Chapter 11
RF cavities for particle
accelerators
Rüdiger Schmidt (CERN) – Darmstadt TU - 2011 –Version E2.2
Accelerating structures in linear and circular accelerators
• Acceleration cavity (cavity)
• Analogy between oscillating circuit and cavity
• Cylindrical cavity
• Shunt impedance and quality factor
2
Acceleration in the cylindrical cavityT=0 (accelerating phase)
(100 MHz)

E (t)
2a
z
E(z)
E0
g
z
3
Linear and circular accelerators
Linear accelerator: Acceleration by traveling once through
many RF
Circular accelerator: Acceleration by travelling many times
through few RF cavities
4
Analogy between cavity and oscillating circuit

E (t)

E (t)
C
L
R
A simple RF accelerator would work with a
capacitor (with an opening for the beam)
and a coil in parallel to the capacitor. The
energy oscillates between electric and
magnetic field.
L
R
5
Analogy between cavity and oscillating circuit
Oscillating circuit with capacitor,
coil and resistance.
Resonance
frequency
Quality factor : Q 
Time constant
: ω res 

E (t)
C
1
L C
L
ω res  L
R
R
for damping :   R  C
6
For a frequency of 100 MHz, a typical value for an accelerator, the inductance of
the coil and the capacity of the condenser must be chosen very small. Example:
A c  100 cm
Capacity of a capacitor with a surface of
two plates of
Capacity
2
and a distance between the
d c  1 cm
C c 

0
Ac
dc
2
A s  100 cm , a length of
Inductivity of a coil with a surface of
N s  10
and a number of turns of
2
Inductivity
L c 

0
N s  As
ls
C c  8.854  pF
L c  12.566   H
Oscillation frequency
f c 
1
2

1
Lc Cc
f c  15.088  MHz
l s  10 cm
From oscillating circuit to the cavity

E (t)

E (t)
C
C

B (t)
L

B (t)

E (t)
L
The fields in the cavity oscillate in TM010
mode (no longitudinal magnetic field).
There are an infinite number of oscilllation
modes, but only a few are used for cavities
(calculation from Maxwells equations,
application for waveguides, for example
K.Wille)
8
Parameter of a cylindrical cavity („pill-box“)

E (t)
2a
z
A cylindrical cavity with the
length of g, the aperture
2*a and the field of E(t)
g
9
Acceleration in a cylindrical cavity

E (t)
2a
z
E(z)
E0
g
z
10
Cavity with rotational symmetry
The cavity parameter depend on the
geometry and the material:
• Geometry
• Material

E (t)
r0
=> Frequency
=> Quality factor
z
gc
Comes from Besselfunction
(Solution of wave equation)
11
Field strength for E010 mode for a „pillbox cavity“

E (t)
r0  0.231
r0
æ 2.40483  r ö
÷
r0
è
ø
Ez ( r)  J0 ç
z
æ 2.40483  r ö
÷
r0
è
ø
Hq ( r)  J1 ç
1
0.8
Ez ( r )
0.6
Hq ( r) 0.4
0.2
0
0
0.029 0.058 0.087
0.12
r
0.14
0.17
0.2
0.23
12
The energy gain of a charged particle is :
g /2
 E  q   E z ( z , t )  dz
g /2
The electrical
field as a function of time is :
E z (t)  E 0  cos (   t   )
with E 0 
U0
g
The particle has a (constant)
Therefore
: E 
e0 U 0
g
velocity v :
g /2
 z
g /2
v
  cos(
)  dz
Integratio n yields :
sin(
E  e0 U 0 
 g
2 v
 g
)
sin(
Definition : Transit time factor T tr 
 g
2 v
 g
2 v
2 v
Remember
:
T tr  1
)
Example for „Transit Time Factor“
14
Illustration for the electric field in the RF cavity
15
Superconducting RF cavity for Tesla and X-ray laser at
DESY
RF cavity with 9 cells
16
Normal-conducting RF cavity for LEP
17
Parameters for Cavities
Shunt impedance (Definition for a
circular accelerator) :
Q factor: 38000
g /2
 E z ( z )  dz
2
R sh 
g /2

Pc
For the DORIS Cavity :
U0
2
2  Pc
Q 0  38000
R sh  3 . 0  10
with Pc  Power loss in Cavity
6
Ohm
PHF  50 kW
U 0  548 kV
Quality factor Q :
Q Factor 
Q0 
W
Pc  1
Stored energy
Energy loss per cycle


 W
Pc
18