1
High-beta Cavity Design
1 SW Operation with SRF RF Cavity2 Pill Box Cavity3 Figure of merits of SRF Cavity Design 4 Criteria for Multi-Cell Structures5 Example of SRF High-beta Cavities
KSaitoHigh Energy Accelerator Research Organization (KEK)
Accelerator Lab
KSaito SRF09 Tutorial on 17 Sept 2009
AcknowledgementIn this lecture I used many slides by Dr JSektuwitz DESY which he made nice lecture in China I would like to thank for him very much
2
1 Sanding Wave (SW) Operation and Standing Wave in SRF Cavity
3
6 Multi-cell Structures and Weakly Coupled Structures Cavities
Synchronic acceleration and max of (RQ)accharr lactive =Nlcell= Ncszlig(2f) and the injection takes place at an optimum phase φopt which ensures that particles will arrive at the mid-plane of the first cell when Eacc reaches its maximum (+q passing to the right) or minimum (-q passing to the right)
Ez(r=0z)
lcell
lactive
v = szligc
s
-
+
-
+
-
π-phase advancecell-to-cell
SW Scheme in SRF Cavity Operation
Standing wave (CW) is used in SRF cavity acceleration
ΤΜ010π-mode
for electron β=1 proton β lt 1β=vc
2cellcLf
β λ λsdot= =
4
Transit Time Factor Due to SW Operation
d=λ2
v=c
0 0t z= =
z c t= sdot
6 6Tt z λ
= = 4 4Tt z λ
= =5 5 12 12Tt z λ
= = 2 2Tt z λ
= =
0 0 00 0 0
0
sin2( 0 ) ( 0 )
2 Transit time factor
2 0637 (for Pill Box Cavity)
z zi id d di t c cz z
dcV E r z e dz E r z e dz E e dz E d E d Td
cT
T
VEacc E Td
ω ωω
ω
ω
π
⎛ ⎞⎜ ⎟⎝ ⎠= = = = = = = sdot
= =
equiv =
int int int
Acceleration efficiency is automatically reduced by ~ 40 in the SW scheme
Ze-e- e- e- e- e-
5
2 Pill Box Cavity
6
LC Equivalent Circuit of Cavity
I = minusdQdt
Q = CV V = LdIdt
+ RI
d2Vdt
+RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dVdt
+1
LC⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0 V(t ) = VO exp(minusα + iω )t
(minusα + iω )2 + (minusα + iω )RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
1LC
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0
α =R2L
ω2 =1
LCminus
R2
4L2
R ltlt L ω O2 =
1LC
rArr f =1
2π LC
Q-value of the circuit
Q proportional to 1R
Stored Energy ( )Powerlosssec ( )
=2
P t LQdP t dt R
ω ω ω
ωα
equiv sdot = sdot = sdot
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
2
1 Sanding Wave (SW) Operation and Standing Wave in SRF Cavity
3
6 Multi-cell Structures and Weakly Coupled Structures Cavities
Synchronic acceleration and max of (RQ)accharr lactive =Nlcell= Ncszlig(2f) and the injection takes place at an optimum phase φopt which ensures that particles will arrive at the mid-plane of the first cell when Eacc reaches its maximum (+q passing to the right) or minimum (-q passing to the right)
Ez(r=0z)
lcell
lactive
v = szligc
s
-
+
-
+
-
π-phase advancecell-to-cell
SW Scheme in SRF Cavity Operation
Standing wave (CW) is used in SRF cavity acceleration
ΤΜ010π-mode
for electron β=1 proton β lt 1β=vc
2cellcLf
β λ λsdot= =
4
Transit Time Factor Due to SW Operation
d=λ2
v=c
0 0t z= =
z c t= sdot
6 6Tt z λ
= = 4 4Tt z λ
= =5 5 12 12Tt z λ
= = 2 2Tt z λ
= =
0 0 00 0 0
0
sin2( 0 ) ( 0 )
2 Transit time factor
2 0637 (for Pill Box Cavity)
z zi id d di t c cz z
dcV E r z e dz E r z e dz E e dz E d E d Td
cT
T
VEacc E Td
ω ωω
ω
ω
π
⎛ ⎞⎜ ⎟⎝ ⎠= = = = = = = sdot
= =
