High-beta Cavity Design

High-beta Cavity Design K.Saito High Energy Accelerator Research Organization (KEK), Accelerator Lab 1. SW Operation with SRF RF Cavity. Pill Box Cavity 3. Figure of merits of SRF Cavity Design 4. Criteria for Multi-Cell Structures 5. Example of SRF High-beta Cavities Acknowledgement: In this lecture, I used many slides by Dr. J.Sektuwitz@ DESY, which he made nice lecture in China. I would like to thank for him very much. K.Saito SRF09 Tutorial on 17 Sept. 009 1

1. Sanding Wave (SW) Operation and Standing Wave in SRF Cavity

SW Scheme in SRF Cavity Operation 6. Multi-cell Structures and Weakly Coupled Structures Cavities E z (r=0,z) - + - + π-phase advance - cell-to-cell ΤΜ 010 π-mode v = ßc l cell l active s Standing wave (CW) is used in SRF cavity acceleration! Synchronic acceleration and max of (R/Q) acc l active =Nl cell = Ncß/(f) and the injection takes place at an optimum phase φ opt which ensures that particles will arrive at the midplane of the first cell when E acc reaches its maximum (+q passing to the right) or minimum (-q passing to the right). L cell β λ =, λ = c f β=v/c for electron β=1, proton β < 1 3

Transit Time Factor Due to SW Operation t = 0, z = 0 t T λ T λ =, z = t =, z = 6 6 4 4 t 5T 5λ T λ =, z = t =, z = 1 1 v=c d=λ/ e - e - e - e - e - Z ωd z z sin d d iω d iω iωt c c c V = E ( 0, ) ( 0, ) 0 z r z e dz E 0 z r z e dz E0 e dz E 0 0d = = = = = = E0d T ωd z= ct c T : Transit time factor T = = 0.637 (for Pill Box Cavity) π V Eacc = E0T d Acceleration efficiency is automatically reduced by ~ 40% in the SW scheme. 4

. Pill Box Cavity 5

LC Equivalent Circuit of Cavity I = dq dt d V dt, Q = CV, V = L di dt + RI + R L dv dt + 1 LC = 0, ( α + iω) + ( α + iω) R + L α = R L, ω = 1 LC R 4L V(t) = V O exp( α + iω)t 1 LC R << L, ω O = 1 LC f = 1 π LC = 0, Q-value of the circuit Stored Energy P( t) L Q ω = ω = ω Powerloss/sec dp( t) / dt R ω = α Q : proportional to 1/R 6

Electro-magnetic field in a waveguide Maxwell equations in a waveguide ω ω E= i B, B=0, B= iμε E, E =0, ρ=0, j=0 c c ω E με + = 0, c B E( xyzt,,, ) = E( xy, )exp( ± ikz iωt), k: wavevector, B(, xyxt,,) = B(, xy)exp( ± ikz iωt), ω Ε t + ( εμ k 0,,, = t E z z t Bz z t c = E= e + E B e + B B z 1 Bz ω Homework I. Bt =, t + iεμ ez tez ω z c Lead this formula. εμ k c 1 Ez ω TM mode: Bz = 0, Ez 0 Et = t i ez tbz ω z c εμ k TE mode : Bz 0, Ez = 0 c 7

TM- Modes B E TMmode : B z = 0, E z 0 [ e E ] t z t z ω εμ k t ω iεμ = c c 1 E =, z t ω z εμ k c ω tez + ( εμ k ) E 0, z = c, Can accelerate beam Beam Solve the eigenvalue problem, get k and Ez Boundary condition E z S = 0 ( Q B z n E= 0 nb = 0 on the surface of perfect conducto n S = 0 ( Q on the surface, but automatically satisfied by the TM - mode condition) Similar for TE modes, which kicks beam 8

Eigen-value problem ψ(x, y ) = E z (x, y ) for TM- mode or B z (x, y) for TE- mode r ( ) t + γ ψ = 0, ψ = 0 (for TM-Modes) or ψ = 0 (for TE-Modes) S S n ω γ = εμ k 0 c From the boundary condition, γ ω = γ λ, ψ = ψ λ (λ =1,,L) k λ = εμ c γ λ γλ If ω < c, then k is an imaginal number. εμ λ The wave is damped/trapped in the waveguide, if surface resistance large/small. γλ ωλ = c Lcutoff frequency εμ When ω ω, wave number k is a real number, λ then the wave can propagate into the waveguide. λ 9

