Lecture 10 Radiofrequency Cavities II

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Lecture 10 - Radiofrequency Cavities II

• Emmanuel Tsesmelis (CERN/Oxford)
• John Adams Institute for Accelerator Science
• 13 November 2009

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Table of Contents II

• Group Velocity
• Dispersion Diagramme for Waveguide
• Resonant Cavities
• Rectangular and Cylindrical Cavities
• Quality Factor of Resonator
• Shunt Impedance and Energy Gain
• Transit-Time Factor
• Iris-loaded Structures
• Synchronizing Particles with Cavities
• Accelerating Structures for Linacs
• Operation of Linac Structure

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Group Velocity

• Energy (and information) travel with wave group
  velocity.
• Interference of two continuous waves of slightly
  different frequencies described by

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Group Velocity

• Mean wavenumber frequency represented by
  continuous wave
• Any given phase in this wave is propagated such
  that kx ?t remains constant.
• Phase velocity of wave is thus
• Envelope of pattern described by
• Any point in the envelope propagates such that x
  dt t d? remains constant and its velocity, i.e.
  group velocity, is

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Dispersion Diagramme for Waveguide

• Description of wave propagation down a waveguide
  by plotting graph of frequency, ?, against
  wavenumber, k 2p/?
• 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 ?/k
• Group velocity slope of tangent

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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 velocity.
• At cut-off frequency, no energy flows along the
  waveguide.
• Also

Dispersion diagramme for waveguide is the
hyperbola
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Resonant Cavities

• General solution of wave equation
• 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, AB, and
• Describes field configuration which has a static
  amplitude 2Acos(kr), i.e. a standing wave.

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Resonant Cavities

• Resonant Wavelengths
• Stable standing wave forms in fully-closed cavity
  if
• where l distance between entrance and exit of
  waveguide after being closed off by two
  perpendicular sheets.
• ? only certain well-defined wavelengths ?r are
  present in the cavity.
• General resonant condition
• 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

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Rectangular Resonant Cavities

• Inserting
• into the resonance condition yields
• 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

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Cylindrical Resonant Cavities

• Inserting the expression for cut-off frequency
  into general resonance condition yields
• where x12.0483 is the first zero of the Bessel
  function.
• For the case of q0, termed the TM010 mode, the
  resonant wavelength reduces to

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Pill-box Cylindrical Cavity
Cylindrical pill-box cavity with holes for beam
and coupler.

• The simplest RF cavity type
• The accelerating modes of this
• cavity are TM0lm
• Indices refer to the polar
• co-ordinates f, r and z

TM010
TM011
Lines of force for the electrical field.
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Pill-box Cylindrical Cavity

• The modes with no f variation are
• 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

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Pill-box Cylindrical Cavity

• TM010 Mode

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Quality Factor of Resonator, Q

• Ratio of stored energy to energy dissipated per
  cycle divided by 2?
• Ws stored energy in cavity
• Wd energy dissipated per cycle divided by 2?
• Pd power dissipated in cavity walls
• ? frequency

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Quality Factor of Resonator, Q

• Stored energy over cavity volume is
• 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.

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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

with s inner surface of conductor
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Quality Factor of Resonator, Q

• Rsurf surface resistance
• d skin depth

For Cu, Rsurf 2.61 ? 10-7 O
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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)

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Shunt Impedance - Rs
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Energy Gain

• Energy gain of particle as it travels a distance
  through linac structure depends only on potential
  difference crossed by particle

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Analogous to Electrical Oscillator

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

Electrical response of cavity described by
parallel circuit containing C, L, and Rs
On resonance the impedance is
?r resonant frequency ?? frequency shift at
which amplitude is reduced by -3 dB relative to
resonance peak
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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
• Makes cavity less efficient and resultant energy
  gain which is only a fraction of the peak voltage

Field is uniform along gap axis and depends
sinusoidally on time Phase ? refers to particle
in middle of gap z0 at t0
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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

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Transit-Time Factor
The Transit Gap Factor is defined as
Defining a transit angle
the Transit Gap Factor becomes
with 0 lt ? lt 1
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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.

