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Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Accelerator PhysicsFundamentals of RF Cavities
Suba De Silva, S. A. Bogacz, G. A. Krafft,
and B. Dhital
Old Dominion University / Jefferson Lab
Lecture 10
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• RF Cavities
– Cavity Basics
– RF Properties
– TM Type Cavities
• Types of Cavities
– Accelerating Cavities
– Low β cavities
– Deflecting/Crabbing Cavities
• Limitations in SRF Cavities
• Losses in RF Cavities
• For normal conducting cavities
• For superconducting cavities
Outline
2
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• H. Padamsee, J. Knobloch, T. Hays “ RF Superconductivity
for Accelerators”, John Wiley & Sons, Inc; ISBN 0-471-
15432-6
• Proceedings of the Workshops on RF Superconductivity
1981–2015 – (ww.jacow.org)
• CERN Accelerator School – 1955 – 2016
(https://cds.cern.ch/collection/CERN%20Yellow%20Report
s?ln=en)
Suggested Literature
3
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
PAS A ccelerato r Phy
RF Cavities
RF cavities made of differentmaterials, in differentshapes and sizes
1500MHz 5-cell
1300 MHz 9-cellQuarter
Wave
Cavity
Half
Wave
Cavity
Triple
Spoke
Cavity
CESR 500
MHz Cavity
(Cornell) LEP 350 MHz 4-cell Nb on Cu
4
Single
Spoke
Cavity
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Space enclosed by conducting walls that can sustain an
infinite number of resonant electromagnetic modes
• Shape is selected so that a particular mode can efficiently
transfer its energy to a charged particle
• An isolated mode can be modeled by an LRC circuit
• Lorentz force
• An accelerating cavity needs to provide an electric field
(E) longitudinal with the velocity of the particle
• Magnetic fields (H) provide deflection but no acceleration
RF Cavities
5
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Simplest form of RF resonator LC circuit
• LC circuit Pill box cavity
– Electric field is concentrated near axis
– Magnetic field is concentrated at outer cylindrical wall
RF Resonator
6
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Fields in an rf cavity are solution to the wave equation
• Subjected to boundary conditions:
– No tangential electric field
– No normal magnetic field
• Two sets of eigenmode solutions with infinite number of
modes
– TM modes Modes with longitudinal electric fields and
no transverse magnetic fields
– TE modes Modes with longitudinal magnetic fields and
no transverse electric fields
Cavity Basics
7
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
TM and TE Modes in a Pill Box Cavity
xmn is the nth root ofJm
x’mn is the n th root ofJ’m
8
L
2R
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• TM010
– Electric field is purelylongitudinal
– Electricand magnetic fields have no angulardependence
– Frequencydependsonlyon radius, independent on length
• TM0np
– Monopolemodes that can coupleto the beamand exchange energy
• TM1np
– Dipolemodes that can deflect the beam
• TE modes
– No longitudinal E field
– Cannotcouple to the beam
– TE-typemodes can deflect thebeam
Modes in Pill Box Cavity
9
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Pill Box Cavity
• Needs longitudinal electric field for acceleration
• Operated in the TM010 mode
• Hollow right cylindrical enclosureE
TM010 mode
10
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
TM010 Mode in a Pill Box Cavity
E
TM010 mode
R
• Frequency scales inversely with cavity radius
11
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
TM010 Field Profile
MagneticField
ElectricField Surface ElectricField
Surface MagneticField
• Peak surface electric
field at end plates
• Peak surface magnetic
field at outercylindrical
surface
12
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity RF Properties
Slide 13
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• For efficient acceleration, choose a cavity geometry and a
mode where:
– Electric field is along the particle trajectory
– Magnetic field is zero along the particle trajectory
– Velocity of the electromagnetic field is matched to
particle velocity
• Accelerating voltage for charged particles
Accelerating Voltage
14
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Accelerating voltage for charged particles
• For the pill box cavity
T is the transit time factor
Accelerating Voltage (Vc)L
Enter Exit
L
2
15
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Accelerating field (gradient): Voltage gained by a particle
divided by a reference length
• For velocity of light particles:
N – no. of cells
• For less-than-velocity-of-light cavities (β < 1), there is no
universally adopted definition of the reference length
