cm 30 GHz structure CAS Darmstadt '09 RF Cavity Design 60
Disc loaded structure with strong HOM damping choke mode cavity Dimensions in mm CAS Darmstadt '09 RF Cavity Design 61
Power coupling with waveguides Input coupler shown: Re {Poynting vector} (power density) Output coupler Travelling wave structure (CTF3 drive beam, 3 GHz) ¼geometry shown CAS Darmstadt '09 RF Cavity Design 6
3 GHz Accelerating structure (CTF3) CAS Darmstadt '09 RF Cavity Design 63
Examples (CLIC structures @ 11.4, 1 and 30 GHz) T18 reached 105 MV/m! HDS novel fabrication technique CAS Darmstadt '09 RF Cavity Design 64
SUPERCONDUCTING ACCELERATING STRUCTURES CAS Darmstadt '09 RF Cavity Design 65
LEP Superconducting cavities 10. MV/ per cavity CAS Darmstadt '09 RF Cavity Design 66
LHC SC RF, 4 cavity module, 400 MHz CAS Darmstadt '09 RF Cavity Design 67
ILC high gradient SC structures at 1.3 GHz 5 35 MV/m CAS Darmstadt '09 RF Cavity Design 68
Small superconducting cavities (example RIA, Argonne) 115 MHz split ring cavity, 17.5 MHz β = 0.19 lollipop cavity 57.5 MHz cavities: β= 0.06 QWR (quarter wave resonator) β = 0.03 fork cavity 345 MHz β = 0.4 spoke cavity β = 0.01 fork cavity pictures from Shepard et al.: Superconducting accelerating structures for a multi beam driver linac for RIA, Linac 000, Monterey CAS Darmstadt '09 RF Cavity Design 69
RFQ S CAS Darmstadt '09 RF Cavity Design 70
Old pre injector 750 kv DC, CERN Linac before 1990 All this was replaced by the RFQ CAS Darmstadt '09 RF Cavity Design 71
RFQ of CERN Linac CAS Darmstadt '09 RF Cavity Design 7
The Radio Frequency Quadrupole (RFQ) Minimum Energy of a DTL: 500 kev (low duty) 5 MeV (high duty) At low energy / high current we need strong focalisation Magnetic focusing (proportional to β) is inefficient at low energy. Solution (Kapchinski, 70 s, first realised at LANL): Electric quadrupole focusing + bunching + acceleration CAS Darmstadt '09 RF Cavity Design 73
RFQ electrode modulation The electrode modulation creates a longitudinal field component that creates the bunches and accelerates the beam. CAS Darmstadt '09 RF Cavity Design 74
A look inside CERN AD s RFQ D CAS Darmstadt '09 RF Cavity Design 75