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The Transit-Time Factor
Field in resonant cavity

• Compromise cavity design
• Increasing ratio of volume/surface area
• Reduces ohmic losses
• Increases Q factor
• Minimise gap factor

Nose-cones
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Iris-loaded Structures

• Accelerating systems for particles travelling
  close to c consist of series of cavities in
  single assembly.
• Structures consist of
• Sequence of pill-boxes.
• Cylindrical waveguide, loaded with number of
  equidistant irises.

• Power from amplifier is coupled into cavity at
  one end and is either absorbed in load at the
  other end or reflected to set up a standing wave.

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Iris-loaded Structures

• Waveguides cannot be used for sustained
  acceleration as all points on dispersion curve
  lie above diagonal in dispersion diagramme.
• Phase velocity gt c
• An iris-loaded structure slows down the phase
  velocity.

Dispersion diagramme for a loaded waveguide
The k-value for each space harmonic is
By choosing any frequency in dispersion diagramme
it will intercept dispersion curve at k values
spaced by 2np/d First rising slope used for
acceleration.
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Synchronising Particles with Cavities

• If accelerator has more than single cavity,
  particles should be bunched to arrive at the same
  phase with respect to the voltage at each cavity.
• Space cavities by distance L that a particle
  travels in one RF period

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Synchronising Particles with Cavities

• Alvarez Structure
• Increasing L between accelerating gaps along
  structure.
• Snapshot of fields across each gap shows them all
  exactly in phase.
• Particles phase advance between cells is 2p
• Wideröe Structure
• Alternate drift tubes grounded.
• Snapshot shows vector alternating in sign from
  gap to gap.
• In these cases, cells oscillate either in phase
  or in antiphase.
• Difficult for power to propagate along the
  waveguide and small errors produce serious
  distortions.

Alvarez Cavity
Wideröe Cavity
Adjacent single-gap cavities in (a) p mode and b)
2p mode
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Multicell Cavities

• Travelling waves can be used instead of standing
  waves.
• Wave travels along a long chain of cavities to be
  absorbed in a load at the other end.
• Structures which repeat every one, two, three or
  four cavities correspond to phase changes of 0,
  p, 2p/3 or p/2 per cell, respectively.

Modes of a multicell cavity
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Accelerating Structures for LINACS

• Acceleration in a waveguide is not possible as
  the phase velocity of the wave exceeds that of
  light.
• Particles, which are travelling more slowly,
  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.

• Need to modify waveguide to reduce
• phase velocity to match that of the particle
• (less than speed of light).
• Install iris-shaped screens with a constant
• separation in the waveguide.

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Accelerating Structures for LINACS

• Recall that the dispersion relation in a
  waveguide is
• With the installation of irises, curve flattens
  off and crosses boundary at vfc at
• kzp/2

With suitable choice of iris separation d the
phase velocity can be set to any value
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Operation of LINAC Structure

• Standard operation of linac structure is in the
  S-band.
• ?0.100m (fRF3 GHz)
• As in radar technology, RF power supplied by
  pulsed power tubes klystrons.
• Power fed into linac structure by TE10 wave in
  rectangular waveguide which is connected
  perpendicular to cylindrical TM01 cavity.

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Operation of LINAC Structure
The two modes of operation of a linac
structure.
Travelling wave mode, in which an absorber is
installed at the end of the structure to prevent
reflections, is more commonly used. In a
standing wave mode, the energy is reflected
virtually without loss.
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Operation of LINAC Structure

• Irises form a periodic structure within cavity,
  reflecting the wave as it passes through and
  causing interference.
• Loss-free propagation only if wavelength is
  integer multiple of iris separation d

• Irises only allow certain
• wavelengths, characterised
• by number p, to travel in
• longitudinal direction.
• These fixed wave configurations
• are termed modes.
• In principle there are arbitrary such
• modes but only three used for
• acceleration.

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Operation of LINAC Structure

• p-mode
• Takes long time for transient oscillations to die
  away and a stationary state to be used.
• Not suitable for fast-pulsed operation.
• p/2-mode
• Low shunt impedance so for fixed RF power energy
  gain per structure is small.
• 2p/3-mode
• Best compromise between p-mode p/2-mode

Field configurations of three most important
modes in linac structures.