July 9

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July 9
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Can we get more insight into how the RFQ
works? We consider electric quadrupole cross
section and the potential function for pure
quadrupole geometry (no z dependence).

• U and are zero on axis no axial
  (z) field is present in this case.
• This gives transverse quadrupole fields but no
  acceleration.

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Check that the potential function gives the right
polarity on the vanes
Potential function
Horizontal vanes
Vertical vanes
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How to get an axial field? Lets see what
happens if we have an electric quadrupole with
unequal apertures?
Assume a trial solution for the potential
function.
There are two unknowns, X and A. What is
the solution for X and A in terms of the new
geometry?
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Find potential function that give the correct
polaritities on the vanes
Our trial solution
where X and A are unknowns.
We have two equations with two unknowns, X and A.
Add and subtract the two equations and we can
solve for the unknowns. The results are
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How to get an axial field for acceleration of
particles?

• Electric quadrupole cross section with unequal
  apertures has nonzero potential on axis.
• The on-axis potential always has the sign of the
  electrodes closest to the axis.

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Getting an axial accelerating field

• To produce the axial field suppose we
  sinusoidally modulate the vanetips along the
  axial direction.
• If this is done with x and y modulations that are
  180 degrees out of phase, the on-axis potential
  will follow the potential variations of the
  vanetips and a sinusoidal on-axis electric field
  is produced.

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RFQ Beam Dynamics Equations
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Physical interpretation of A A equals the
fraction of the intervane voltage across the
unit cell
On axis r0, and
Potential difference across the bl/2 cell length
is
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RFQ transit-time factor for the synchronous
particle
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Synchronous acceleration in the RFQ is really the
same as for conventional linac structures
Interpretation A is the fraction of the
intervane voltage V0 that is applied across a
cell of length bl/2.

• Note that E0 decreases as b increases.Thus the
  acceleration becomes less efficient
• with increasing b.

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RFQ transverse focusing

• RFQ electric quadrupole focusing is just another
  example of alternating gradient focusing such as
  what we have from quadrupole lenses.
• The beam particles see focusing with alternating
  polarity. As the beam sees it, it is just FD
  quadrupole focusing.
• As with magnetic lenses, you get net focusing
  because the beam has a larger displacement in the
  focusing lens than in the defocusing lens.
• What is different is that in the RFQ the focusing
  alternates in time rather than in space.

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Transverse Equation of motion
Averaging over a cell of length bl/2, assuming x
is constant over a cell, we get
quadrupole
RF defocusing (when f is negative)
This result has the form of the Mathieu equation,
also known as the Mathieu-Hill equation.
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m0 is the mass.
Substitute
Also known as Mathieu-Hill equation.
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Transverse equation of motion is the Mathieu-Hill
equation (now shown in dimensionless form)
w RF frequency
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(No Transcript)
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Take average after substituting the trial
solution into the equation of motion
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W/bc is phase advance per unit length, And bl is
length (distance synch particle travels) per
focusing period.
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Transverse Equation of Motion in Smooth
Approximation(Good approximation for s0lt90 deg)
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Advantages of RFQ focusing

• Use of electric rather than magnetic fields is
  superior for low velocity particles.
• Use of RF focusing fields rather than DC fields
  allows higher peak fields.
• The focusing alternates in time but is spatially
  uniform. When the fields are focusing they focus
  everywhere in the RFQ. When the fields are
  defocusing they focus everywhere in the RFQ.
• Spatially uniform quadrupole focusing in the RFQ
  increases the fraction of space used for focusing
  to 100.
• The short focusing period (bsl) keeps the phase
  advance per focusing period small which helps
  keep the beam away from the unstable limit at
  sp.

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Frequency Choice Issues and RFQ Structure Types
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RFQ Frequency Choice Issues

• Lower frequency allows larger aperture, higher
  beam current limit, lower power density, more
  relaxed vane alignment tolerances.
• Higher frequency gives higher peak surface field
  limits, lower charge per bunch for a given beam
  current, and less emittance growth.
• RFQ frequency is always constrained by
  availability of RF tubes.
• The frequency choice also determines what type of
  RFQ structure, 4-vane (above 200 MHz), or 4-rod
  (below 200 MHz). These are the most common RFQ
  structures.

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4-vane and 4-rod are the most common RFQ
structures
4-vane
4-rod
The choice of RFQ structure affects things
like power dissipation, cooling, and ease of
tuning.
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Four-vane RFQ
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Four-vane cavity overview

• Used in the high frequency range above about 200
  MHz.
• Most common structure for light ions especially
  protons.
• Built with two specially configured end cells to
  produce a longitudinally uniform fields
  throughout interior of cavity.
• Transverse electric field is localized near vane
  tips.
• Magnetic field is longitudinal localized in four
  outer quadrants.
• Efficiency is high because vane charging currents
  are uniform along the length of the vanes.

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Lumped circuit model of RFQ four-vane structure
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The separation of E and B field regions suggest a
description based on a simple lumped circuit
model.

• Each quadrant is analyzed as a resonant cavity
  with capacitance C and inductance L.
• The four gaps provide separate parallel paths
  between vanes of opposite potential. Total
  capacitance per unit length is

where the vane length is lv.

• We can derive some useful formulas in terms
  of Cl.

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Equivalent circuit for the quadrupole mode of a
four-vane cavity
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Determine inductance

• Assume B is uniform over outer part of quadrants.
  Write magnetic flux as

where I is total transverse current over vane
length lV, and A is effective cross sectional
area per quadrant for magnetic field.

• Inductance of each quadrant is ratio of flux to
  current .