TM mn,, p mode ρmn, Ez = Eo cos( kz) Jm( r)exp( imθ ), Bz = 0 a ie pπ pπ J ( ρ) Emεμω ρ E = cos( z) exp( im ), B = cos( kz) J ( r)exp( imθ ) 0 m 0 mnp,, mn, r θ r m γmnp,, d ρ γm,n,p a Empπ pπ ρmn, ie0 εμωmn,, p J m ( ρ) Eθ = cos( z) Jm ( r)exp( imθ), Bθ = cos( kz)exp( imθ) γ dc d a γ c ρ 0 mnp,, mn,, p resonace frequency ω c mn, p π mnp,, fmnp,, = + εμ ρ a d TE mnp Modes E z E E r θ = 0 iωε m = E 0J k r iωε ρ' mn = E k a m 0 ρ' mn pπz r sin( mθ )sin( ) a d ρ' mn pπz J m r cos( mθ )sin( ) a d H H r H z θ = E 0 1 = k J m 1 = k pπ E d ρ' a mn 0 J m pπ m d r pπz r cos( mθ )sin( ) d ρ' a E 0 J mn m pπz r cos( mθ )cos( ) d ρ' a mn pπz r sin( mθ )cos( ) d Resonance frequency ω ρ' mn pπ c ρ' mn ε 0μ0 = + f = a d π a + pπ d 10

TM mnp Modes Acceleration Modes d a r=a J0(ρ01) J0(ρ0) J1(ρ11) Ez 11

TE mnp Modes Kick Modes Z 1

4. Figure of merits of Cavity Design 13

Figure of Merits of The RF Cavity Design Items 1) Surface resistance R S [Ω] ) RF dissipation P loss [W] 3) Acceleration gradient V[V] 3) Unloaded Q Q O 4) Geometrical factor Γ[Ω] 5) Shunt Impedance Rsh[Ω] 6) R over Q (R/Q) [Ω] Notation 7) Acceleration gradient Eacc[V/m] 8) Ratio of the surface peak electric field vs Eacc Ep/Eacc [Oe/(MV/m) 9) Ratio of the surface magnetic field vs Eacc Hp/Eacc 10) HOM loss factor 11) Field flatness factor a ff 1) Cell to cell coupling k CC κ //, κ

Maxwell Equations. r r r H B = 0, E + μ = 0 t r r r E r D = 0, H ε σe = 0 t r r J = σ E (Ohm's Law) Surface Resistance r r r r r r E(x,t) = E l (x,t) + E t (x,t), r r r r r r H(x,t) = H (x,t) + H (x,t) l For the transvers, r r r r r Plane wave : E (x,t) = E (0) exp( ik x - ωt) t r 1 r r r H t(x,t) = [k E t(x,t)], μω r r E(x,t) εμω + μωσ r r H(x,t) t t [ k ( i ) ] = 0 t t Rs: Surface resistance Z R ix E t s + s = Ht Surface Surface Impedance μω k Homework II Lead the formula of Rs for good electric conductor R s = μω σ = 1 σ μσω = 1 σδ 1 P R H loss = S S S ds 15

Charismatic RF Parameter of Cavity U stored energy, Ploss:Surface RF Heating Unloaded Q O 1 1 1 Ploss = R S HS ds S ωu QO P For acceleration mode (TM010), the higher Qo is better. U = ε E dv = μ H dv V V loss Acceleration Voltage 0 L eff V[V] = Edz on the beam axis. Shunt Impedance R [ ] sh Ω V P loss This means the efficiency of the acceleration. 16

Charismatic RF Parameter of Cavity, continued. 3. RF Parameters Cavities (R/Q) ( R/ Q) ( R / Q ) = sh O V ωu This means how much energy is concentrated on beam axis. This does not depend on material but only cavity shape. This means the goodness of the cavity shape. Geometrical factor ω ΓΩ [ ] QO RS = 1 V S H dv H ds When you know the Γ using a computer code, you can calculate the surface resistance R S from the measured Q O value. Γ is about 70Ω for β=1 cavities with elliptical or spherical shape. S 17

Charismatic RF Parameter of Cavity Gradient ( R/ Q) E = P Q = Z P Q acc loss O loss O Leff E E peak acc l active Contour plot of /E/ B E peak acc Contour plot of /B/ Smaller value is preferred against electron emission phenomenon. Ratio shows the limit in E acc due to the break-down of superconductivity (Nb ~000 Oe ). 18

