Introductory Comments by eL3VEnx

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									Lecture 14 -
Radiofrequency Cavities II
        Professor Emmanuel Tsesmelis
           Directorate Office, CERN
   Department of Physics, University of Oxford


      Accelerator Physics Graduate Course
   John Adams Institute for Accelerator Science
               24 November 2011
Table of Contents II

   Group Velocity
   Dispersion Diagramme for Waveguide
   Iris-loaded Structures
   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
Iris-loaded Structures
   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.
Iris-loaded Structures

   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
Iris-loaded Structures

Waveguides cannot
 be used for sustained
 acceleration as all
 points on dispersion
 curve lie above                       Dispersion diagramme
                                       for a loaded waveguide
 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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