equiv =
int int int
Acceleration efficiency is automatically reduced by ~ 40 in the SW scheme
Ze-e- e- e- e- e-
5
2 Pill Box Cavity
6
LC Equivalent Circuit of Cavity
I = minusdQdt
Q = CV V = LdIdt
+ RI
d2Vdt
+RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dVdt
+1
LC⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0 V(t ) = VO exp(minusα + iω )t
(minusα + iω )2 + (minusα + iω )RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
1LC
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0
α =R2L
ω2 =1
LCminus
R2
4L2
R ltlt L ω O2 =
1LC
rArr f =1
2π LC
Q-value of the circuit
Q proportional to 1R
Stored Energy ( )Powerlosssec ( )
=2
P t LQdP t dt R
ω ω ω
ωα
equiv sdot = sdot = sdot
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
3
6 Multi-cell Structures and Weakly Coupled Structures Cavities
Synchronic acceleration and max of (RQ)accharr lactive =Nlcell= Ncszlig(2f) and the injection takes place at an optimum phase φopt which ensures that particles will arrive at the mid-plane of the first cell when Eacc reaches its maximum (+q passing to the right) or minimum (-q passing to the right)
Ez(r=0z)
lcell
lactive
v = szligc
s
-
+
-
+
-
π-phase advancecell-to-cell
SW Scheme in SRF Cavity Operation
Standing wave (CW) is used in SRF cavity acceleration
ΤΜ010π-mode
for electron β=1 proton β lt 1β=vc
2cellcLf
β λ λsdot= =
4
Transit Time Factor Due to SW Operation
d=λ2
v=c
0 0t z= =
z c t= sdot
6 6Tt z λ
= = 4 4Tt z λ
= =5 5 12 12Tt z λ
= = 2 2Tt z λ
= =
0 0 00 0 0
0
sin2( 0 ) ( 0 )
2 Transit time factor
2 0637 (for Pill Box Cavity)
z zi id d di t c cz z
dcV E r z e dz E r z e dz E e dz E d E d Td
cT
T
VEacc E Td
ω ωω
ω
ω
π
⎛ ⎞⎜ ⎟⎝ ⎠= = = = = = = sdot
= =
equiv =
int int int
Acceleration efficiency is automatically reduced by ~ 40 in the SW scheme
Ze-e- e- e- e- e-
5
2 Pill Box Cavity
6
LC Equivalent Circuit of Cavity
I = minusdQdt
Q = CV V = LdIdt
+ RI
d2Vdt
+RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dVdt
+1
LC⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0 V(t ) = VO exp(minusα + iω )t
(minusα + iω )2 + (minusα + iω )RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
1LC
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0
α =R2L
ω2 =1
LCminus
R2
4L2
R ltlt L ω O2 =
1LC
rArr f =1
2π LC
Q-value of the circuit
Q proportional to 1R
Stored Energy ( )Powerlosssec ( )
=2
P t LQdP t dt R
ω ω ω
ωα
equiv sdot = sdot = sdot
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
4
Transit Time Factor Due to SW Operation
d=λ2
v=c
0 0t z= =
z c t= sdot
6 6Tt z λ
= = 4 4Tt z λ
= =5 5 12 12Tt z λ
= = 2 2Tt z λ
= =
0 0 00 0 0
0
sin2( 0 ) ( 0 )
2 Transit time factor
2 0637 (for Pill Box Cavity)
z zi id d di t c cz z
dcV E r z e dz E r z e dz E e dz E d E d Td
cT
T
VEacc E Td
ω ωω
ω
ω
π
⎛ ⎞⎜ ⎟⎝ ⎠= = = = = = = sdot
= =
equiv =
int int int
Acceleration efficiency is automatically reduced by ~ 40 in the SW scheme
Ze-e- e- e- e- e-
5
2 Pill Box Cavity
6
LC Equivalent Circuit of Cavity
I = minusdQdt
Q = CV V = LdIdt
+ RI
d2Vdt
+RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dVdt
+1
LC⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0 V(t ) = VO exp(minusα + iω )t
(minusα + iω )2 + (minusα + iω )RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
