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```									Lecture 14 -
Professor Emmanuel Tsesmelis
Directorate Office, CERN
Department of Physics, University of Oxford

John Adams Institute for Accelerator Science
24 November 2011

   Group Velocity
   Dispersion Diagramme for Waveguide
   Resonant Cavities
 Rectangular and Cylindrical Cavities

   Quality Factor of Resonator
   Shunt Impedance and Energy Gain
   Transit-Time Factor
   Kilpatrick Limit
   Software for Cavity Design
Group Velocity

   Energy (and
information) travel
with wave group
velocity.
   Interference of two
continuous waves of
slightly different  E  E sin k  dk x    d  t   E sin k  dk x  (  d  )t 
0                      0

 E sin kx   t  cosdk x  d  t 
0
frequencies          2 E f x, t  f ( x, t )0 1   2

described by:
Group Velocity
   Mean wavenumber & frequency represented by continuous wave
f 1 ( x , t )  sin  k x   t 
   Any given phase in this wave is propagated such that kx – ωt remains
constant.
   Phase velocity of wave is thus
 f1  x , t   t 
vp                       
 f1 ( x , t )  x k
   Envelope of pattern described by
f 2 ( x, t )  cos[dkx  dt ]
   Any point in the envelope propagates such that x dt – t dω remains
constant and its velocity, i.e. group velocity, is
f 2 ( x, t ) t d
vg                        
 f 2 ( x , t )  x dk
Dispersion Diagramme for Waveguide

Description of wave propagation down a
waveguide by plotting graph of frequency, ω,
against wavenumber, k = 2π/λ
Imagine experiment in which signals of different
frequencies are injected down a waveguide and
the wavelength of the modes transmitted are
measured.
Measurables
Phase velocity for given frequency: ω/k
Group velocity: slope of tangent
Dispersion Diagramme for Waveguide

Observations
 However small the k,
the frequency is always
greater than the cut-off
frequency.
 The longer the
wavelength or lower
the frequency, the
slower is the group        Dispersion diagramme for waveguide
velocity.                  is the hyperbola
 At cut-off frequency, no
          c 
2             2
energy flows along the
   k  
2
waveguide.                     c       c 
 Also v p h v g  c
2
   Acceleration in a                •Need to modify waveguide to reduce
waveguide is not possible        phase velocity to match that of the particle
as the phase velocity of         (less than speed of light).
the wave exceeds that of
light.                           •Install iris-shaped screens with a constant
   Particles, which are         separation in the waveguide.
travelling slower, undergo
acceleration from the
passing wave for half the
period but then experience
an equal deceleration.
   Averaged over long time
interval results in no net
transfer of energy to the
particles.

   Recall that the
dispersion relation in
a waveguide is
2
 2   
  c kz  
2
     

 c    
   With the installation of
irises, curve flattens
off and crosses
boundary at vφ=c at         With suitable choice of iris separation
d the phase velocity can be set to
kz=π/2                     any value

Waveguides cannot
be used for sustained
acceleration as all
points on dispersion
curve lie above                       Dispersion diagramme
diagonal in dispersion
diagramme.                 The k-value for each space harmonic is
2n
 Phase velocity > c                       k k
n    0

d

An iris-loaded           By choosing any frequency in dispersion
diagramme it will intercept dispersion curve
structure slows down     at k values spaced by 2nπ/d
the phase velocity.
First rising slope used for acceleration.
Resonant Cavities

   General solution of wave equation
W (r , t )  Aei (t  k r )  Be i (t k r )
   Describes sum of two waves – one moving in one
direction and another in opposite direction
   If wave is totally reflected at surface then both
amplitudes are the same, A=B, and
W (r , t )  Aeit (eikr  e ikr )
 2 A cos( k  r )eit

   Describes field configuration which has a static
amplitude 2Acos(k·r), i.e. a standing wave.
Resonant Cavities
   Resonant Wavelengths
   Stable standing wave forms in fully-closed cavity if
z
lq        with q  0, 1, 2,...
2
   where l = distance between entrance and exit of waveguide after being
closed off by two perpendicular sheets.
    only certain well-defined wavelengths λr are present in the cavity.
   General resonant condition                1    11q
2

 2  
r c 4  l 
2

   Near the resonant wavelength, resonant cavity behaves
like electrical oscillator but with much higher Q-value and
corresponding lower losses of resonators made of
individual coils and capacitors.
   Exploited to generate high-accelerating voltages
Rectangular Resonant Cavities

   Inserting                   2
 2  m n
2          2

      
   a  b
 c
into the resonance condition yields
2
r                               with m, n, q  int egers
2           2   2
m n q
     
 a  b  l 

   Integers m,n,and q define modes in resonant
cavity.
   Number of modes is unlimited but only a few of them
used in practical situations.
   m,n,and q between 0 and 2
Cylindrical Resonant Cavities

   Inserting the expression for cut-off frequency
into general resonance condition yields
2          2
1    x  1q
 1             with q  0,1,2,...
r  D  4  l 
2

   where x1=2.0483 is the first zero of the Bessel
function.
   For the case of q=0, termed the TM010 mode,
the resonant wavelength reduces to
D
r 
x1
Bessel Functions
Pill-box Cylindrical Cavity
Cylindrical pill-box cavity with holes
 The simplest RF cavity type
for beam and coupler.
 The accelerating modes of this
cavity are TM0lm