• However multi-cell elliptical cavities with β < 1
Length per cell
Accelerating Gradient (Eacc)
2L
N
Vacc
accEL
2L
16
[MV/m]
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Energy density in electromagnetic field:
• Because of the sinusoidal time dependence and the 90º
phase shift, the energy oscillates back and forth between
the electric and magnetic field
• Total energy content in the cavity:
Stored Energy (U)
17
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Surface current results in power dissipation proportional to
the surface resistance (Rs)
• Power dissipation per unit area
• Total power dissipation in the cavity walls
Power Dissipation (Pdiss)
18
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• For normal conductors
– per unit length
– per unit area
• For superconductors
– per unit length
– per unit area
Power Dissipation (Pdiss)
19
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Quality Factor (Q0)
• Measures cavity performance as to how lossy cavity
material is for given stored energy
• For normal conducting cavities ~ 104
• For superconducting cavities ~ 1010
20
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Geometrical factor [Ω]
– Product of the qualify factor (Q0) and the surface
resistance (Rs)
– Independent of size and material
– Depends only on shape of cavity and electromagnetic
mode
– For superconducting elliptical cavities
Geometrical Factor (G)
21
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Shunt impedance (Rsh) [Ω]
• Maximize shunt impedance to get maximum acceleration
• Note: Sometimes the shunt impedance is defined as or
quoted as impedance per unit length (Ω/m)
• R/Q [Ω]: Measures of how much of acceleration for a
given power dissipation
Shunt Impedance (Rsh) and R/Q
22
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Optimization parameter:
• R/Q and RshRs
– Independent of size (frequency) and material
– Depends on mode geometry
– Proportional to no. of cells
• In practice for elliptical cavities
– R/Q ~ 100 Ω per cell
– RshRs ~ 33,000 Ω2 per cell
RshRs and R/Q
R R Rsh QR
RGsh s s
Q Q
23
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
TM010 Mode in a Pill Box Cavity
Energy content
Geometrical factor
Power dissipation
2
2 2 2
0 0 1 01J (x )LR2
U E
2
0 1 012 J (x )(R L)RRs
P E
x01
2 (R L)
LG
01
1 01
x 2.40483
J (x ) 0.51915
24
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
TM010 Mode in a Pill Box Cavity
Energy Gain
Gradient
Shunt impedance
Slide 25
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Accelerating Cavities
Slide 26
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Beam tubes reduce the electric field on axis
– Gradient decreases
– Peak fields increase
– R/Q decreases
Real Cavities
27
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Pill Box to Elliptical Cavities
Slide 28
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Single Cell Cavities
Electric field high at iris
Magnetic field high at equator
• Important parameters: Ep/Eacc and Bp/Eacc
• Must minimize the ratios as smaller as possible
29
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Single Cell Cavities
Cornell SC 500 MHz
270 Ω
88 Ω/cell
2.5
52 Oe/MV/m
30
1 Oe = 1 G = 0.1 mT
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• What is the purpose of the cavity?
• What EM parameters should be optimized to meet the
design specs?
• Beam aperture
• Peak surface field ratios – Ep/Eacc, Bp/Eacc
• Shunt impedance – RshRs
• Higher Order Mode (HOM) extraction
Cell Shape Design
The “perfect” shape does not exist, it all
depends on your application
31
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• “High Gradient” shape: lowest Ep/Eacc
• “Low Loss” shape: lowest cryogenic losses RshRs = G(R/Q)
Example: CEBAF Upgrade
32
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
TM-Cavity Design
• The field emission is not a hard limit in the performance of
sc cavities if the surface preparation is done in the right way
• Unlikely this, magnetic flux on the wall limits performance
of a sc cavity (Q0 decreases or/and quench). Hard limit ~180
mT for Nb
Bpeak / Eacc should be low
1. Cavities may operate at
higher gradients.
2. Cavities may operate at
lower cryogenic load.
V 2 G (R / Q)
Pdiss Rs
acc
33
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
New Shapes for ILC
TTF
1992
LL
2002/2004
RE
2002
f = 1300 MHz
r iris [mm] 35 30 33
kcc [%] 1.9 1.52 1.8
Epeak/Eacc - 1.98 2.36 2.21
Bpeak/Eacc [mT/(MV/m)] 4.15 3.61 3.76
R/Q [Ω] 113.8 133.7 126.8
G [Ω] 271 284 277
R/Q*G [Ω*Ω] 30840 37970 35123
Slide 34
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF Simulation Codes for Cavity Design
• 2D is fast and allowsto define geometry ofa cylindrical symmetric body(inner
and end-cells)ofthe cavity.