1LC
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0
α =R2L
ω2 =1
LCminus
R2
4L2
R ltlt L ω O2 =
1LC
rArr f =1
2π LC
Q-value of the circuit
Q proportional to 1R
Stored Energy ( )Powerlosssec ( )
=2
P t LQdP t dt R
ω ω ω
ωα
equiv sdot = sdot = sdot
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
5
2 Pill Box Cavity
6
LC Equivalent Circuit of Cavity
I = minusdQdt
Q = CV V = LdIdt
+ RI
d2Vdt
+RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dVdt
+1
LC⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0 V(t ) = VO exp(minusα + iω )t
(minusα + iω )2 + (minusα + iω )RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
1LC
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0
α =R2L
ω2 =1
LCminus
R2
4L2
R ltlt L ω O2 =
1LC
rArr f =1
2π LC
Q-value of the circuit
Q proportional to 1R
Stored Energy ( )Powerlosssec ( )
=2
P t LQdP t dt R
ω ω ω
ωα
equiv sdot = sdot = sdot
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
6
LC Equivalent Circuit of Cavity
I = minusdQdt
Q = CV V = LdIdt
+ RI
d2Vdt
+RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dVdt
+1
LC⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0 V(t ) = VO exp(minusα + iω )t
(minusα + iω )2 + (minusα + iω )RL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
1LC
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 0
α =R2L
ω2 =1
LCminus
R2
4L2
R ltlt L ω O2 =
1LC
rArr f =1
2π LC
Q-value of the circuit
Q proportional to 1R
Stored Energy ( )Powerlosssec ( )
=2
P t LQdP t dt R
ω ω ω
ωα
equiv sdot = sdot = sdot
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
7
Electro-magnetic field in a waveguideMaxwell equations in a waveguide
22
2
=0 =0 =0 =0
0
( ) ( )exp( ) wavevector( ) ( )exp( )
i ic c
cx y z t x y ikz i t kx y x t x y ikz i t
ω ωμε ρ
ωμε
ωω
nablatimes = nablasdot nablatimes = minus nablasdot
⎛ ⎞⎧ ⎫nabla + =⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠
= plusmn minus= plusmn minus
E B B B E E j
EB
E EB B
2 22 2 2 2
2 2
22
2
22
2
( 0 =
1
1
t t z z t z z t
zt t z t z
zt t z t z
k E Bc z
B i Ez c
kc
E i Bz c
kc
ωεμ
ωεμωεμ
ωωεμ
⎡ ⎤ part⎧ ⎫nabla + minus = nabla equiv nabla minus = + +⎨ ⎬⎢ ⎥ part⎩ ⎭⎣ ⎦⎡ part ⎤⎛ ⎞= nabla + timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠⎡ part ⎤⎛ ⎞= nabla minus timesnabla⎜ ⎟⎢ ⎥part⎛ ⎞ ⎝ ⎠⎣ ⎦minus⎜ ⎟
⎝ ⎠
Ε E e E B e BB
B e
E e
Homework ILead this formula
mode 0 0mode 0 = 0
z z
z z
TM B ETE B E
= nene
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
8
TM- Modes TM-mode
Bz = 0 Ez ne 0 Can accelerate beam Beam
Solve the eigenvalue problemget k and Ez
[ ]22
2
22
2
22 2
2
1
( ) 0
t z t z
zt t
t z z
ic E
kc
Ez
kc
E k Ec
ωεμ
ωεμ
ωεμ
ωεμ
= timesnabla⎛ ⎞
minus⎜ ⎟⎝ ⎠
part⎛ ⎞= nabla ⎜ ⎟part⎛ ⎞ ⎝ ⎠minus⎜ ⎟⎝ ⎠
⎡ ⎤nabla + minus =⎢ ⎥
⎣ ⎦
B e
E
Boundary condition E z S = 0 ( Q on the surface of perfect conductoBzn S = 0 ( on the surface
but automatically satisfied by the TM- mode condition)
0times =n E
0sdot =n BQ
Similar for TE modes which kicks beamhelliphellip
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
9
Eigen-value problemψ(x y ) = Ez(x y ) for TM- mode or Bz(x y) for TE- mode
From the boundary condition
γ2 = γ λ
2 ψ = ψ λ (λ = 1 2L) kλ2 = εμ
ω 2
c 2 minus γ λ2
If then is an imaginal number
