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Lecture 1 Introduction to RF
for Accelerators
Dr G Burt
Lancaster University
Engineering
Electrostatic Acceleration
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Van-de Graaff - 1930s
A standard
electrostatic
accelerator is
a Van de
Graaf
These devices are limited to
about 30 MV by the voltage hold
off across ceramic insulators
used to generate the high
voltages (dielectric breakdown).
RF Acceleration
By switching the charge on the plates in phase with the particle
motion we can cause the particles to always see an
acceleration
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You only need to hold off the voltage between two plates not the full
accelerating voltage of the accelerator.
We cannot use smooth wall waveguide to contain rf in order to accelerate a
beam as the phase velocity is faster than the speed of light, hence we
cannot keep a bunch in phase with the wave.
Early Linear Accelerators (Drift
Tube)
• Proposed by Ising (1925)
• First built by Wideröe (1928)
• Alvarez version (1955)
Replace static fields by time-varying fields by only exposing the
bunch to the wave at certain selected points. Long drift tubes shield
the electric field for at least half the RF cycle. The gaps increase
length with distance.
Cavity Linacs
• These devices store large amounts of
energy at a specific frequency allowing low
power sources to reach high fields.
Cavity Quality Factor
• An important definition is the cavity Q factor, given
by
U
Q0 
Pc
Where U is the stored energy given by,
1
2
U  0  H dV
2
The Q factor is 2p times the number of rf cycles it
takes to dissipate the energy stored in the cavity.
 t 
U  U 0 exp   
 Q0 
• The Q factor determines the maximum energy the
cavity can fill to with a given input power.
Cavities
• If we place metal walls at
each end of the waveguide
we create a cavity.
• The waves are reflected at
both walls creating a standing
wave.
• If we superimpose a number
of plane waves by reflection
inside a cavities surface we
can get cancellation of E|| and
BT at the cavity walls.
• The boundary conditions
must also be met on these
walls. These are met at
discrete frequencies only
when there is an integer
number of half wavelengths in
all directions.
a
L
The resonant frequency of a rectangular
cavity can be given by
(/c)2=(mp/a)2+ (np/b)2+ (pp/L)2
Where a, b and L are the width, height
and length of the cavity and m, n and p
are integers
Pillbox Cavities
Wave equation in cylindrical co-ordinates
1     1 
2
2
r




k
z   0
 r r  r  r 2  2




Solution to the wave equation
  A1 J m (k t r )e  im
• Transverse Electric (TE) modes
  ' m,n r  im
e
H z r ,    A1 J m 
 a 
ik z a 2
H t  2 t H z
 'm , n
ia 2
E t   2  zˆ   t H z 
 ' m,n
• Transverse Magnetic (TM) modes
  m,n r  im
e
E z r ,    A1 J m 
 a 
Et 
ik z a 2

2
m, n
t Ez
Ht 
ia 2

2
m,n
zˆ   t E z 
Bessel Function
Jm(kTr)
1.0
m=0
0.8
m=1
m=2
0.5
m=3
0.3
0.0
kTr
0
2
-0.3
4
6
8
10
• Ez (TM) and Hz (TE)
vary as Bessel
functions in pill box
cavities.
• All functions have
zero at the centre
except the 0th order
Bessel functions.
-0.5
First four Bessel functions.
One of the transverse fields varies with the differential of the
Bessel function J’
All J’ are zero in the centre except the 1st order Bessel
functions
Cavity Modes
rθ
TE2,1
TE1,1
TE0,1
TM0,1
TEr,θ
Cylindrical (or pillbox) cavities are more common than rectangular cavities.
The indices here are
m = number of full wave variations around theta
n = number of half wave variations along the diameter
P = number of half wave variations along the length
The frequencies of these cavities are given by f = c/(2p * (z/r)
Where z is the nth root of the mth bessel function for TE modes or the nth root of
the derivative of the mth bessel function for TE modes or
TM010 Accelerating mode
Electric Fields
Almost every RF cavity operates
using the TM010 accelerating mode.
This mode has a longitudinal electric
field in the centre of the cavity which
accelerates the electrons.
The magnetic field loops around this
and caused ohmic heating.
Magnetic Fields
TM010 Monopole Mode
H
 2.405r   it
Ez  E0 J 0 
e
 R 
Hz  0
Hr  0
i
 2.405r   it
H 
E0 J1 
e
Z0
 R 
E  0
E
Beam
Er  0
Z0=377 Ohms
A standing wave cavity
Accelerating Voltage
Ez, at t=0
Ez, at t=z/v
Normally voltage is the
potential difference between
two points but an electron can
never “see” this voltage as it
has a finite velocity (ie the field
varies in the time it takes the
electron to cross the cavity
Position, z
The voltage now depends
on what phase the electron
enters the cavity at.
Position, z
If we calculate the voltage at
two phases 90 degrees
apart we get real and
imaginary components
Accelerating voltage
• An electron travelling close to the speed of light traverses through a
cavity. During its transit it sees a time varying electric field. If we use
the voltage as complex, the maximum possible energy gain is given
by the magnitude,
L/ 2
E  eVb  e