• 3D is much more time consumingbut necessary for modelingof full equipped
cavity with FPC and HOM couplersand if needed to model fabrication errors.
Also couplingstrength for FPC and dampingof HOMs can be modeled only3D.
2 2We need numerical methods:( ) A 0
The solutionto 2D (or 3D) Helmholtzequation can be analyticallyfound onlyfor
very few geometries (pillbox, sphericalresonatorsor rectangular resonator).
Approximating operator
(Finite Difference Methods)Approximating function
(Finite Element Methods)
35
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Free, 2D finite-difference code to design cylindrically
symmetric structures (monopole modes only)
• Use symmetry planes to reduce number of mesh points
SUPERFISH
File di SuperFish Generato da BuildCav F = 1472.6276 MHz
12 14 16
C:\LANLV7\HALFCEBSC.AF 11-27-2006 16:58:22
0
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
4
5
12
11
10
9
8
7
6
0 2 4 6 8 10
File di SuperFish Generato da BuildCav F = 1472.6276 MHz
2.8 2.9 3 3.1 3.2
C:\LANLV7\HALFCEBSC.AF 11-27-2006 16:58:22
7.30
7.40
7.50
7.60
7.70
7.80
7.90
8.00
8.10
7.30
7.40
7.50
7.60
7.70
7.80
7.90
8.00
8.10
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7
http://laacg1.lanl.gov/laacg/serv
ices/download_sf.phtml
36
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Expensive, 3D finite-element code, used to design complex
RF structure
• Runs on PC
• Perfect boundary approximation
CST Microwave Studio
https://www.3ds.com/products-services/simulia/products/cst-studio-suite/
Hexahedral mesh
37
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• SLAC, 3D code, high-order Parallel Finite
Element (PFE) method
• Runs on Linux
• Tetrahedral conformal mesh
• High order finite elements (basis order p
=1 – 6)
• Separate software for user interface
(Cubit/Trelis), and visualization and
post processing (ParaView)
Omega3P – ACE3P (SLAC Code Suite)
38
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cell Shape Parametrization• Full parametric model of thecavity in terms
of 7 meaningful geometrical parameters:
Ellipseratio at theequator (R=B/A) ruled by mechanics
Ellipseratio at the iris (r=b/a)Epeak
Side wall inclination( ) and position (d)Epeak vs. Bpeak tradeoffand couplingkcc
Cavity iris radiusR iris
couplingkcc
Half-cell Length L/2=/4
Cavity radiusD
used for frequencytuning
• Behavior of all EM and mechanical properties has been found as a function ofthe above parameters L/2
39
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Reducing Peak Surface Fields
Add “electric volume” at the
iris to reduce Epeak
• “Rule of thumb” for Optimizing Peak Surface Fields
Add “magnetic
volume” at the equator
to reduce Bpeak
40
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Multi-Cell Cavities
Single-cell is attractive from the RF-point of
view:
• Easier to manage HOM damping
• No field flatness problem
• Input coupler transfersless power
• Easy for cleaningand preparation
• But it is expensive to base even a small
linear accelerator on the single cell. We do
it only for very high beam current
machines.