The wave is dampedtrapped in the waveguide if surface resistance largesmall
cutoff frequency
When wave number is a real number then the
c k
c
k
λλ
λλ
λ λ
γω
εμ
γω
εμω ω
lt
=
ge
L
wave can propagate into the waveguide
( )2 2
22 2
2
0 0 (for TM-Modes) or 0 (for TE-Modes)
0
t S Sn
kc
γ ψ ψ ψ
ωγ εμ
partnabla + = = =
part
= sdot minus ge
r
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
10
0 0
mnp
02
mode
cos( ) ( )exp( ) 0
( )cos( ) exp( ) cos( ) ( )exp( )
cos( ) ( )exp(
m n p
m nz o m z
m n p m nmr r m
m n p
m nm
m n p
TM
E E kz J r im Ba
E miE p JpE z im B kz J r imd a
E mp pE z J r id adcθ
ρθ
εμω ρπ ρπ θ θγ ρ γ
ρπ πγ
minus
= minus =
part= minus = minus minus
part
= minus 0
2 2 2
2 2
( )) cos( )exp( )
resonace frequency
m n p m
m n p
m nm n p m n p
iE Jm B kz im
c
c pfa d
θ
εμω ρθ θ
γ ρ
ρ πωεμ
part= minus
part
rArr = +
TEmnp Modes
)sin()sin(02 d
zpmra
JErm
kiE mn
mrπθρωε
⎟⎠⎞
⎜⎝⎛=
)sin()cos(02 d
zpmra
JEak
iE mnm
mn πθρρωεθ ⎟
⎠⎞
⎜⎝⎛=
0=zE
)cos()cos(102 d
zpmra
JEdp
kH mn
mrπθρπ
⎟⎠⎞
⎜⎝⎛=
)cos()sin(102 d
zpmra
JErm
dp
kH mn
mπθρπ
θ ⎟⎠⎞
⎜⎝⎛minus=
)sin()cos(0 d
zpmra
JEH mnmz
πθρ⎟⎠⎞
⎜⎝⎛=
2222
002
2
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=rArr⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
dp
acf
dp
amnmn πρ
ππρμεωResonance frequency
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
11
TMmnp Modes
d
2a
Ez
r=a
J0(ρ01) J0(ρ02) J1(ρ11)
Acceleration Modes
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
12
TEmnp Modes
Z
Kick Modes
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
13
4 Figure of merits of Cavity Design
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
Figure of Merits of The RF Cavity DesignItems Notation
1) Surface resistance RS [Ω]2) RF dissipation Ploss[W]3) Acceleration gradient V[V]3) Unloaded Q QO
4) Geometrical factor Γ[Ω]5) Shunt Impedance Rsh[Ω]6) R over Q (RQ) [Ω]7) Acceleration gradient Eacc[Vm]
8) Ratio of the surface peak electric field vs Eacc EpEacc [Oe(MVm)
9) Ratio of the surface magnetic field vs Eacc HpEacc
10) HOM loss factor11) Field flatness factor aff
12) Cell to cell coupling kCC
κ κperp
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
15
Surface Resistance
Surface Impedance
ts s
t Surface
EZ R iXH k
μωequiv + equiv =
Rs =μω2σ
=1σ
μσω2
=1
σδ2
S Sloss1P R H2 S
dS= sdot int
Homework IILead the formula of Rs for good electric conductor
Rs Surface resistance
HB = 0 E + 0tED = 0 H E 0t
J = E (Ohms Law)
μ
ε σ
σ
partnabla sdot nablatimes =
partpart
nabla sdot nablatimes minus minus =part
rr r
rr r r
r rt t
2 2 t
t
1H (xt) = [k E (xt)]
E (xt)[ k ( ) ] = 0
H (xt)i
μω
εμω μωσ
times
⎧ ⎫⎪ ⎪minus + ⎨ ⎬⎪ ⎪⎩ ⎭
rr r r
r r
r r
t
t
E(xt) = E (xt) + E (xt)
H(xt) = H (xt) + H (xt)l
l
r r rr r r
r r rr r r
t t
For the transvers
Plane wave E (xt) = E (0) exp( k x - t)i ωsdot sdotrr rr r
Maxwell Equations
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
16
U equiv stored energy PlossSurface RF Heating
Unloaded QO
For acceleration mode (TM010) the higher Qo is better
2 2
2
1 12 2
12
V V
loss S SS
Oloss
U E dV H dV
P R H dS
UQP
ε μ
ω
= sdot = sdot
= sdot
equiv
int int
int
Charismatic RF Parameter of Cavity
Shunt Impedance2
[ ]shloss
VRP
Ω equiv
Acceleration Voltage0