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Formulas from lumped circuit model of four vane
resonator

• Effective cross sectional area per quadrant as
  function
• of radius of quadrant r

• Resonant frequency

• Assuming ejwt time dependence, the peak
  transverse current on outer wall and
  Bm0Hm0I/lv in quadrants are

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RF power Pl dissipated per unit length, stored
energy per unit length Wl ,and Q
where V is intervane voltage and s is conductivity
Wl Cl V2/2
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Estimating the value of capacitance per unit
length Cl
First approximation electrostatic calculation
for four rods of circular cross section whose
radius of curvature equals aperture radius gives
Cl90 pF/m
Better approximation electromagnetic code
SUPERFISH for a four-vane cavity is a weak
function of vane radius whose value is about
Cl120pF/m
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Summary of lumped circuit model of four-vane
resonator

• The model can be used to estimate properties of a
  four-vane cavity and show approximate dependence
  of cavity properties on the parameters.
• For accurate calculation of cavity properties for
  any specific geometry, an electromagnetic-field-
  solver code should be used.

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4-vane RFQ structure characteristics

• Vanetip charging currents are distributed
  uniformly along the vanes so the wall currents
  see less resistance and less power loss. The
  4-vane RFQ is an efficient structure.
• Accidental mode degeneracy can occur between the
  quadrupole mode we want and the dipole modes that
  we dont want.
• The need to suppress dipole modes may complicate
  4-Vane tuning.
• The 4-vane structure is generally chosen for RF
  frequencies above 200 MHz where this structure
  is more compact, efficient, and vanes are easier
  to cool.

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RF electric and magnetic fields in the central
region of the RFQ (away from the ends)
These are the E fieldsnear the beam axis From
the two-term potential function.
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gt Special end geometry termination makes the
central region of the RFQ look like a uniform
structure. The termination at the ends creates an
LC circuit with the same frequency as the central
region gt End region provides open-circuit
boundary conditions maintains continuity of the
magnetic flux as the flux turns around at the
ends.
But what happens on the ends? End magnetic field
lines split and go from a quadrant into the two
adjacent ones.
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But there will be distortion of the fields from
mode-mixing induced by perturbations and other
errors of fabrication

• Errors or perturbations induce mixing of fields,
  especially from other modes nearby in frequency.
  Fields from these modes can be added as
  corrections to the unperturbed quadrupole
  fields.
• The amplitudes of these corrections depend on
  size and location of local frequency errors.
  Perturbations include RF drive ports, pickup
  ports, vacuum ports, and other fabrication errors
  induce these corrections.
• Do not think of the perturbations as exciting
  other modes! Field corrections are induced by
  errors and perturbations (called mode mixing).

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Modes in 4-vane RFQ

• Azimuthal modes, particularly dipole modes.
  Unique problem for four-vane resonator.
• Higher longitudinal quadrupole modes.
  Longitudinal modes are a potential problem for
  any RFQ structure whose length is large compared
  with the wavelength, including the four-rod RFQ.
• Problem modes need to be displaced in frequency
  from the operating mode by use of appropriate
  tuners.

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RFQ dispersion curve example for a 4-vane
resonator showing the desired TE210 operating
mode at 425 MHz, other longitudinal TE21p modes
(p1, 2, ), and the family of TE11p dipole
modes.
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The are an infinite number of quadrupole modes,
each having a different guide wavelength. The
ideal quadrupole mode dispersion curve of the
four-vane cavity has the classic hyperbolic shape
that is characteristic of a uniform structure.
wn is the 2p times the frequency of the nth
mode w0 is the operating mode frequency , also
called the cutoff frequency. lv is the vane
length.
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Dipole modes are a special problem for the 4-vane
resonator

• If the dipole mode fields are added to the
  quadrupole mode fields, the beam will see
  electric deflecting forces.
• The lowest frequency dipole mode lies lower in
  frequency than the quadrupole mode. For certain
  vane lengths a dipole modes can have nearly the
  same frequency as the operating mode. (This is
  known as accidental degeneracy).
• Accidental degeneracy is a problem because the
  mixing of modes occurs more readily when modes
  are close in frequency.
• The problem is not that the modes are being
  excited by the generator but that errors are
  inducing mixing of mode fields resulting in
  distortion of fields for the operating mode.

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Dipole mode configurations
The operating mode will be mostly an admixture of
these three modes if they are all close in
frequency. The 4-vane RFQ must be designed and
tuned to minimize the contributions of the two
dipole modes.
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• Recall the TE mode nomenclature.
• Modes are labeled TEmnp where m, n, p
  corresponds to q, r, z.
• m number of full period variations in q of
  the field components.
• n number of zeros of axial field component Bz
  in the radial direction in range excluding
  r0.
• p number of half period variations in z of
  the field components. p0, 1, 2,
• The 4-vane operating mode TE210 has p0. Higher
  longitudinal
• modes correspond to p1, 2, 3,
• Dipole modes are TE11p, p0, 1, 2, .
• Whenever any of these modes lie close in
  frequency to the RFQ operating mode, we have to
  find ways of suppressing them.

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Several methods have been devised to suppress the
effects of unwanted modes.

• Vane coupling rings that electrically connect
  opposite vanes ensuring the same vane potentials.
  Shifts the dipole frequencies upwards eliminating
  their effect.
• Tuning rods that shift the dipole mode
  frequencies upwards. The simplest approach is
  rods attached to the end plates that extend into
  the midplane of each quadrant.
• Adjustable slug tuners in all four quadrants
  along the outer walls. These also allow us to
  adjust the longitudinal vane voltage profile and
  compensate for nearby longitudinal modes.