 Indices refer to the polar
co-ordinates φ, r and z

TM010                 TM011

Lines of force for the
electrical field.
Pill-box Cylindrical Cavity

   The modes with no φ variation are:
 2 E  2 E  0
1   E  1 E  2 E
r    2   2  2 E  0
r r  r  r r z
P          m    
E z  E0 J 0  0 l
 r     r  cos
  h    z
 0                
m r0  P0l        m 
E r  E0        J1       r  sin 
  h z
P0l h  r0
               
2
P       m 
2

2
0 lm     0l
 r     
  h 
 0          

   l indicates the radial variation while m controls
the number of wavelengths in the z-direction.
   P0l is the argument of the Bessel function when
it crosses zero for the lth time.
   J0(P0l) = 0 for P0l = 2.405
Pill-box Cylindrical Cavity

   TM010 Mode

 2.405            2.405   010
 r r ;  010  r ; 010  
E  E0 J 0         
 0                   0

010               1
 010       ; 010 
2             010 
Quality Factor of Resonator, Q

   Ratio of stored energy to energy dissipated
per cycle divided by 2

Q  W s W      s

W   d    P   d

Ws = stored energy in cavity
Wd = energy dissipated per cycle divided by 2
Pd = power dissipated in cavity walls
 = frequency
Quality Factor of Resonator, Q

   Stored energy over cavity volume is

 E
2                                 2
         H
0
W 2
s
dv         W   s
2
0
dv

where the first integral applies to the time the
energy is stored in the E-field and the second
integral as it oscillates back into the H-field.
Quality Factor of Resonator, Q

 Losses on cavity walls are introduced by
taking into account the finite conductivity  of
the walls.
 Since, for a perfect conductor, the linear
density of the current j along walls of
structure is
j = n H
we can write
R2
              H
surf
P   d
2          s
ds   with s = inner surface of conductor
Quality Factor of Resonator, Q

Rsurf = surface resistance
δ = skin depth


f                 
1
R   surf
0
   
For Cu, Rsurf = 2.61  10-7  Ω
Shunt Impedance - Rs

Figure of merit for an accelerating cavity
Relates accelerating voltage to the power Pd to be
provided to balance the dissipation in the walls.
Voltage along path followed by beam in
electric field Ez is
V = path |Ez(x,y,z)| dl
from which (peak-to-peak)
2

 V
R s 2 Pd
Shunt Impedance - Rs
 z 1   2
2
 sin D / 2 
Rs  5.12 10 8
           
p  2.61 z 1     D / 2 
with

z         (phase velocity)
c
h
  (h  thickness, d  iris separation )
d
p  number of irises per wavele ngth (equal to mode number)
2
D      1   
p
Energy Gain

   Energy gain of particle as it travels a distance
through linac structure depends only on
potential difference crossed by particle:
U  K PRF lRs
where
PRF  supplied RF power
l  length of linac structure
Rs  shunt impedance
K  correction factor ( 0.8)
Analogous to Electrical Oscillator

   Cavity behaves as an
electrical oscillator but
with very high quality
factor (sharp resonance)


Electrical response of cavity

Q         r
 R    s
described by parallel circuit
containing C, L, and Rs
         Z                    On resonance the impedance is
r resonant frequency
1
Δ = frequency shift at which amplitude            Z  L 
is reduced by -3 dB relative to resonance peak              C
Transit-Time Factor
The RF Gap

 Accelerating gap
 Space between drift tubes
in linac structure
 Space between entrance
and exit orifices of cavity
resonator
 Field is varying as the
particle traverses the gap        E   z
   E   0
c o s(  t   )
 Makes cavity less efficient
and resultant energy gain     Field is uniform along gap axis
which is only a fraction of   and depends sinusoidally on time
the peak voltage
Phase  refers to particle in middle of
gap z=0 at t=0
Transit-Time Factor

   Transit-Time Factor is ratio of energy actually
given to a particle passing the cavity centre at
peak field to the energy that would be
received if the field were constant with time at
its peak value
   The energy gained over the gap G is:
G
sin(  G 2  c )
E 0 G cos  
2
V   E 0 cos( t   ) dz 
G                          G / 2c
2
Transit-Time Factor
The Transit Gap Factor is defined as

sin(  G 2  c )
Transit G ap Factor  
G / 2c
Defining a transit angle

T ra n sit A n g le     G  c  2  G  

the Transit Gap Factor becomes

sin  2
                   with 0 <  < 1
 2
The Transit-Time Factor

Observations
At relativistic energies, cavity dimensions are
comparable with /2
 Reduction in efficiency due to transit-time factor is
acceptable.
At low energies, this is not the case
Cavities have strange re-entrant configuration to
keep G short compared to dimensions of its
resonant volume.
The Transit-Time Factor
Field in resonant cavity

 Compromise cavity
design
‘Nose-cones’
 Increasing ratio of
volume/surface area
 Reduces ohmic losses
 Increases Q factor
 Minimise gap factor
Kilpatrick Limit

   RF breakdown observed at very high fields.
   Kilpatrick Limit expresses empirical relation
between accelerating frequency and E-field
   f = 1.64Ek2e − 8.5 / Ek
Software for Cavity Design

Structures usually solved by Finite Element Analysis

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