1 H. Padamsee Superconducting RF Cavities Cavity Design and Ancillaries I and 2.
Accelerating (S)RF Cavities - Cornell Universityhoff/LECTURES/08S_688/08S_688_080324.pdfTaiwan Light...
Transcript of Accelerating (S)RF Cavities - Cornell Universityhoff/LECTURES/08S_688/08S_688_080324.pdfTaiwan Light...
1Matthias Liepe; 03/24/2008
Accelerating (S)RF Cavities
Matthias Liepe
2Matthias Liepe; 03/24/2008
Outline
• Accelerating cavities– DC accelerators– RF cavities– Examples
• RF Cavity Fundamentals– Pill box cavity– Figures of merit– Accelerating mode– Multicell cavities and circuit models
• RF Cavity Design– NC vs. SC– Objectives and cell shapes– RF codes
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Accelerating cavities
• Accelerating cavities– DC accelerators– RF cavities– Examples
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DC Accelerators
• Use high DC voltage to accelerate particles
• No work done by magnetic fields
Cockroft and Walton's electrostatic accelerator (1932)
Protons were accelerated and slammed into lithium atoms
producing helium and energy.
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DC Accelerators: Limitations
⇒⇒⇒⇒ Use time-varying fields!
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Example of Time Varying Fields: Drift Tube Linac
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Example of Time Varying Fields: Drift Tube Linac
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RF Cavities
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RF Cavities
• Time dependent electromagnetic field inside metal box
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Taiwan Light Source cryomodule
Introduction to RF Cavities
• The main purpose of using RF cavities in accelerators is to provide energy gain to charged-particle beams
• The highest achievable gradient, however, is not always optimal for an accelerator. There are other factors (both machine-dependent and technology-dependent) that determine operating gradient of RF cavities and influence the cavity design, such as accelerator cost optimization, maximum power through an input coupler, necessity to extract HOM power, etc.
• In many cases requirements are competing.
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CW High-Current Storage Rings (colliders and light sources)
• NC or SC• Relatively low
gradient (1…9 MV/m)
• Strong HOM damping (Q ~ 102)
• High average RF power (hundreds of kW)
CESR cavities
KEK cavity
PEP II Cavity
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Pulsed Linacs (ILC, XFEL, …)
• High gradients
• Moderate HOM damping reqs.
• High peak RF power
ILC: 21,000 cavities!
12
12
ILC / XFEL cavities
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Traveling Wave Cavities
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CW low-current linacs (CEBAF, ELBE)
• SRF cavities• Moderate to low
gradient (8…20 MV/m)
• Relaxed HOM damping requirements
• Low average RF power (5…13 kW)
CEBAF cavities
ELBE cryomodule
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CW High-Current ERLs
• SRF cavities• Moderate
gradient (15…20 MV/m)
• Strong HOM damping (Q = 102…104)
• Low average RF power (few kW)
Cornell ERL cavities
BNL ERL cavity
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RF Cavity Fundamentals
• RF Cavity Fundamentals (Standing wave cavities)– Cavity eigenmodes– Figures of merit– Accelerating mode– Multicell cavities and circuit model
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RF Cavities and their Eigenmodes I
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RF Cavities and their Eigenmodes II
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RF Cavities and their Eigenmodes III
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RF Cavities and their Eigenmodes IV
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Accelerating Mode
• Time dependent electromagnetic field inside metal box
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Figures of Merit - 1Accelerating Voltage & Accelerating Field (v = c f or Particles)
Enter Exit
, here, is flight time factor (for pill-box with this length d only, other shapes can have a different value of T ).
• Frequency • For maximum acceleration we need
so that the field always points in the same direction as the bunch traverses the cavity•Accelerating voltage then is
•Accelerating field is
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Figures of Merit - 2Dissipated Power, Stored Energy, Cavity Quality (Q)
•Surface currents (∝ H) result in dissipation proportional to the surfaceresistance (Rs):
•Stored energy is:
•Dissipation in the cavity wall given bysurface integral:
UTrf Pc
= 2 π•Define Quality (Q) as
which is ~ 2 π number of cycles it takes to dissipate the energy stored in the cavity Easy way to measure Q• Qnc ≈ 104, Qsc ≈ 1010
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Figures of Merit for Cavity Design - 3Geometry Factor - G
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Accelerating ππππ-mode:
Accelerating voltage:
(((( ))))∫∫∫∫
++++
−−−−========
2/
2/
/
chargegain energy maximum L
L
czizacc dzeEV ωωωω
Shunt-impedance: Quality factor:
( )dis
acca P
VR
2
=dis
L PU
Qωωωω====
( )U
V
Q
R
Q
R acc
L
a
ω
2
==
Figures of Merit for Cavity Design - 4Shunt Impedance (R a)
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Figures of Merit for Cavity Design - 4Shunt Impedance (R a)
•Ra/Q only depends on the cavity geometry This quantity is also used for determining the level of mode excitation by charges passing through the cavity.
