TM110 Deflecting/Crabbing Cavity for Muon
Emittance Exchange ?
Haipeng Wang, Robert Rimmer
Jefferson Lab
Muon Collider Design
Workshop, 12/8~12/2008
Talk Outlines
• Motivation: After muon 6D emittance cooling in helix
channel, using TM110 mode of RF cavities instead of
absorbers combining with dipole chicanes to exchange
transverse emittance (too large) to longitudinal emittance
(too small) before (pre) acceleration.
• This is an open question to implement this technique.
• Review principle of TM110 mode RF cavity
• Examples of past and present applications of
deflecting/crabbing cavities in different projects.
• Design challenge and limitation of practicable cavity.
Muon Collider Design
Workshop, 12/8~12/2008
Principle of Deflecting Mode of a RF cavity
Cylindrical pillbox
Panofsky-Wenzel Theorem:
d
 e d
e
P    E   v  B  dz     i   E z dz
 v 0
 0  0
a
B
E
TM110 mode
W. K. H. Panofsky and W. A. Wenzel, Review of Scientific Instruments, Nov. 1956, p967.
also M. J. Browman, LANL, PAC93, May 17-20, 1993, Washington D.C. USA.
• Panofsky’s theorem implies that for any given RF mode, no matter
who (E or B) deflecting the beam, there is must an non-zero transverse
gradient of longitudinal component of the electric field.
• TM110 is one of such modes. Two rod type, TEM mode is another
one. There are also other “exotic” modes, like off-axis TM010 mode,
sideway TM012 mode.
• Transverse verses longitudinal impedance based
on Panofsy’s:
Rt/Q(R///Q)/(ka)2 k=/c a=off-axis
distance where to assess the R//.
• Deflecting force:
k2
2
2
Fx   evz By   eE0 (1 
•
d 2 x eE0

sin t
dz 2 mc 2
8
(3x  y ) 
/2
Ez  2 Eo J1 (kr ) cos  e it
Er  E  cBz  0
  3.832
2iE0
J1 (kr ) sin  e it
kr
cB  2iE0 J1 (kr ) cos  e it
cBr 
Deflecting
B
crabbing
)sin t
Aberration terms
Muon Collider Design
Workshop, 12/8~12/2008
c
a
Scaling laws of RF deflecting cavities
Two-rod transmission line
Cylindrical pillbox
d0
a
dc
TM110 mode
/2
R
1920

Q  u11 J 2 u11 2
 J1  

   J 2  
2


Here =u11r/a, u11=3.832, is root of J1, J1/J2
is first/second order of Bessel function.
b  0.5 dc2  d02
for r0, R/Q=64.16 
which is wavelength independent.
U
H
2
max


 7.5u112 2  3462
m 
2

  64.16  48.59  3117.4


a  0.5dc  d0 
for a 800MHz cavity, d0=2cm, dc=5cm R/Q= 3091.2 
which is wavelength dependent.
d
ln  c 
U
 d0 
3

30

m 2
2
2
H max
 1  1

 d
d 0  d c 
 0
 R  U 
2

 2   3091.2  0.206  636.8 m
 Q  H max 

For a 800 MHz cavity,
 R  U
  2
 Q  H max
TEM dipole mode
~/2
Reference: C. Leemann and C. G. Yao, LINAC 1990,
Albuquerque, NM, p233.
d
ln  c 
2
R 960
 d0 


Q
 3 b 2 ln 2 b  a
ba
m2

Muon Collider Design
Workshop, 12/8~12/2008
Scaling laws of RF deflecting cavities
Two-rod transmission line
Cylindrical pillbox
a
Vdef
Bmax