A multi-cell structure is less expensive and offers
higher real-estategradient but:
• Field flatness (stored energy)in cells
becomes sensitive to frequencyerrors of
individualcells
• Other problemsarise:HOMtrapping…
41
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cell to Cell Coupling
+ +
+ -
Symmetryplane for
the H field
Symmetryplane for
the E field
which is an additional
solution
ωo
ωπ
0
0
kcc
2
The normalized
difference between
these frequencies is a
measure of the energy
flow via the coupling
region
Slide 42
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Cost of accelerators are lower (less auxiliaries: He vessels,
tuners, fundamental power couplers, control electronics)
• Higher real-estate gradient (better fill factor)
• Field flatness vs. N (N – no. of cells)
• HOM trapping vs. N
• Power capability of fundamental power couplers vs. N
• Chemical treatment and final preparation become more
complicated
• The worst performing cell limits whole multi-cell structure
Pros and Cons of Multi-Cell Cavities
43
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Multi-Cell Cavities
Mode frequencies:
ω0 – accelerating mode frequency
Voltages in cells:
Cb Ck / 2k
Ck
C
44
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cell Excitations in Pass-Band Modes9 Cell, Mode 1
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
9 Cell, Mode 2
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
9 Cell, Mode 3
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
9 Cell, Mode 4
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
9 Cell, Mode 5
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
9 Cell, Mode 6
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
9 Cell, Mode 7
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
9 Cell, Mode 8
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
9 Cell, Mode 9
-1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
45
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Couplingof TM010 modes of the individual cells via the iris (primarily electric
field) causes them to split into a passband ofclosely spaced modes equal in
number to the numberof cells
Pass-Band Modes Frequencies
0 1 2 3 4 5 6 7 8 9
9-cell cavity
1/2
0 (1 4k )
46
ω0
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Geometrical differences between cells causes a mixing of
the eigenmodes
• Sensitivity to mechanical deformation depends on mode
spacing
Field Flatness in Multi-Cell Cavities
47
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Low Beta Accelerating Structures –
TEM Class Cavities
Slide 48
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Two main types of structure geometries
– TEM class (QW, HW, Spoke)
– TM class (Elliptical)
Classification of Structures
49
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Low β cavities: Cavities that accelerateparticleswith β < 1 efficiently
• Increased needsfor reduced-beta (β < 1) SRF cavity especially in CW
machines or high duty pulsed machine (duty > 10%)
• Reduced betaElliptical multi-cell SRF cavity
– For CW, prototypingby several R&D groups have demonstrated
as lowas β=0.47
– For pulsed,SNS β=0.61,0.81 cavities & ESS
• Ellipticalcavity has intrinsicproblemas β goes down
– Mechanicalproblem,multipacting, low rf efficiency
Low β Cavities
50
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Applications of Low β Cavities
High Current Medium/Low Current
CW
Pulsed
Acceleratordriven systems
• Waste transmutation
• Energy production
Beam: p, H-, d
Production of radioactive ions
Nuclear Structure
Beam; p to U
Pulsed spallation sources
Beam: p, H-
51
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Technical Issues and Challenges
High Current Medium/Low Current
CW
Pulsed
• Beam losses (~ 1 W/m)
• Activation
• High cw rf power
• Higher ordermodes
• Cryogenics losses
• Beam losses (~ 1 W/m)
• Activation
• High cw rf power
• Higher ordermodes
• Cryogenics losses
• Beam losses (~ 1 W/m)
• Activation
• Higher ordermodes
• High peak rf power
• Dynamic Lorentz detuning
52
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Design Considerations
• Cavities with high acceptance
• Development of high cw
power couplers
• Extraction of HOM power
• Cavities with high shunt
impedance
Medium/Low Current
• Cavities with low sensitivity to vibration
• Development of microphonicscompensation
• Cavities with high shunt impedance
• Cavities with large velocity acceptance (few cells)
• Cavities with large beam acceptance (low frequency, small frequency transitions)
Note:
Large beam acceptance
• Large aperture (transverse acceptance)
• Low frequency (longitudinal acceptance)
• Cavities with high acceptance
• Development of high peak
power couplers
• Extraction of HOM power
• Development of active
compensation of dynamic
Lorentz detuning
High Current
CW
Pulsed
Slide 53
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Resonant transmission lines
– λ/4
Quarter wave
Split ring
Twin quarter wave
Lollipop
– λ/2
Coaxial half wave
Spoke
H-type
• TM type
– Elliptical
– Reentrant
• Other
– Alvarez
– Slotted iris
Basic Geometries
Slide 54
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Many different shapes and sizes
Low β Cavities
ANL cavities for RIA
SpokeQW HW
Half-wave cavity
RFQ
Slide 55
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• The transit time factor is theratio of theacceleration voltage to the
(non‐physical)voltage a particlewith infinite velocity would see