• Shunt impedance (Ra) determines how much acceleration one gets for a given dissipation (analogous to Ohm’s Law)
Another important figure of merit is
To maximize acceleration (Pc given), must maximize shunt impedance.
To minimize losses (Pc) in the cavity, we must maximize G*Ra/Q0:
)/()/)(()/( 0
2
00
2
00
22
QRG
RV
RQRQR
V
QRQ
V
R
VP
a
sc
sas
c
a
c
a
cc ⋅
⋅=⋅
=⋅
==
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Typical Values for Single Cells
270 Ω88 Ω /cell
2.552 Oe/MV/m
Cornell SC 500 MHz
Current is high, it excites a lot of Higher Order Modes, so the hole is made big to propagate HOMs, and this is why Hpk and Epk grew, and R/Q drops.
Diff. applications – diff. trade-offs.
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inputcoupler
inputcoupler
inputcoupler
inputcoupler
inputcoupler
inputcoupler
inputcoupler
inputcoupler
inputcoupler
active (acceleration)passive
higher fill-factor:
F =
⇒ lower costs
⇒ better beam
active length
total length
fewer
• input couplers
• waveguide elements
• RF control systems
• …
Multicell Cavities: Why?
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Multicell Cavities: Why?
Example: 500 GeV Linear Collider
21,024 9-cell cavities: 27.8 km (17.3 miles)
189,216 1-cell cavities: 75.4 km (46.8 miles)
IP
linac 1 linac 2
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Multicell Cavities and Cell-to-Cell Coupling
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Cell-to-Cell Coupling
Single Cell:
TM 010 mode
Coupled Cells
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Two Coupled Cells: TM010 Modes
2 coupled cells 2 TM010 modes
n coupled cells n TM010 modes
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Two Coupled Cells: TM010 Modes
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Cell-to-Cell Coupling
Coupling factor: ( ) dsuhec
k zS
cci
rrr ⋅×= ∫0ω
electric boundarycondition
magnetic boundarycondition
z z
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A Circuit Model: Let’s start at the Beginning:Step 1.1
Solve Maxwell’s equationsfor given boundary conditions
⇓⇓⇓⇓orthogonal eigenmodes
⇓⇓⇓⇓orthononal eigenfunctions
1,2,... m , )(),(
)(),(
2/)()(
)()(
)(
)(
========
====
++++ ππππωωωω
ωωωω
itimm
timm
m
m
erHtrH
erEtrE
rrrr
rrrr
)(2
)(e (m)(m)0(m) rE
Ur
rrrr εεεε====
mnV
dvrr δδδδ====∫∫∫∫ )(e)(e (n)(m) rrrr
)(2
)(h (m)(m)0(m) rH
Ur
rrrr µµµµ====
mnV
dvrr δδδδ====∫∫∫∫ )(h)(h (n)(m) rrrr
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A Circuit Model: Step 1.2
)(),(
)(),(
2/)()(
)()(
)(
)(
ππππωωωω
ωωωω
itimm
timm
m
m
erHtrH
erEtrE
++++====
====rrrr
rrrr
)(2
)(e (m)(m)0(m) rE
Ur
rrrr εεεε====)(2
)(h (m)(m)0(m) rH
Ur
rrrr µµµµ====
With Maxwell’s equations in vacuum:
and the eigenmodes
and the normalized eigenfunctions
we get the relations
tE
H∂∂∂∂∂∂∂∂====××××∇∇∇∇r
rr
0εεεεt
HE
∂∂∂∂∂∂∂∂−−−−====××××∇∇∇∇r
rr
0µµµµ
)()(
)( mm
m ec
hrrr ωωωω====××××∇∇∇∇ )(
)()( m
mm h
ce
rrr ωωωω====××××∇∇∇∇
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A Circuit Model: Step 1Eigenmodes: Example: Pillbox-Cavity
electric field magnetic field
⇒⇒⇒⇒ acceleration
lowest frequency mode:
transverse magnetic
0 ⇒⇒⇒⇒ monopole mode
z
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A Circuit Model: Step 1Eigenmodes: Example: Single Cell TESLA Cavity
magnetic fieldelectric fieldTM010 :
Accelerating mode:
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Circuit Model Step 2: Expansion in Eigenmodes
Each time dependent field in a cavity can be written as a sum of the cavity eigenfunctionswith time dependent amplitudes:
∑∑∑∑ΕΕΕΕ====m
mm rettrE )()(ˆ),( )()( rrrr
time dependent amplitude
eigenfunction
∑∑∑∑====m
mm rhtHtrH )()(ˆ),( )()( rrrr
time dependent amplitude
eigenfunction
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Circuit Model: Step 3.1
Insert expansion of fields into Maxwell’s equations in vacuum:
tE
H∂∂∂∂∂∂∂∂====××××∇∇∇∇r
rr
0εεεε
⇓⇓⇓⇓