[V] on the beam axiseffLV Edz= int
This means the efficiency of the acceleration
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
17
3 RF Parameters Cavities
(RQ)2
( ) ( )sh OVR Q R Q
Uωequiv =
This means how much energy is concentrated on beam axisThis does not depend on material but only cavity shapeThis means the goodness of the cavity shape
Geometrical factor
2
2[ ]
12
VO S
SS
H dVQ R
H dS
ωΓ Ω equiv sdot = int
intWhen you know the Γ using a computer code you can calculate the surface resistance RS from the measured QO value
Γ is about 270Ω for β=1 cavities with elliptical or spherical shape
Charismatic RF Parameter of Cavity continued
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
18
Con
tour
plo
t of
E
Con
tour
plo
t of
B
lactive
acc
peak
EE
acc
peak
EB
Smaller value is preferred againstelectron emission phenomenon
Ratio shows the limit in Eacc due tothe break-down of superconduc-tivity (Nb ~2000 Oe )
( )acc loss O loss O
eff
R QE P Q Z P Q
L= sdot sdot = sdot sdotGradient
Charismatic RF Parameter of Cavity
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
19
When ultra relativistic point charge q passes empty cavity
Longitudinal and Transverse Loss Factors
Er ~1r
Cone ~1γro
HOM modes are excited in the cavityIf those are damped fast enough the following beam feel them and can be kicked Thus the beam can be degradated
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
20
HOM nth (RQ)n a ldquomeasurerdquo of the energy exchange between point charge and mode n
mode n ωn En Hn Trajectory of the point charge qassumed here to be a straight line
Wn
a b
2z
za
nzn
2z
za
nznn
b
a
b
a
dz))zz(c
cos(Edz))zz(c
sin(EV⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛minus= intint β
ωβω
nn
2n
n WV)QR(
ωequiv
3 RF Parameters Cavities(RQ)n of n-th HOM
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
21
The amount of energy lost by charge q to the cavity isΔUq = kmiddotq2 for monopole modes (max on axis)ΔUq = kmiddotq2 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 En(rφz) field along the trajectory
4)QR(k nn
n||sdot
=ωp
For individual mode n and point-like charge
Similar for other loss factorshelliphellip
3 RF Parameters CavitiesLongitudinal and Transverse Loss Factors JSekutwitzrsquos Slide
These harmful HOM power should be take out from the cavityDr Noguchi will make a lecture about the HOM coupler design
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
22
This is an empirical correction based on intuition
szligkNacc
2
ff sdot=
Field flatness factor for elliptical cavities with arbitrary szlig=vc
1
2
2
1
szligxszligx
ff
sdotpartsdotpart
=partpart
szlig1 szlig2
The same error partx causes bigger partf
when a cell is thinner
Cells which geometric szlig lt1 are more sensitive to shape errors
Field Flatness Factor affJSekutwitzrsquos Slide
ffA fa
A fΔ Δ
= sdot
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
23
The last parameter relevant for multi-cell accelerating structures is the coupling kcc
between cells for the accelerating mode pass-band (Fundamental Mode pass-band)
2
k0
0cc ωω
ωωπ
π
+minus
=
+ + + -
no Er (in general transverse E 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)