1
0
 R  U
  2
 Q  H max



d0
dc
/2
~/2
Vdef / Bmax (MV/mT)
0.04
0.035
0.03
0.025
0.02
0
5
10
15
20
Rod Gap Distance dc-d0(mm)
Two-Rod, d0=2cm
Two-Rod, d0=5cm
Pillbox
25
30
For a 800 MHz cavity
with a 50mm beam
aperture, two–rod type is
only about 45% in
efficiency of pillbox type,
and even less than the
elliptical cavity. But its
transverse dimension is
55% or less than the
pillbox type.
Squashing elliptical
cavity in transverse
dimension is in wrong
direction for the
transverse kick (will give
vertical kick instead).
Pillbox verses Two Rod of 2.8GHz Cavity
0.05
0.0429
0.0357
Vdef / Bmax (MV/mT)
Pillbox verses Two Rod of 2.8GHz Cavity
0.045
0.0286
0.0214
0.0143
0.0071
0
0
20
40
60
Rod Diameter d0 (mm)
Two-Rod, dc-d0=2cm
Two-Rod, dc-d0=5cm
Pillbox
Muon Collider Design
Workshop, 12/8~12/2008
80
100
Application Examples of Deflecting/Crabbing Cavities
• Particle separation: (CEBAF separator)
• Temporal beam diagnostics: (injector/gun emittance
measurement, BPM, BCM)
• Crab-crossing in colliders (KEK B Factory, LHC, ILC, ELIC,
eRHIC…)
• X-ray pulse compression: (APS crab cavity R&D)
• Emittance exchange: (AWA, FELs, Muon preacceleration?...)
Most technical challenge to those designs are for high current
accelerators (circular) which require heavy damping on
parasitic modes (LOM, SOM, HOM, SPBM) and single highQ deflecting mode CW operation, so SRF structure.
For muon (single pass) EMX, the damping might not required.
Muon Collider Design
Workshop, 12/8~12/2008
CEBAF Normal Conducting Separator Cavity
Quick fact and number:
• Qcu is only ~5000 (structure wise), the stainless
steel cylinder only takes less than 5% of total
loss.
• Each cavity is two-cell, ~ long, can produce
400kV deflecting voltage with 1.5kW input RF
power.
• The maximum surface magnetic field at the rod
ends is ~14.3mT.
• Need water cooling on the rods.
• Can kick beam into three experiment halls
simultaneously.
Muon Collider Design
Workshop, 12/8~12/2008
Crab Crossing in Linear and Circular Colliders
Robert Palmer invented “Crab crossing” in Feb. 1988 at SLAC to reduce head-on
collision luminosity loss due to bremsstrahlung. Just the second day after Peshi Chen
reported this possible mechanism.
Since then, the first group to use SRF cavities to do the “crab crossing” in a circular
collider is KEKB. A global crabbing scheme to increase luminosity has shown a good
result recently.
Other crab cavities for LHC, ILC are aggressively
Muon Collider Design
Workshop, 12/8~12/2008
KEKB Crab Cavity Developments
elliptical squashed shape
Muon Collider Design
Workshop, 12/8~12/2008
KEKB Crab Cavity Commissioning
Curtsy of K. Hosoyama: KEK elliptical crab type cavity, 508.9MHz,
Started from 1994 Superconducting Nb, one cavity per ring, global
crab scheme in KEKB operation.
Muon Collider Design
Workshop, 12/8~12/2008
ILC Crab Cavity Developments
(FNAL/SLAC/Cockcroft Intitutes)
• Collaboration has been worked
on this project for many years. So
far the 3.9 GHz 9cell, slightly
squashed elliptical
superconducting cavity has been
chosen for the ILC local crabbing
scheme.
• Cavity VTA test has been done
and to be integrated into a
cryomodule.
• A lot of bead-pulls, simulation of
HOM/LOM/SOM work have been
done.
• All LOM/SOM/LOM damping
by coaxial couplers have been
designed and simulated.
Prototyping in on going.
Muon Collider Design
Workshop, 12/8~12/2008
Optimize Crab Cavity’s Squash Ratio
Dispersion Curve
1.10E+09
1.05E+09
Crab cavity for LHCs’ squash
ratio is chosen to optimize
mode separation
HOM(TE111)
1.00E+09
9.50E+08
SOM
Curtsy of L. Xiao and Z. Li of SLAC.
F (Hz)
9.00E+08
8.50E+08
Dy
FM
8.00E+08
7.50E+08
7.00E+08
6.50E+08
6.00E+08
TM010-1
TM110-1-H (opt.mode)
TM010-2
TM110-2-H
TM110-1-V (SOM)
TE111-1-H
TM110-2-V
TE111-2-H
TE111-1-V
TE111-2-V
Dx
LOM
5.50E+08
0.7
0.75
0.8
0.85
0.9
Cross Section Elliptical Ratio
0.95
1
Fc=1.2GHz@R_beampipe=70mm
Muon Collider Design
Workshop, 12/8~12/2008
Crab Cavities for Light Sources
•
•
•
•
Use transverse-deflecting rf cavities to impose a correlation (“chirp” between the longitudinal position of a particle
within the bunch and the vertical momentum.
The second cavity is placed at a vertical betatron phase advance of n downstream of the first cavity, so as to
cancel the chirp.
With an undulator or bending magnet placed between the cavities, the emitted photons will have a strong
correlation among time and vertical slope.
This can be used for either pulse slicing or pulse compression.