(((( )))) (((( )))) (((( )))) (((( ))))redt
tEdre
ctH m
m
mm
m
m
m rrrr )()(
0)(
)()( ˆ
ˆ ∑∑∑∑∑∑∑∑ ==== εεεεωωωω
⇓⇓⇓⇓
(((( )))) (((( )))) 1,2...m , 0ˆˆ )(0
)()(
========−−−− tEdtd
tHc
mmm
εεεεωωωω
insert eigenmode-expansion of fields…
use orthogonality of eigenmodes:
…and the equations from page 37
multiply by e(m) and integrate over cavity volume
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Circuit Model: Step 3.2
Insert expansion of fields into Maxwell’s equations:
tH
E∂∂∂∂
∂∂∂∂−−−−====××××∇∇∇∇r
rr
0µµµµ
⇓⇓⇓⇓
(((( )))) (((( )))) (((( )))) (((( ))))rhdt
tHdrh
ctE m
m
mm
m
m
m rrrr )()(
0)(
)()( ˆ
ˆ ∑∑∑∑∑∑∑∑ −−−−==== µµµµωωωω
⇓⇓⇓⇓
(((( )))) (((( )))) 1,2...m , 0ˆˆ )(0
)()(
========++++ tHdtd
tEc
mmm
µµµµωωωω
insert eigenmode-expansion of fields…
use orthogonality of eigenmodes:
…and the equations from page 37
multiply by h (m) and integrate over cavity volume
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Circuit Model: Step 3.3Differential Equation for the Eigenmode Amplitudes
(((( )))) (((( )))) 1,2...m , 0ˆˆ )(0
)()(
========−−−− tEdtd
tHc
mmm
εεεεωωωω
(((( )))) (((( )))) 1,2...m , 0ˆˆ )(0
)()(
========++++ tHdtd
tEc
mmm
µµµµωωωω
⇓⇓⇓⇓
(((( )))) (((( )))) (((( )))) 1,2...m , 0ˆ ˆ )(2 )()(2
2========++++ tEtE
dt
d mmm ωωωω
amplitude of eigenmode #m
⇒⇒⇒⇒⇒⇒⇒⇒ Differential equation of an oscillator (we Differential equation of an oscillator (we will add losses and a generator later)!will add losses and a generator later)!
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Circuit Model: Step 4Coupled Cells
TM010 modes in couples cells:
N coupled cells ⇒⇒⇒⇒ N coupled oscillators
⇒⇒⇒⇒ N coupled differential equations:
TM 010 field amplitude in
cell #n
TM 010eigenfrequency
of cell #n
cell-to-cell coupling factor
N-cell structure
⇒⇒⇒⇒ N coupled differential equations
(((( )))) (((( )))) N1,2...,n , 0ˆˆ2
ˆˆ2
ˆˆ1
21
222
2========−−−−++++−−−−++++++++ ++++−−−− nnnnnnnnn EE
KEE
KEE
dt
d ωωωωωωωωωωωω
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Circuit Model: Step 5Equivalent Circuit Model (TM010 Modes only)
(((( )))) (((( )))) N1,2...,n , 0ˆˆ2
ˆˆ2
ˆˆ1
21
222
2========−−−−++++−−−−++++++++ ++++−−−− nnnnnnnnn EE
KEE
KEE
dt
d ωωωωωωωωωωωω
substitute:
nn LC
12 ====ωωωω 1,1,
1, 22 ++++++++
++++ ======== nnnn
nnn k
CC
K
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Circuit Model: Step 6Eigenmode Equation for the TM010 Modes
(((( )))) (((( )))) N1,2...,n , 0ˆˆ2
ˆˆ2
ˆˆ1
201
20
202
2========−−−−++++−−−−++++++++ ++++−−−− nnnnnn EE
KEE
KEE
dt
d ωωωωωωωωωωωω
steady state ansatz
)(12
0
)()(
1
,1,1
,13,22,11,2
3,23,22,12,1
2,12,1
1
1
1
1j
N
jj
N
NNNN
NNNN A
A
A
A
kk
kkkk
kkkk
kk
====
++++++++++++
−−−−++++++++−−−−−−−−++++
−−−−−−−−
−−−−−−−−−−−−
MMOωωωω
ωωωω
eigenvector #j eigenfrequency of mode #j
Eigenvector: TMEigenvector: TM010010 cell amplitudes of a cell amplitudes of a TMTM010010 eigenmode of a multicell cavity.eigenmode of a multicell cavity.
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TM010 EigenmodesExample: 9-Cell Cavity (1)
An
An
An
An
An An
An
cell #
cell #
cell #
cell #
cell #
cell #
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TM010 EigenmodesExample: 9-Cell Cavity (2)
An
An An
accelerating mode: ππππ cell-to-cell phase advance
cell #
cell #
cell #
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Circuit Model: Step 7Add Losses and a Generator
generator current
losses:
ndis
nn P
UR
LQ
,0
ωωωωωωωω ========
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Simulation Example:TM010 Eigenmode-Spectrum of a 9-Cell Cavity
ampl
itude
(ce
ll #1
) [
dBA
]
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Dispersion Relation
The working point. If it is too close to the neighbor point this neighbor mode can also be excited. To avoid this, more cell-to-cell coupling is needed: broader aperture.
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Mode Beating during Cavity Filling