Longitudinal and Transverse Loss Factors When ultra relativistic point charge q passes empty cavity, Cone ~1/γ E r ~1/r r o HOM modes are excited in the cavity. If those are damped fast enough, the following beam feel them and can be kicked. Thus the beam can be degradated. 19

3. RF Parameters Cavities (R/Q) n of n-th HOM HOM nth (R/Q) n, a measure of the energy exchange between point charge and mode n. mode n : {ω n,e n, H n }. Trajectory of the point charge q, a W n b assumed here to be a straight line. V n zb zb ωn = En,z sin( ( z za ))dz + βc za za E n,z ωn cos( βc ( z z a ))dz (R / Q ) n V n n ω W n 0

3. RF Parameters Cavities Longitudinal and Transverse Loss Factors The amount of energy lost by charge q to the cavity is: J.Sekutwitz s Slide ΔU q = k q for monopole modes (max. on axis) ΔU q = k q for non monopole modes (off axis) where k and k (r) are loss factors for the monopole and transverse modes respectively. The induced E-H field (wake) is a superposition of cavity eigenmodes (monopoles and others) having the E n (r,φ,z) field along the trajectory. For individual mode n and point-like charge: k p,n n = ω (R / 4 Q ) n Similar for other loss factors. These harmful HOM power should be take out from the cavity. Dr. Noguchi will make a lecture about the HOM coupler design. 1

Field Flatness Factor a ff J.Sekutwitz s Slide Field flatness factor for elliptical cavities with arbitrary ß=v/c ΔA A = a ff Δf f a ff = k N cc ß This is an empirical correction, based on intuition. The same error x causes bigger f ß 1 when a cell is thinner ß f f 1 = x ß x ß 1 Cells which geometric ß <1 are more sensitive to shape errors

Cell to cell coupling k CC The last parameter, relevant for multi-cell accelerating structures, is the coupling k cc between cells for the accelerating mode pass-band (Fundamental Mode pass-band). ω o ω π + + + - no E r (in general transverse E field) component at the symmetry plane no H φ (in general transverse H field) component at the symmetry plane The normalized difference between these frequencies is a measure of the Poynting vector (energy flow via the coupling region) k cc = ω ω π π ω0 + ω0 3

Spherical/Elliptical Shape in SRF cavity design Design of SRF cavity is usually done with spherical or Elliptical shapes. Choose spherical shape Multipacting to reduce multipacting R. Parodi (1979) presented first spherical C-band cavity with much less multipacting barrier than other cavities at that time. P. Kneisel (early 80 s) proposed for the DESY experiment the elliptical shape of 1 GHz cavity preserving good performance of the spherical one and stiffer mechanically. The RF cavity design tool will be given by Dr. U. van Rienen In this tutorial. 4

5. Geometry and Criteria for Cavity Design Cavities We begin with inner cells design because these cells dominate parameters of a multi-cell superconducting accelerating structure. RF parameters summary: J.Sekutwitz s Slide FM : (R/Q), G, E peak /E acc, B peak /E acc, k cc HOM : k,k. There are 7 parameters we want to optimize for a inner cell Geometry : Optimization of Cell Shape iris ellipsis : half-axis h r,h z iris radius : r i equator ellipsis : half-axis h r,h z There is some kind of conflict 7 parameters and only 5 variables to tune 5

5. Geometry Effect and Criteria of Cavity for Aperture Design on RF ParametersCavities Example: f = 1.5 GHz A. Mosnier, E. Haebel, SRF Workshop 1991 6

Aperture Effects on κ // and κ ᅩ ) 8.0 J.Sekutwitz s Slide k_mon(r)/k_mon(40) k_dipole(r)/k_dipole(40) 6.0 4.0.0 0.0 0 5 30 35 40 ri [mm] (R/Q) = 15 Ω B peak / E acc = 3.5 mt/(mv/m) E peak / E acc = 1.9 (R/Q) = 86 Ω B peak / E acc = 4.6 mt/(mv/m) E peak / E acc = 3. 7

Aperture Effect on Cell to Cell Coupling (κ CC ) 1.0 J.Sekutwitz s Slide 0.8 kcc(r)/kcc(40) 0.6 0.4 0. 0.0 0 5 30 35 40 ri [mm] (R/Q) = 15 Ω B peak / E acc = 3.5 mt/(mv/m) E peak / E acc = 1.9 (R/Q) = 86 Ω B peak / E acc = 4.6 mt/(mv/m) E peak / E acc = 3. 8