X-ray pulse compression
Slitting
y

A. Zholents, P. Heimann, M. Zolotorev, J. Byrd, NIM A 425(1999), 385
Muon Collider Design
Workshop, 12/8~12/2008
Squashed elliptical cavity shape optimization
sqrt ( Rt/Q * G ) with Rcav=44.7mm, Rbp=25mm,
Zcav=53.28mm
Rt/Q with Rcav=44.7mm, Rbp=25mm, Zcav=53.28mm
94
36
Rcav
mm
10
34
11
33
12
13
32
Rcav
mm
92
sqrt(Rt/Q*G) (Ohm)
Rt/Q=Vt^2/P (Ohm)
35
14
31
90
10
88
11
12
86
13
14
84
82
30
80
7
8
9
10
11
12
13
7
8
9
10
rcon (mm)
11
12
13
rcon (mm)
MWS ,ANSYS, HFSS and Gdfidl simulation by J. Shi and G. Waldschmidt
Bmax/Vdef with rbp=23mm, rcon=8mm, Rcav=10mm
Bmax/Vdef with rbp=23mm, rcon=8mm,
Rcav=10mm
Bmax/Vdef with Rbp=25mm, Rarc=44.74mm,
Zcav=53.24mm
182
190
180
175
170
165
160
155
185
Bmax/Vdef (mT/(Mv/m))
185
Bmax/Vdfe (mT/MV/m))
Bmax/Vdef (mT/(MV/m))
190
180
175
170
165
160
155
150
150
1.4
1.5
1.6
1.7
1.8
1.9
Racetrack long/short axis ratio
2
2.1
2.2
180
178
176
174
172
170
38
40
42
44
46
Rarc (mm)
48
50
52
6
7
8
9
rcon (mm)
Muon Collider Design
Workshop, 12/8~12/2008
10
11
12
Squashed elliptical cavity shape comparison
racetrack radius
beam pipe radius
cavity equator radius
cavity iris radius
cavity iris-to-iris distance
cavity racetrack half straight length
optimized squashed dimensions
mm
Rarc
44
Rbp
25
rcav
14
rcon
9
zcav
53.24
yline
33.66
scaled to 800MHz KEK crab dimensions scaled to 800MHz
JLab-ANL-LBNL
KEK
154.9
241.5
153.6
88.0
94
59.8
49.3
90
57.3
31.7
20
12.7
187.4
294.5
187.3
118.5
191.5
121.8
Scaled KEK and JLab-ANL-LBNL’s crab cavity shapes to 800MHz
Muon Collider Design
Workshop, 12/8~12/2008
Elliptical squashed SRF cavity R&D for APS
(JLab/LBNL/AL/Tsinghua Univ.)
rcav
Rarc
optimized squashed dimensions:
rcon
Rbp
Rarc
Rbp
rcav
rcon
zcav
yline
zcav
44
25
14
9
53.24
33.66
First time vertical test achieved design gradient!
mm
mm
mm
mm
mm
mm
Single-cell 2.815GHz Nb crab cavity
Crab Cavity Test #1
Qo
1.00E+10
1.00E+09
RF System unstable
1.00E+08
0
20
40
60
Bpeak [mT]
80
100
120
Single-cell structure with beam pipes
TM110-y mode frequency
Rt/Q include TTF (Rt=Vt^2/P)
Geometry factor G
sqrt((Rt/Q)*G)
Bsmax/Vt
Esmax/Vt
Transverse Gradient Et=Vt/d
Bsmax/Et
Esmax/Et
cavity effective gap d
BCS surface resistance Rbcs of Nb at 2K
Residual resistance R0
Q0 at 2K
BCS surface resistance Rbcs of Nb at 4.2K
Q0 at 4.2K
MHz
Ohm
Ohm
Ohm
mT/MV
1/m
2815.76
35.27
232.29
90.51
157.15
75.60
mT/(MV/m)
8.367
4.025
53.24
51.29
20.00
3.3E+09
2498.33
9.2E+07
mm
nOhm
nOhm
nOhm
Muon Collider Design
Workshop, 12/8~12/2008
Waveguide HOM Damped Cavity Structure for APS
R///Q and Rt/Q
Calculated from
MWS eigen solver
Bench Qext measurement by using
• RF absorbers on WG ports
• Clamping copper parts (low contact loss)
• Weak coupling to VNA
• Rotatable antennas to suppress the unwanted
modes.
Muon Collider Design
Workshop, 12/8~12/2008
TM110 Cavity Replace Wedge Absorber?
• No gas or liquid to vacuum interface windows.
• No scattering, no straggling
Original Efrom Y. Derbenev and
R. P. Johnson EPAC 2006, WEPLS019
E
TM110
TM110
Muon Collider Design
Workshop, 12/8~12/2008
TM110 cavity used in Trans/Long Emittance Exchange
M in (x, x', z, ) phase space
k
eV0
aE
a is cavity radius
 and  are dispersion
and momentum compaction
Factor respectively
1. M. Cornacchia and P. Emma, Phys. Rev. ST Accel. Beams 5,
084001 (2002).
2. P. Emma, Z. Huang, K.-J. Kim and P. Piot, Phy. Rev. ST Accel.
Beams 9, 100702, (2006).
Muon Collider Design
Workshop, 12/8~12/2008
Emittance Exchange Simulations and Experiments
(x,y, z)=
Curtsy of G. Wei and J. Power
Muon Collider Design
Workshop, 12/8~12/2008
TM110/TE111 Modes Cell-to-Cell Coupling and Double-Chain Model
B field enhancement
when operates in pi mode
Bane, K. L. & Gluckstern, R. L. (1993), 'The Transverse
Wake Field of a Detuned X-band Accelerator Structure', Part.
Accel. 42, 123-169. (SLAC-PUB-5783)
Mode1
dispersion curve
Mode2
4000
TE11
3800
frequency
3600
3400
3200
TM11
3000
2800
TM11
TE11
2600
0
30
60
90
120
150
180
phase adv per cell (Deg)
    