no Hφ (in general transverse H field) component at the symmetry plane
ωπωo
Cell to cell coupling kCC
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
24
SphericalElliptical Shape in SRF cavity design
Choose spherical shape to reduce multipacting
Multipacting
Design of SRF cavity is usually done with spherical or Elliptical shapes
R Parodi (1979) presented first spherical C-band cavity with much less multipactingbarrier than other cavities at that timeP Kneisel (early 80rsquos) proposed for the DESY experiment the elliptical shape of 1 GHzcavity 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
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
25
5 Geometry and Criteria for Cavity Design Cavities
RF parameters summary
FM (RQ) G EpeakEacc BpeakEacc kcc
HOM k k
There are 7 parameters we want to optimize for a inner cell
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
Geometry
We begin with inner cells design because these cells ldquodominaterdquo parameters of a multi-cell superconducting accelerating structure
There is some kind of conflict 7 parameters and only 5 variables to ldquotunerdquo
Optimization of Cell Shape
JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
26A Mosnier E Haebel SRF Workshop 1991
Examplef = 15 GHz
5 Geometry and Criteria for Cavity Design Cavities Effect of Cavity Aperture on RF Parameters
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
27
00
20
40
60
80
20 25 30 35 40 ri [mm]
k_m
on(r
)k_m
on(4
0)
k_di
pole
(r)k
_dip
ole(
40)
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
Aperture Effects on κ and κᅩ)JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
28
(RQ) = 86 Ω Bpeak Eacc = 46 mT(MVm)Epeak Eacc = 32
(RQ) = 152 Ω Bpeak Eacc = 35 mT(MVm)Epeak Eacc = 19
00
02
04
06
08
10
20 25 30 35 40 ri [mm]
kcc(
r)k
cc(4
0)
Aperture Effect on Cell to Cell Coupling (κCC)JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
29
Criteria RF-parameter Improves when Cavity examples
Operation at high gradient
Epeak Eacc
Bpeak Eacc
ri
Iris Equator shape
TESLA
HG CEBAF-12 GeV
Low cryogenic losses (RQ) Γri
Equator shape
LL CEBAF-12 GeV
LL- ILC cavity
High Ibeam harrLow HOM impedance
k k riB-Factory
RHIC cooling
5 Geometry and Criteria for Cavity Design Cavities General Trends of Cavity Optimization on RF Geometrical Parameters
iris ellipsis half-axis hr hz
iris radius ri
equator ellipsis half-axis hr hz
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
30
What about accelerating mode frequency of a superconducting cavity
fπ [MHz] 2600
RQ [Ω] 57
rq=(RQ)l [Ωm] 2000
G [Ω] 271
fπ [MHz] 1300
RQ [Ω] 57
rq=(RQ)l [Ωm] 1000
G [Ω] 271
x 2 =
rq=(RQ)l ~ f
Choice of the RF Frequency
JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
31
Frequency Choice of the SRF Cavity
2 21 ( ) exp( )2loss s S acc SS
B
AP R H dS E R BCS fT k T
Δ= prop = sdot sdot minusint
Cryogenic power
Cavity fabrication cost
f
f
350MHz500MHz
(U-band)
13GHz
(L-band)
2896GHz
(S-band)
Cost + Existing RF source
L-band cavity is preferred and it is operated at 2K to reduce the cryogenic power
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
32
f π [MHz] 13000
riris [mm] 35
kcc [] 19
EpeakEacc - 198
BpeakEacc [mT(MVm)] 415
RQ [Ω] 1138
G [Ω] 271
RQG [ΩΩ] 30840