5. Geometry General Trends and Criteria of Cavity for Optimization Cavity Design on RF Geometrical Parameters Cavities iris ellipsis : half-axis h r,h z iris radius : r i equator ellipsis : half-axis h r,h z Criteria RF-parameter Improves when Cavity examples Operation at E peak / E acc r i TESLA, high gradient B peak / E acc Iris, Equator shape HG CEBAF-1 GeV Low cryogenic losses (R/Q) Γ r i Equator shape LL CEBAF-1 GeV LL- ILC cavity High I beam Low HOM impedance B-Factory k, k r i 9 RHIC cooling

Choice of the RF Frequency What about accelerating mode frequency of a superconducting cavity? J.Sekutwitz s Slide x = f π [MHz] 600 R/Q [Ω] 57 r/q=(r/q)/l [Ω/m] 000 G [Ω] 71 f π [MHz] 1300 R/Q [Ω] 57 r/q=(r/q)/l [Ω/m] 1000 G [Ω] 71 r/q=(r/q)/l ~ f 30

Frequency Choice of the SRF Cavity 1 A Δ Ploss = R s HS ds Eacc, RS ( BCS) = f exp( ) S T k T B Cryogenic power Cavity fabrication cost 350MHz 500MHz (U-band) f 1.3GHz (L-band).896GHz (S-band) L-band cavity is preferred and it is operated at K to reduce the cryogenic power. Cost + Existing RF source f 31

TESLA Cavity Inner Cell Shape The inner cell geometry was optimize with respect to: low E peak /E acc and coupling k cc. At that time (199) the field emission phenomenon and field flatness were of concern, no one was thinking about reaching the magnetic limit. f π [MHz] 1300.0 r iris [mm] 35 k cc [%] 1.9 E peak /E acc - 1.98 B peak /E acc [mt/(mv/m)] 4.15 R/Q [Ω] 113.8 G [Ω] 71 R/Q*G [Ω*Ω] 30840 Inner cell; Contour of E field 3

High Gradient Shapes Cavity shape designs with low Hp/Eacc from J.Sekutowicz lecture Note TTF: TESLA shape Reentrant (RE): Cornell Univ. Low Loss(LL): JLAB/DESY Ichiro ー Single(IS): KEK TESLA LL RE IS Diameter [mm] 70 60 66 61 Ep/Eacc.0.36.1.0 Hp/Eacc [Oe/MV/m] 4.6 36.1 37.6 35.6 R/Q [W] 113.8 133.7 16.8 138 G[W] 71 84 77 85 Eacc max 41.1 48.5 46.5 49. 33

Eacc vs. Year Eacc,max [MV/m] 70 60 50 40 30 1 st Breakthrough! High pressuer water rinsing (HPR) Electropolshing(EP) + HPR + 10 O C Bake nd Breakthrough! RE,LL,IS shape New Shape 0 Chemical Polishing 10 '91 '93 '95 '97 '99 '00 '03 '05 '07 34 Date [Year]

4. Criteria for Multi-cell Structures Single-cell is attractive from the RF-point of view: Easier to manage HOM damping No field flatness problem. Input coupler transfers less power Easy for cleaning and preparation But it is expensive to base even a small linear accelerator on the single cell. We do it only for very high beam current machines. A multi-cell structure is less expensive and offers higher real-estate gradient but: Field flatness (stored energy) in cells becomes sensitive to frequency errors of individual cells Other problems arise: HOM trapping How to decide the number of cells? 35

HOM Parameters considered with N RF Field flatness factor : HOM trapped mode : Input handling power : Handling : Δ A Δ f N Δ f A f k f i i i = a ff = i i cc i a ff = N k Beam pipe has no acceleration beam. BP reduce the efficiency. Multi-cell is more efficient. cc More serious within creasing N PINPUT N Easy to bend, Difficult to preparation.. 36

6. Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity Factor Field flatness vs. N J.Sekutwitz s Slide Original Cornell N = 5 High Gradient N =7 Low Loss N =7 TESLA N=9 SNS ß=0.61 N=6 SNS ß=0.81 N=6 RIA ß=0.47 N=6 RHIC N=5 year 198 001 00 199 000 000 003 003 a ff 1489 59 388 4091 3883 94 5040 850 Many years of experience with: heat treatment, chemical treatment, handling and assembly allows one to preserve tuning of cavities, even for those with bigger N and weaker k cc For the TESLA cavities: field flatness is better than 95 % 37