1 

 k1 k 2  1 





cos  
2
2
 1 
  
1  2 k 2  1  22 k1
  
  
2
1
2
2
2
2
Curtsey of J. Shi, Tsinghua Univ. Beijing, China
frequency / MHz
Dispersion of Dipole
2820
2810
2800
2790
2780
2770
2760
2750
2740
2730
simulation
single chain
double chain
0
50
100
150
200
phase adv / DEG
Muon Collider Design
Workshop, 12/8~12/2008
Magnetic Field Enhancement at Iris of TM110 Multi-cell Cavity
Thanks K. Tian at JLab
Magnitude of the
magnetic field on the 3cell cavity. Note the large
field enhancement along
the iris.
Thanks to G.
Waldschmidt
Muon Collider Design
Workshop, 12/8~12/2008
Multi-cell TM110 and Loaded Structure of Crabbing Cavities
APS 4-cell crab cavity, 2.815GHz,
0 mode, 8MV total needed periodic
damping LOM/SOM/SPBM/HOM modes
Curtsy of Z. Li and L. Xiao from SLAC:LHC
crab cavity in IP4 GC scheme, 800MHz
prototype phase I with
LOM/SOM/SPBM/HOM modes couplers
HOM coupler
200MHz for LHC LC scheme
LOM/SOM coupler
JLab/Cockcroft Inst./Lancaster Univ. UK
Parallel Bar advanced , 400MHz for LHC, 499MHz for CEBAF 11GeV.
Muon Collider Design
Workshop, 12/8~12/2008
TEM Parallel Bar (Half-Wave) Deflecting Structure
• recent study for low frequency application
• more efficient J. Delayen, H. Wang, LINAC 2008’s paper.
• more compact
• no LOM but acceleration mode in HOM
Parameter
Ω3P
Analytical
model
Unit
Frequency of -mode
400
374.7
414.4
374.7
500
381.9
100
200
100
0.375
4.09
13.31
0.215
96.0
260
400
374.7
400
∞
∞
374.7
100
200
0
0.375
4.28
14.25
0.209
112
268
MHz
mm
MHz
mm
mm
mm
mm
mm
mm
MV
MV/m
mT
J
Ω
Ω
λ/2 of -mode
Frequency of 0-mode
E field
B field
Cavity length
Cavity width
Bar length
Bar diameter (2R)
G*Rt/Q
Ep/Et
10
Bar axes separation (2A)
50000
9
45000
8
40000
7
35000
6
 2 30000
5
Aperture diameter
Deflecting voltage Vt *
Ep *
Bp *
25000
A/R= 1.6
4
A/R= 1.8
3
A/R= 1.6
20000
2
A/R= 2.0
A/R= 2.2
1
A/R= 2.4
0
0
0.02
U *
A/R= 1.8
15000
A/R= 2.0
0.04
0.06
R/
0.08
0.1
0.12
0.14
10000
A/R= 2.2
5000
A/R= 2.4
0
0.00
0.02
0.04
G
Rt/Q
0.06
0.08
0.10
0.12
0.14
* at Et=1MV/m
R/
Muon Collider Design
Workshop, 12/8~12/2008
Summary
• Using crab cavity for muon emittance exchange is an
interesting idea. Detail study is just starting. We need
simulations with a real field map including fringe field of
cavity and dipole magnets.
• If technical feasible, this scheme will solve absorber’s
problem and lower cost.
• NC and SC deflecting/crabbing cavity development
experience in other projects can be brought in to see the
technical limitation.
• Low frequency, larger aperture crab cavity structure
without HOM damping is needed for the emittance
exchange section.
Muon Collider Design
Workshop, 12/8~12/2008