The inner cell geometry was optimize with respect to low EpeakEacc and coupling kcc
At that time (1992) the field emission phenomenon and field flatness were of concern no onewas thinking about reaching the magnetic limit
Inner cell Contour of E field
TESLA Cavity Inner Cell Shape
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
33
High Gradient Shapes
Cavity shape designs with low HpEacc
TTF TESLA shapeReentrant (RE) Cornell Univ Low Loss(LL) JLABDESYIchiroーSingle(IS) KEK
from JSekutowicz lecture Note
TESLA LL RE IS
Diameter [mm] 70 60 66 61
EpEacc 20 236 221 202
HpEacc [OeMVm] 426 361 376 356
RQ [W] 1138 1337 1268 138
G[W] 271 284 277 285
Eacc max 411 485 465 492
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
34
Eacc vs Year
10
20
30
40
50
60
70
Eac
cm
ax [M
Vm
]
Date [Year]91 0095 0593 97 03
High pressuer water rinsing
(HPR)
Electropolshing(EP)
+ HPR + 120OC Bake
New Shape
Chemical Polishing
RE LL IS shape
99 07
2nd Breakthrough1st Breakthrough
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
35
4 Criteria for Multi-cell StructuresSingle-cell is attractive from the RF-point of view
Easier to manage HOM dampingNo field flatness problemInput coupler transfers less powerEasy for cleaning and preparationBut it is expensive to base even a smalllinear accelerator on the single cell We doit 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 becomessensitive to frequency errors of individual cellsOther problems arise HOM trappinghellip
How to decide the number of cells
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
36
2i i i
ffi i cc i
A f fNaA f k f
Δ Δ Δ= = sdot
Field flatness factor
Beam pipe has no acceleration beamBP reduce the efficiency
Multi-cell is more efficient
HOM RF
2
ffcc
Nak
=
Parameters considered with N
HOM trapped mode
Input handling power
More serious within creasing N
NINPUTP prop
Handling Easy to bend Difficult to preparation hellip
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
37
Many years of experience with heat treatment chemical treatment handling and assembly allowsone to preserve tuning of cavities even for those with bigger N and weaker kcc
For the TESLA cavities field flatness is better than 95
OriginalCornellN = 5
HighGradient
N =7
LowLossN =7
TESLA
N=9
SNSszlig=061
N=6
SNSszlig=081
N=6
RIAszlig=047
N=6
RHIC
N=5
year 1982 2001 2002 1992 2000 2000 2003 2003
aff 1489 2592 3288 4091 3883 2924 5040 850
Field flatness vs N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Effect of N on Field Flatness Sensitivity FactorJSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
38
No fields at HOM couplers positions which are always placed at end beam tubes
e-m fields at HOM couplers positions
N = 17
N = 13
N = 9
N = 5
HOM trapping vs NJSekutwitzrsquos Slide
Cure for the trapped mode Make the bore radius largerBreak mirror symmetry
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
39
HOM couplers limit RF-performance of sc cavities when they are placed on cells no E-H fields at HOM couplers positions which
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 kccHOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they hardly resonate
togetherf = 2415 MHz