Ez  z, t  ei z / c dz
L/ 2
•
To receive the maximum kick with multiple cells the particle should
traverse the cavity in a half RF period (see end of lecture).
c
L
2f
Transit time factor
•
An electron travelling close to the
speed of light traverses through a
cavity. During its transit it sees a time
varying electric field. If we use the
voltage as complex, the maximum
possible energy gain is given by the
magnitude,
L/ 2
E  eV  e

Ez  z, t  ei z / c dz  E0 LT
L/ 2
Where T is the transit time factor given
by
L/2
T

Ez  z , t  e
L/ 2
L/2

i z / c
dz

Ez  z , t  dz

sin p g
pg
 

L/ 2
•
•
For a gap length, g.
For a given Voltage (=E0L) it is clear
that we get maximum energy gain for a
small gap.
1.2
1
Transit time factor, T
•
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
-0.2
-0.4
g/
2
2.5
Overvoltage
• To provide a stable bunch you often will accelerate off
crest. This means the particles do not experience the
maximum beam energy.
• Vb=Vc cos(fs) = Vc q
• Where Vc is the cavity voltage and Vb is the voltage
experienced by the particle, f is the phase shift and q is
known as the overvoltage.
V
Vp
fs
Stable
region
f
Phase stability is given by
off-crest acceleration
For TM010 mode
Ez, at t=z/v
 L / 2

i z / c
V     Ez  z , t  e dz 
 L / 2

L/2
 E0

cos  z / c  dz
L/ 2
Position, z
L/2
 sin  z / c  
  E0


/
c

L/ 2
2sin  L / 2c 
 E0
/c
Hence voltage is maximised when L=c/2f
This is often approximated as
Where L=c/2f, T=2/p
V  Ez 0 LT cos  
V  E0 cos  
2
p
L
Peak Surface Fields
• The accelerating gradient is the average gradient seen by an
electron bunch,
Eacc 
Vc
L
• The limit to the energy in the cavity is often given by the peak
surface electric and magnetic fields. Thus, it is useful to
introduce the ratio between the peak surface electric field and
the accelerating gradient, and the ratio between the peak
surface magnetic field and the accelerating gradient.
Emax p

Eacc 2
For a pillbox
H max
A/ m
 2430
Eacc
MV / m
Electric Field Magnitude
Surface Resistance
As we have seen when a time
varying magnetic field impinges
on a conducting surface current
flows in the conductor to shield
the fields inside the conductor.
However if the conductivity is
finite the fields will not be
completely shielded at the
surface and the field will .
penetrate into the surface.
This causes currents to flow
and hence power is absorbed
in the surface which is
converted to heat.
Skin depth is the distance in
the surface that the current has
reduced to 1/e of the value at
the surface, denoted by
Current
Density, J.
x

2
r
The surface resistance is defined as
Rsurf 
1

For copper 1/ = 1.7 x 10-8 Wm
Power Dissipation
• The power lost in the cavity walls due to ohmic heating is given by,
Pc 
1
2
Rsurface  H dS
2
Rsurface is the surface resistance
• This is important as all power lost in the cavity must be replaced by
an rf source.
• A significant amount of power is dissipated in cavity walls and hence
the cavities are heated, this must be water cooled in warm cavities
and cooled by liquid helium in superconducting cavities.
Capacitor
The electric field of the
TM010 mode is contained
between two metal
plates
E-Field
–
This is identical to a capacitor.
This means the end plates
accumulate charge and a
current will flow around the
edges
Surface
Current
Inductor
B-Field
Surface
Current
–
The surface current travels
round the outside of the cavity
giving rise to a magnetic field
and the cavity has some
inductance.
Resistor
Surface
Current
This can be accounted for by
placing a resistor in the circuit.
In this model we assume the
voltage across the resistor is the
cavity voltage. Hence R takes the
value of the cavity shunt
impedance (not Rsurface).
Finally, if the cavity has
a finite conductivity, the
surface current will flow
in the skin depth
causing ohmic heating
and hence power loss.
Equivalent circuits

