-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 1
5. RF Systems and Particle Acceleration5.1 Waveguides
5.1.3 Cylindrical Waveguides5.2 Accelerating RF Cavities
5.2.1 Introduction5.2.2 Traveling wave cavity: disk loaded
waveguide5.2.3 Standing wave cavities5.2.4
Higher-Order-Modes5.2.5 The pillbox cavity5.2.6 SRF primer
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 2
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 3
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 4
-
Mode for particle acceleration: TM01 )cos()()(00 tzkJExE zrrzz
=
)sin()(')(101 tzkJkrExE zrrzzr =
0)( =xE 0)( =xBr
)sin()(')(12 01 tzkJrExB zrrcz =
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 5
inRS
nzz erJBxBxE nm )()(,0)( 0==
R)(
)(
2222
222
zczzki
zzzki
EBkB
BEkE
zc
zc
=
+=
( )( )
( )( )
)cos()()cos()(
)sin()('0
)sin()(')cos()(
0
10
12
10
2
10
2
10
12
2
2
tzknrJBBtzknrJRnkBBikB
tzknrJRkBBikBE
tzknrJRBBiE
tzknrJRnBBiE
zRS
nz
zRS
nrR
SzzrSR
z
zRS
nSzzrSR
zr
z
zRS
nSzrSR
zRS
nrR
SzrSR
r
nm
nm
nmnm
nmnmnm
nmnmnm
nm
nmnm
+=
+==
+==
=
+==
+==
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 6
-
0 1 2 3 4 5 6-1
-0.5
0
0.5
1
E
B
Bx
z
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 7
0)(,)()( 0 == xBerJExE zinRZnz nm
R)(
)(
2222
222
zczzki
zzzki
EBkB
BEkE
zc
zc
=
+=
0)sin()('
)cos()()cos()(
)cos()()sin()('
2
2
0
10
10
10
1
0
=
+==
+==
+=
+==
+==
z
zRZ
nczrZR
zRZ
nrR
ZczrZR
r
zRZ
nz
zRZ
nrR
ZzzrZRk
zRZ
nzzrZRk
r
BtzknrJEEiB
tzknrJnEEiBtzknrJEE
tzknrJnkEEiEtzknrJkEEiE
nmnm
nmnmnm
nm
nmnmnm
z
nmnmz
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 8
-
0 1 2 3 4 5 6
-1
-0.5
0
0.5
1
E
B
Bx
z
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 9
0 1 2 3 4 5 6-1
-0.5
0
0.5
1
TES
TMS
TESx
z
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 10
-
Energy density for TE mode 2,2
-1-0.5
00.5
1 -1
-0.5
0
0.5
1
01000200030004000
-1-0.5
00.5
Energy density for TM mode 2,2
-1-0.5
00.5
1 -1
-0.5
0
0.5
1
02000400060008000
-1-0.5
00.5
Energy density for TM mode 2,2
-1-0.5
00.5
1 0
2
4
6
02000400060008000
-1-0.5
00.5
TEU TMU TMU
x x
xy y z
x
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 11
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 12
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 13
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.
DC Accelerators:
Use time-varying fields!
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 14
-
Taiwan Light Source cryomodule
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.
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 15
NC or SC Relatively low
gradient (19 MV/m)
Strong HOM damping (Q ~ 102)
High average RF power (hundreds of kW)
CESR cavities
KEK cavity
PEP II Cavity
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 16
-
NC or SC High gradients Moderate HOM
damping reqs. High peak RF power
ILC: 21,000 cavities!ILC / XFEL cavities
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 17
SRF cavities Moderate to low
gradient (820 MV/m)
Relaxed HOM damping requirements
Low average RF power (513 kW)
CEBAF cavities
ELBE cryomodule
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 18
-
SRF cavities Moderate
gradient (1520 MV/m)
Strong HOM damping (Q = 102104)
Low average RF power (few kW)
Cornell ERL cavities
BNL ERL cavity
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 19
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 20
Standing wave cavity. Traveling wave cavity (wave guide).
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 21
Cylindrical Waveguide: TM01 has longitudinal electric field and
could in principle be used for particle acceleration
But: phase velocity of wave > c > speed of particle->
no average energy transfer to beam !
Solution: Disc Loaded Waveguide Iris shaped plates at constant
separation in waveguide lower phase
velocity Iris size is chosen to make the phase velocity equal
the particle
velocity
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 22
Vgroup = Vparticle
operation
koperation
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 23
Irises form periodic structure in waveguide-> Irises reflect
part of wave-> Interference-> For loss free propagation: need
disk spacing d
p=integer
d
pdz=
pdkz2
=
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 24
Selection of integer p:
Long initial settling or filling time,not good for pulsed
operation with very short pulses.
Small shunt impedance per length (shunt impedance determines how
much acceleration a particle can get for a given power dissipation
in a cavity).
Common compromise.
p=2
p=3
p=4
ccsh PVR
2=
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 25
Time dependent electromagnetic field inside metal box
Energy oscillates between electric and magnetic field!
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 26
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 27
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 28
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 29
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 30
500MHz Cavity of DORIS:GHz4967.01.23 )(010 == Mfcmr
l The frequency is increased and tuned bya tuning plunger.
l An inductive coupling loop excites themagnetic field at the
equator of the cavity.
-
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 31
LC1
=
TE TM
TM010
0,0 == HE nn
0122 =
H
Etc
00010
10
00
,405.2
405.2
405.2
==
=
=
Rc
eRrJEiH
eRrJEE
ti
tiz
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 32
-
Higher Frequency (Order) Standing Wave Modes
TMmnp (TEmnp),m, n, p Ez (Hz) , r, z
TM010
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 33
T
T = 2/ Eacc Eacc = Vc/d d = /2
( )
== dzezEV czizc 0,0
TdE
cdcd
dEdzeEVd
czic =
== 00
0
00
0
2
2sin0
dEVdTttT centerexittransit 00222 ====
t
E z
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 34
-
Surface currents ( H) result in dissipation proportional to the
surfaceresistance (Rs):
Dissipation in the cavity wall given bysurface integral:
Energy density in electromagnetic field:
Because of the sinusoidal time dependence and 90 phase shift,
the energy oscillates back and forth between the electric and
magnetic field. The stored energy in a cavity is given by
== VV dvdvU2
02
0 21
21 EH
( )2221 HE += u
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 35
Quality factor
2 ~104 ~1010
( )0
0000
00012losspower average
energy stored
====
cc PU
TPUQ
=
Ss
VdsR
dvQ 2
2000 H
H
-1000 -500 0 500 10000
1
cavit
y fie
ld [a
rb. u
nits
]
Frequency 1.3 GHz [Hz]
Bandwidth
(1) The RF system has a resonant frequency
(2) The resonance curve has a characteristic width
0
Q20 =
-
Geometry factor
G
G = 257Ohm
sRGQ =0
=
S
Vds
dvG 2
200
H
H
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 37
Shunt impedance and R/Q
Pc
R/Q = 196 Ohm
ccsh PVR 2
2=
ccsh PVR
2=
caccsh P
Er4
=2
UV
QR csh
0
2
0 =
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 38
-
GR/Q
)/()/)(()/( 02
00
2
00
22
QRGRV
RQRQRV
QRQV
RVP
shsc
sshsc
shc
shcc
=
=
==
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 39
270 88 /cell
2.552 Oe/(MV/m)
Cornell SC 500 MHz
Hpk Epk R/Q
Pillbox vs. real life cavity
Matthias Liepe, P4456/7656, Spring 2010, Cornell University
Slide 40