12 Trapping of Modes within Cavities HOM HOM Issue with Multi-Cell StructureJSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
40
6 Multi-cell Structures and Weakly Coupled Structures Cavities Adjustment of End-Cells
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 modefield strength of the accelerating mode at FPC location enabling operation with matched Qextfields strength of dangerous HOMs ensuring their required damping by means of HOM couplers orand beam line absorbers
All three make design of the end-cells more difficult than inner cells
+ +
JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
41
Power capability of fundamental power couplers vs N
When Ibeam and Eacc are specified and a superconducting multi-cell structure does not operate in the energy recovery mode
PINPUT ~ N
6 Multi-cell Structures and Weakly Coupled Structures Cavities Capable Input Power on N
Coupler handling power limits the small N in the intensity frontier machine like KEK B(N=1)
KEKB Acceleration beam current gt 1 A
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
42
2 TESLA Cavities and Auxiliaries as ILC Baseline Design
TTF 9-cells Contour of E field
7 identical inner cellsEnd-cell 1 End-cell 2
f π [MHz] 130000
f π-1 [MHz] 129924
RQ [Ω] 1012
G [Ω] 271
Active length [mm] 1038
The cavity was designed in 1992 (A Mosnier D Proch and JS )
TESLAILC Baseline Cavity
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
43
CEBAFOriginalCornell
szlig=1
CEBAF -12High
Gradientszlig=1
CEBAF -12LowLossszlig=1
TESLA
szlig=1
SNS
szlig=061
SNS
szlig=081
RIA
szlig=047
RHICCoolerszlig=1
fo [MHz] 14483 14689 14751 12780 7928 7928 7930 6830
fπ [MHz] 14970 14970 14970 13000 8050 8050 8050 7037
kcc [] 329 189 149 19 152 152 152 294
EpeakEacc - 256 196 217 198 266 214 328 198
BpeakEacc [mT(MVm)] 456 415 374 415 544 458 651 578
RQ [Ω] 965 112 1288 1138 492 838 285 802
G [Ω] 2738 266 280 271 176 226 136 225
RQG [ΩΩ] 26421 29792 36064 30840 8659 18939 3876 18045
k (σz=1mm) [VpCcm2] 022 032 053 023 013 011 015 002
k (σz=1mm) [VpC] 136 153 171 146 125 127 119 085
new new newnewExamples of Inner cells
Overview of CavitiesJSekutwitzrsquos Slide
β vs RF parameters β smallerEpEacc largerHpEacc larger(RQ) smaller
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
44
5 Example of SRF High-beta Cavities
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
45
Type No of cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 8Single-cell with max
Ibeam
Ibeam= 134 A1389 bunches
cw350 16
x 16
Multi-cell with max
Ibeam
Ibeam le 40 mA180 bunches
cw60 013
x 1 Multi-cell with max Eacc
Eacc= 35 MVm13mspulse 1Hz
PRF
~100Almost no beam
loading0
HERA 05 GHz
KEK-B 05 GHz
TTF-I 13 GHz
Cavities operating with highest Ibeam or Eacc
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
46
Type No cavities Pbeamcavity[kW]
PHOM cavity[kW]
x 33 Multi-cell with max
Ibeam
Ibeam=38 (59 ) mA13mspulseDF = 6
240 (366) 006 peak
x 48 482 006 peak
x 8Multi-cell with max
Eacc
Eacc= 35 MVm13mspulse 10Hz
PRF146 lt 002gt
Cavities which will operate with high Ibeam in the near future
SNS szlig= 061 0805 GHz
SNS szlig= 0805 GHz
TTF-II ep 13 GHz
6 Multi-cell Structures and Weakly Coupled Structures Cavities JSekutwitzrsquos Slide
Top Related