HOM trapping vs. N J.Sekutwitz s Slide No fields at HOM couplers positions, which are always placed at end beam tubes N = 17 N = 13 N = 9 N = 5 e-m fields at HOM couplers positions Cure for the trapped mode: Make the bore radius larger. Break mirror symmetry. 38

1. Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell Structure HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions, which J.Sekutwitz s Slide are always placed at end beam tubes The HOM trapping mechanism is similar to the FM field profile unflatness mechanism: weak coupling HOM cell-to-cell, k cc,hom difference in HOM frequency of end-cell and inner-cell f = 385 MHz That is why they hardly resonate together f = 415 MHz 39

6. Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells J.Sekutwitz s Slide The geometry of end-cells differs from the geometry of inner cells due to the attached beam tubes + + Their function is multi-folded and their geometry must fulfill three requirements: field flatness and frequency of the accelerating mode field strength of the accelerating mode at FPC location enabling operation with matched Qext fields strength of dangerous HOMs ensuring their required damping by means of HOM couplers or/and beam line absorbers. All three make design of the end-cells more difficult than inner cells. 40

6. Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N Power capability of fundamental power couplers vs. N When I beam and E acc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode: P INPUT ~ N Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1). KEKB: Acceleration beam current > 1 A. 41

. TESLA Cavities and Auxiliaries as ILC Baseline Design TESLA/ILC Baseline Cavity The cavity was designed in 199 (A. Mosnier, D. Proch and J.S. ). End-cell 1 7 identical inner cells End-cell TTF 9-cells; Contour of E field f π [MHz] 1300.00 f π-1 [MHz] 199.4 R/Q [Ω] 101 G [Ω] 71 Active length [mm] 1038 4

Examples of Inner cells Overview of Cavities J.Sekutwitz s Slide new new new new CEBAF Original Cornell ß=1 CEBAF -1 High Gradient ß=1 CEBAF -1 Low Loss ß=1 TESLA ß=1 SNS ß=0.61 SNS ß=0.81 RIA ß=0.47 RHIC Cooler ß=1 f o [MHz] 1448.3 1468.9 1475.1 178.0 79.8 79.8 793.0 683.0 f π [MHz] 1497.0 1497.0 1497.0 1300.0 805.0 805.0 805.0 703.7 k cc [%] 3.9 1.89 1.49 1.9 1.5 1.5 1.5.94 E peak /E acc -.56 1.96.17 1.98.66.14 3.8 1.98 B peak /E acc [mt/(mv/m)] 4.56 4.15 3.74 4.15 5.44 4.58 6.51 5.78 R/Q [Ω] 96.5 11 18.8 113.8 49. 83.8 8.5 80. G [Ω] 73.8 66 80 71 176 6 136 5 R/Q*G [Ω*Ω] 641 979 36064 30840 8659 18939 3876 18045 k (σ z =1mm) [V/pC/cm ] 0. 0.3 0.53 0.3 0.13 0.11 0.15 0.0 k (σ z =1mm) [V/pC] 1.36 1.53 1.71 1.46 1.5 1.7 1.19 0.85 β vs RF parameters β smaller Ep/Eacc larger Hp/Eacc larger (R/Q) smaller 43

5. Example of SRF High-beta Cavities 44

6. Multi-cell Structures and Weakly Coupled Structures Cavities Cavities operating with highest I beam or E acc J.Sekutwitz s Slide Type /No. of cavities P beam /cavity [kw] P HOM /cavity [kw] KEK-B 0.5 GHz x 8 Single-cell with max I beam I beam = 1.34 A 1389 bunches cw 350 16 HERA 0.5 GHz x 16 Multi-cell with max I beam I beam 40 ma 180 bunches cw 60 0.13 TTF-I, 1.3 GHz x 1 Multi-cell with max E acc E acc= 35 MV/m 1.3ms/pulse 1Hz PRF ~100 Almost no beam loading 0 45

6. Multi-cell Structures and Weakly Coupled Structures Cavities Cavities which will operate with high I beam in the near future J.Sekutwitz s Slide Type /No. cavities P beam /cavity [kw] P HOM /cavity [kw] SNS ß= 0.61, 0.805 GHz x 33 40 (366) 0.06 peak SNS ß= 0.805 GHz Multi-cell with max I beam I beam =38 (59 ) ma 1.3ms/pulse DF = 6 % x 48 48 0.06 peak TTF-II ep, 1.3 GHz x 8 Multi-cell with max E acc E acc = 35 MV/m 1.3ms/pulse 10Hz PRF 146 < 0.0> 46