To increase the
frequency
the
inductance
and
capacitance has to
be increased.
1
LC
2
Vc
Pc 
2R
CVc
U
2
2
The stored energy is just the stored energy in the capacitor.
The voltage given by the equivalent circuit does not contain the transit
time factor, T. So remember
Vc=V0 T
Shunt Impedance
• Another useful definition is the shunt impedance,
2
1 Vc
Rs 
2 Pc
• This quantity is useful for equivalent circuits as it
relates the voltage in the circuit (cavity) to the
power dissipated in the resistor (cavity walls).
• Shunt Impedance is also important as it is
related to the power induced in the mode by the
beam (important for unwanted cavity modes)
TM010 Shunt Impedance
Vc 
2 E0 L
Pc 
p
i
 2.405r 
H 
E0 J1 

Z0
R


1
2
Rsurface  H dS
2
Pc ,ends
Pc , walls
2
E0
 2.405r 
 2 Rsurface  2p r J1 
 dr
Z0
 R 
2
2
E0 2
 p RL 2 Rsurface J1  2.405 
Z0
2
E0 2
Pc  p R  R  L  2 Rsurface J1  2.405 
Z0
2Z 0 L 

Rs 
2
3
p R  R  L  Rsurface J1  2.405
2
5 x104

Rsurface
Geometric shunt impedance,
R/Q
• If we divide the shunt impedance by the
Q factor we obtain,
2
R Vc

Q 2U
• This is very useful as it relates the
accelerating voltage to the stored
energy.
• Also like the geometry constant this
parameter is independent of frequency
and cavity material.
TM010 R/Q
V
2 E0 L
p
i
 2.405r 
H 
E0 J1 

Z0
R


1
2
U  0  H dV
2
2
E0
 2.405r 
U  2 L  0  p r J1 
 dr
Z0
 R 
2
U
p 0 E0 2
2
R L J1  2.405 
2
8Z0 2
R
L
L

 150  196Ohms
2
Q p 0c  2.405 J1  2.405 R
R
2
Geometry Constant
• It is also useful to use the geometry constant
G  RsurfaceQ0
• This allows different cavities to be compared
independent of size (frequency) or material, as it
depends only on the cavity shape.
• The Q factor is frequency dependant as Rs is
frequency dependant.
Q factor Pillbox
2
E0 2
Pc  p R  R  L  2 Rsurface J1  2.405
Z0
U
p 0 E0 2
2
Q0 
R L J1  2.405
2
0 RL
2  R  L  Rsurface
453L / R
G
 260
1  L / R 
2
453L / R

Rsurface 1  L / R 
Equivalent circuits
These simple circuit equations
can now be used to calculate
the cavity parameters such as Q
and R/Q.
U
C
Q0 

R
Pc
L
R
V2
1
L



Q0 2U C
C
In fact equivalent circuits have been proven to accurately
model couplers, cavity coupling, microphonics, beam loading
and field amplitudes in multicell cavities.
Cavity geometry
• The shunt impedance is
strongly dependant on
aperture
Similarly larger
apertures lead to higher
peak fields.
Using thicker walls has
a similar effect.
Higher frequencies
needborrowed
smaller
apertures
Figures
from
Sami
Tantawi
as well
Frequency Scaling
• Rsurf ~ f0.5 normal conducting
• Rsurf ~ f2 superconducting
• Qo ~ f-0.5 normal conducting
• Qo ~ f-2 superconducting
• Rs ~ f-0.5 normal conducting
• Rs ~ f-2 superconducting
• R/Q ~ f0 normal conducting
• R/Q ~ f0 superconducting
Multicell
• It takes x4 power to double the voltage in one cavity but only x2 to
use two cavities/cells to achieve the same voltage (Rs ~number of
cells).
• To make it more efficient we can add either more cavities or more
cells. This unfortunately makes it worse for wakefields (see later
lectures) and you get less gradient per unit power.
• In order to make our accelerator more compact and cheaper we can
add more cells. We have lots of cavities coupled together so that we
only need one coupler. For N cells the shunt impedance is given by
Rtotal  NRsin gle
This however adds
complexity in tuning,
wakefields and the gradient of
all cells is limited by the worst
cell.
Synchronous particle
• Imagine we have a series of gaps. The phase change
between two gaps when the beam arrives is given by
fn  fn 1  
ln 1
 a
 n 1c
• Where a is the phase advance, (the phase difference
between adjacent coupled cavities)
• Hence the distance between cells should be
 a  n 1c
d

• In a linac we choose a synchronous phase fs and design the
lengths so that the synchronous particle sees the desired
phase (not always constant)
• For a standing wave structure the synchronous phase occurs
when the cavity is half a free space wavelength long.