• Energy gain (W):
• Assuming constantvelocity
• Transit time factor:
Transit Time Factor
56
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Velocity Acceptance
• Velocity acceptance
• Lower thevelocity of the
particleor cavity β
o Faster the velocity of the
particlewill change
o Narrower the velocity range of
a particularcavity
o Smaller the numberof cavities
of that β
o More important:Particle
achieve design velocity
57
Velocity acceptance for sinusoidal field profile
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
QWR A capactively loaded λ/4 transmission line
Maximum voltage builds up on the open tip and
maximum current at the plate connecting the center
conductor
Beam tube is placed near the end of the tip to
produce a high voltage double gap acceleration
geometry
Quarter Wave Resonator (QWR)
58
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Quarter Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β
Vp – voltage on the center conductor with outer conductor at ground
Peak Magnetic Field
Geometrical Factor Energy Content
59
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Quarter Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β
Vp – voltage on the center conductor with outer conductor at ground
Shunt Impedance
Power Dissipation (Ignore losses in the shorting
end plate)
R/Q
60
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Half–Wave Resonator
61
2b
2a
L
E
B
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Half Wave Resonator (HWR)HWR A λ/2 transmission line
Equivalent to 2 QWR facing each other and connected
Magnetic field loop around the inner conductor with peak
fields at the shorted ends
Beam tube is placed at the center of the inner conductor to
coincide with the maximum voltage
Same accelerating voltage is obtained at about 2 times larger
power in QWR (PHWR ~ 2PQWR)
62
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Half Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β
Vp – voltage on the center conductor with outer conductor at ground
Geometrical Factor
Peak Magnetic Field
Energy Content
63
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Half Wave ResonatorOptimizing the expected performance of a resonator for given frequency and β
Vp – voltage on the center conductor with outer conductor at ground
Shunt Impedance
Power Dissipation (Ignore losses in the shorting end plates)
R/Q
64
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
There have been extensive efforts for design optimization especiallyto reduce
the ratios of
Ep/Eacc and Bp/Eacc
• ControllingA/B (Ep/Eacc)and C/D (Bp/Eacc) Shapeoptimization
• Flat contactingsurface at spoke base will also help in minimization of Bp/Eacc
• For thesecavities:
– Calculationsagree wellEp/Eacc~3,Bp/Eacc ~(7~8)mT/(MV/m)
– Though it is tricky to obtainprecise surface field information from the
3D simulation
Spoke Resonators
325 MHz, β=0.17 (FNAL)
65
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Ep/Ea ~ 3.3, independent of β
• Bp/Ea ~ 8 mT/(MV/m), independent of β
Spoke Resonators
Surface electric field Surface magnetic field
Lines: Elliptical Squares:Spoke
66
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• QRs ~ 200 β [Ω]
• Rsh/Q ~ 205 [Ω] independent of
Spoke Resonators
Lines: Elliptical Squares:Spoke
67
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
•
• U/E2 ~ 200 β2
Rsh Rs ~ 40000 β [Ω2]
[mJ]
Spoke Resonators
Rsh Rs U/E2
Lines: Elliptical Squares:Spoke
0
5000
10000
15000
20000
25000
30000
35000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Beta
Rs
hR
sp
er
ce
ll
2
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Beta
U/E
2p
er
ce
ll(m
J)
@1
MV
/m,
500
MH
z
68
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Features of Spoke Cavities
• Small Size
– About half of TM cavity of same
frequency
• Allows low frequency at reasonable
size
– Possibility of 4.2 K operation
– High longitudinal acceptance
• Fewer number of cells
– Wider velocity acceptance
350 MHz, =
0.45
69
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Deflecting / Crabbing Cavities
70
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Tail of the bunch
deflected down
• Both deflectingand crabbingresonantcavities are required to generate a
transverse momentum
• Can be produced by eitheror by both transverse electric (E t) and magnetic
(Bt) fields
• Lorentzforce:
• Types of designs:
– TM-typedesignsMain contributionfromBt
– TE-typedesignsMain contribution fromE t
– TEM-typedesigns Contributionfromboth E t and B t
Deflecting/Crabbing ConceptComplete bunch is deflected
Head of the bunch
deflected up
71
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Deflecting Cavities
f0 = 1497 MHz
fd = 499 MHz
• RF frequency(fd) = 499 MHz
• Beam energy (E) = 11.023GeV
• Deflecting angle (θ)
• Transverse voltage (Vt)
θ
E0
Beam direction
eVT
• Completebunch is deflected with the
transverse kick applied at the centerof the
bunch
72
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Luminosity with no crabbing
system:
• Luminosity with crabbing
system:
– For head-on collision
• Transverse voltage
Crabbing Cavities
crossing angle
Crabbing
Cavity
crossing angle
Head-on collision
73
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• For particlesmoving virtually at v=c, the integrated transverse force
(kick) can be determined from the transverse variation of the integrated
longitudinal force
• Accordingto the theorem:
– In a pureTE modethe contribution to thedeflection fromthe
magnetic field is completelycancelled by thecontributionfromthe
electric field
Panofsky–Wenzel Theorem
c
• Transverse momentumis related to the gradient of the longitudinal
electric field alongthebeam axis
j Ft t Fz
74
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Deflecting/Crabbing CavitiesKEK TM110 crabbingcavity Superconducting4-rod cavity
BNL doublequarter-wavecavity
75
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Deflecting/Crabbing Cavities
76
RF-Dipole Cavity
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
LHC High Luminosity Upgrade
77
SPS Cryomodule with 2 DQW crab cavities –
Demonstrated crabbing of first proton beam
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity Limitations
78
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity Performance
79
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Multipacting
80
• Multipacting condition – A large amount of secondary
electrons are emitted from the cavity surface by the
incident primary electrons
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Resonant condition
– The secondaryelectronshave localized and
sustainableresonanttrajectorieswith thecavity rf
fields
– Impact energies correspondsto a secondary
emission yield (SEY) greater than one
Multipacting
δmax
EIEmax EII
Impact Energy [eV]
Sec
ond
ary
Em
issi
on
Yie
ld(δ
)
EI 150eV EII 2000eVFor Nb:
81
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Field Emission• Characterized by exponential decayin Q0
• Exponential increaseof losses due to acceleration ofelectron field
emission
• With the increasingrf field the field emitters generate an electron current
leadingto excessive heatingand x-rays producedby bremsstrahlung
• It is a general difficulty in acceleratingstructures,but doesnot present an
ultimate fundamental limit to the maximum surface electric field
• Main cause ofFE is particulatecontamination
Temperature map shows line
heating along the longitude at
the location of the emitter
82
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Fowlerand Nordheim(FN) showed that when thework function barrier
at themetal surface is lowered by an applied surface electric field,
electronscan tunnelthrough
• Electrostaticpotential of the metal-vacuuminterface
• Tunnelingcurrentdensity
Field Emission
No electric field applied Electric field applied
83
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Modified Fowler Nordheimequationfor rf fields
• Field emitters
Field Emission
• FE can be prevented by propersurface
preparation and contaminationcontrol
• Possible to reduce if not completely
eliminateFE using CW RF
processing, High-power Pulsed
Processing(HPP) and/orHelium
processing
84
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Ponderomotive effects: changes in frequency caused by the
electromagnetic field (radiation pressure)
– Static Lorentz detuning (CW operation)
– Dynamic Lorentz detuning (pulsed operation)
• Microphonics: changes in frequency caused by connections
to the external world
– Vibrations
– Pressure fluctuations
• Note: The two are not completely independent. When phase
and amplitude feedbacks are active, the ponderomotive
effects can change the response to external disturbances
Ponderomotive Effects
85
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Lorentz Force Detuning
-0.006
020
z [mm]40
60
P[N
/mm
^2
]
• Electromagnetic fields in a cavity exert Lorentzforces on the cavity wall. The
force per unit area (radiation pressure)is given by
92 kA/m
50 MV/m0.003
-0.003
0
E and H at Eacc =
25 MV/m in
TESLAinner-cup
• Residual deformationof the cavity shapeshifts the
resonant frequency
kL – Lorentzcoefficient
86
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Localized heating
• Thermal breakdown occurswhen theheat generated at thehot spot is
larger than that can be evacuated to the heliumbath
• Both the thermal conductivityand thesurface resistanceof Nb are
highly temperaturedependentbetween2 and 9K
Thermal Breakdown
87
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Losses in RF Cavities
88
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Losses are given by Ohm’s Law where σ is the
conductivity
• In a cavity, rf magnetic field drives an oscillating current in
the cavity wall
• Following Maxwell’s equations
• Neglecting displacement current
Losses in Normal Conducting Cavities
89
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Considering the cavity as wall as a local plane surface, solve
one dimensional problem at the surface for uniform magnetic
field in y direction
• with field decaying into the conductor over the skin depth
Losses in Normal Conducting Cavities
90
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• From Maxwell equation
– A small tangential electric field component decays into
the conductor
• Surface impedance
• Surface resistance is the real part of surface impedance
Losses in Normal Conducting Cavities
91
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Anomalous Skin Effect
• Surface resistance becomes independent of the conductivity at low
temperatures
• As the temperature decreases, the conductivity σ increases− Skin depth decreases
− Skin depth (the distance over which fields vary) can become less then the
mean free path of the electrons (the distance they travel before being
scattered)
− Electrons do not experience a constant electric field over a mean free path
− Local relationship between field and current is not valid
Slide 92
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Anomalous Skin Effect• Theory introduces a dimensionless parameter αs, which depends on the
temperature-independent product of the mean free path l and the resistivity ρ = 1/σ
• αs strongly depends on mean free path l
Slide 93
• Classical expression for the surface resistivity
is valid when a αs ≤ 0.016. In the anomalous
limit, when αs ∞
• For intermediate values
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Superconductivity – Discovered in 1911 Kammerlingh-Onnes, is
a phenomenon where below a certain temperature, called the
critical temperature (Tc), some materials show a sudden drop of
the dc electrical resistance to zero
Superconductivity
Kamerlingh Onnes and van der Waals in
Leiden with the helium 'liquefactor' (1908)
94
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Meissner Effect
Meissner Effect – Discovered by Meissner and Ochsenfeld in 1933
• Ability to completelyexpel an externallyappliedmagnetic field when cooled
down belowthecritical temperature(Tc)
• Superconductorbehaves as a perfect diamagnet
• Surface currentscreated at the
surface generates a magnetic field
that cancelsthe external magnetic
field inside the superconductor
• These surface currentsdo not
decay with time dueto thezero
resistance in thesuperconductor
95
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Critical magnetic field (Bc) – Field beyond at which
superconductivity is destroyed
Critical Magnetic Field
Type I – Soft Superconductors Type II – Hard Superconductors
96
Tc – critical temperature
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Surface Resistance of Superconductors
• Superconductors are free of power dissipation at static fields
• In microwave fields with the time dependent magnetic field in
the penetration depth will generate an electric field
• Electric field will induce oscillations in the normal electrons,
which will lead to power dissipation
Slide 97
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Macroscopic theory of superconductivity
• Coexistence of:
– super electrons (ns)
– normal electrons (nn)
– Total density
• Only super electrons are accelerated by the
constant electric field (E)
• Super current density
• Super electrons are not affected by the normal
electrons
Two Fluid Model
98
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Using Maxwell’s equations
• Field penetration in the superconductor
• London penetration depth
Distance to which a magnetic field
penetrates into a superconductor
London Equations
99
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Fundamental Lengths
• London penetration depth (λL)
– Distance over which magnetic fields decay in superconductors
• Pippard coherence length (ξ0)
– Distance over which the superconducting state decays
100
• Type II
superconductors –
λL >> ξ0
• Type I
superconductors –
ξ0 >> λL
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Material Parameters
101
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Bardeen-Cooper-Schrieffer Theory (1957) – Nobel prize in 1972
• Macroscopical Microscopical representation
BCS Theory
102
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Cooper pairs – Pair of electrons formed due to electron-phonon
interaction that dominates over the repulsive Coulomb force
• Moving electron distorts the lattice and leaves behind a trail of
positive charge that attracts another electron moving in
opposite direction
• Has lower energy than the two separate electrons
• Therefore, electron pairs form bound states of lower energy
which are stable than the Fermi ground state
• Strong overlap of many Cooper pairs
results in the macroscopic phase coherence
Cooper Pairs
103
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Energy Gap
• Energy gap (Δ) – gap around Fermi level between ground
state and excited state• At 0 < T < Tc not all electrons are
bounded in to Cooper pairs
• Density of unpaired electrons is given
by
Normal
conductorSuperconductor
104
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• At the presence of an rf field
– Cooper pairs move without resistance Do not
dissipate power
– Due to inertial mas of Cooper pairs they cannot follow an AC
electromagnetic fields instantly and do not shield it perfectly
– Remaining residual field accelerates the normal electrons that
dissipate power
• More normal electrons The material is more lossy
– Losses decrease with temperature below Tc
• Faster the field oscillates the less perfect the shielding
– Losses increase with frequency
Losses in Superconductor
105
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Surface Impedance
• Following two fluid model
• Total current density
• Surface impedance:
• Skin depth
106
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
BCS Surface Resistance
• Surface impedance
• Taking may parameters into
account Mattis and Bardeen
developed theory based on
BCS theory
107
A – material parameters
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• Surface resistance of superconductors (Rs)
• Residual resistance (Rres) due to:
– Dielectric surface contaminants (gases, chemical residues, ..)
– Normal conducting defects, inclusions
– Surface imperfections (cracks, scratches, ..)
– Trapped flux during cool down through critical temperature
– Hydrogen absorption during chemical processing
• An approximation of RBCS for Nb:
Surface Resistance
108