Radio Frequency Quadrupole (RFQ)

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Radio Frequency Quadrupole(RFQ)

• S. A. Pande

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Introduction

• The first linac was built in 1928 by Widröe

1MHz 25 kV

50 keV K ions
K ions
d??/2
3
The Sloan Lawrence Structure

• E. O. Lawrence in association with Sloan built an
  improved version of Widröes linac
• They used an array of 30 DTs excited by a 42 kV,
  7 MHz oscillator to accelerate Hg ions to 1.26
  MeV.
• RFQ is also a Sloan-Lawrence kind of accelerator
  in which the successive accelerating gaps are
  ??/2 apart.

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The RFQ

• It was first proposed by I. Kapchinskii and V.
  Teplyakov from ITEP Moscow for heavy ions.
• The first RFQ was built and tested at LANL to get
  2 MeV protons.
• Though invented in the last, the RFQ forms the
  first accelerator in a chain of heavy ion
  (including proton) accelerators in recent times.

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• Before 80s almost all of the accelerator
  facilities for protons and heavy ions, invariably
  used DC accelerators from few 100 keVs to few
  MeVs as injectors for linear accelerators which
  in turn formed the main injectors for the bigger
  circular machines or acted as sources of charged
  particle beams.
• The DC accelerators have certain inherent
  limitations and difficulties associated with
  handling of high voltages.
• The beam has to be bunched before injecting into
  the linac in order to avoid energy spread in the
  out coming beam and also to avoid the loss of
  particles.

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• There was another severe problem associated with
  the focusing of the beams. The defocusing due to
  space charge is more severe in the low energy
  beams.
• The invention of RFQ, the low energy high current
  accelerator, helped in overcoming all the
  difficulties we have seen above.
• The RFQ simultaneously
• Focuses
• Bunches and
• Accelerates the beam
• This avoided the need for large DC accelerators
  and avoided the problems to great extent.
• Almost all of the DC accelerators were later
  replaced by RFQ after its invention.

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Principle of Operation

• As its name suggests, the RFQ provides electric
  quadrupole focusing with the electric field
  oscillating at Radio Frequency

Four equispaced conducting electrodes with
alternating polarity as we move from one
electrode to the next forms the electric
quadrupole. Voltage ?1/2V0cos(?t) is applied in
quadru-polar symmetry
-1/2Vcos(?t)
1/2Vcos(?t)
1/2Vcos(?t)
-1/2Vcos(?t)
The electric Quadrupole
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• The off axis particles will experience a
  transverse force which is alternating in time and
  this transverse force provides Alternating
  Gradient focusing.
• The advantage of RFQ is that it provides electric
  focusing for low velocity particles which is
  stronger than conventional magnetic focusing.
• A structure with uniform electrodes along its
  length will have no component of electric field
  along the axis and thus will not work as an
  accelerator.
• To generate an axial electric field component,
  the quadrupole electrodes are modulated
  longitudinally. One pair of electrodes is shifted
  longitudinally wrt the other pair by 180? so that
  when the distance from the axis of vertical vanes
  is at its minimum a, the horizontal vanes will
  be maximum apart at ma.

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Modulation
A
??/2 One unit cell
x
ma
a
a
Beam axis
ma
z
Cross section through AA
m ? 1
A

• Modulation of electrodes to generate longitudinal
  field component

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• The axial electric field component is generated
  due to the potential difference between the point
  of minimum separation from axis of vertical vanes
  (or horizontal vanes) and the point of minimum
  separation from the axis of the horizontal vane
  (or vertical vane).
• In RFQ, the field in successive gaps is in
  opposite direction and therefore when it is
  accelerating in one cell, it is decelerating in
  the next.
• There are two unit cells per structure period. At
  a given time every alternate cell will have a
  particle bunch.

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The general potential function

• In RFQ the electrodes in the form of rods or
  vanes are placed in cavity resonators to
  prevent the RF fields from radiating.
• The issues related to the electrodynamics are
  distinct from those associated with the beam
  dynamics. The beam dynamics is confined to a
  region of small radius near axis as compared to
  the cavity radius which is proportional to the
  wavelength.
• Due to the symmetry property the magnetic field
  is zero on the axis and also for the region rltlt?.

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• The consequences are-
• The wave equation in this region can be replaced
  by Laplace equation
• The vanes present well defined boundaries with a
  potential from which we can analytically derive
  the fields or
• We can ask for specific fields and then determine
  the corresponding vane boundaries.
• Starting with the Laplace equation in cylin.
  Coordinates
• Where U(r,?,z) electric field potential.

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• Solving the above equation by the method of
  separation of variables, we obtain
• This is the general K-T potential function a
  doubly infinite terms.
• K-T considered only the lowest order terms and
  proposed to construct the electrode shapes that
  conform to the resulting equipotential surface.
• Retaining only s0 from the first and s0, n1
  terms from the second summation, we have

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The two term potential

• Retaining only s0 from the first and s0, n1
  terms from the second summation, we have
• where k2?/?? ?velocity of synchronous
  particle
• and I is the modified Bessel function.
• The potential given by this equation is known
  as Two Term Potential and the dynamics in the
  RFQ is studied with this potential function.

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• By assuming the horizontal and vertical vanes at
  V0/2 and V0/2 respectively and putting the
  boundary conditions at the vane tips, we have
• We define two dimensionless quantities

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• With these two dimensionless quantities,
  A0XV0/2a2 and A10AV0/2, the two term time
  dependent potential is written as
• ---------------- ---------------
• I II
• The first term gives the potential of an
  electric quadrupole and the second term gives the
  accelerating potential.
• The quantities X and A are known as focusing
  parameter and acceleration parameter
  respectively.
• From the defining equations of X and A we can
  write
• X 1 AI0(ka)

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• By rearranging the last equation, we can write
• XV AI0(ka)V V
• This tells us that the inter-vane voltage V is
  composed of a part required for focusing (XV) and
  another required for acceleration (AI0(ka))
• Similarly, if we put m1 in the last equation,
  the vanes are unmodulated and the acceleration
  parameter goes to zero.
• A 0 for m 1
• The RFQ will be just a focusing device.
• As m increases the acceleration parameter
  increases and the focusing parameter X decreases.

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The Field Components

• The field components are derived from the
  potential function
• Er - ?U/?r -V0/22(X/a2)rcos2?-kAI1(kr)coskz
  
• E? -(1/r) ?U/??(XV/a2)rsin2?
• Ez - ?U/?z(kAV/2)I0(kr)sinkz
• I1 is the modified Bessel function of first
  order
• The first term in Er and E? is the quadrupole
  focusing field
• The second term of Er is the gap defocusing term
  which applies a radial defocusing impulse
• Since I1(kr)?kr/2, the radial impulse is
  proportional to the displacement from the axis.

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Voltage and energy gain across a unit cell

• The voltage across a unit cell can be calculated
  by
• where we have used Ez as defined earlier and
  Lc??/2
• The energy gain is given by
• ?WqeAVTcos?s
• For RFQ the transit time factor is
• T?/4

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The Vane tip profiles

• With time dependent voltages on horizontal
  vertical electrodes as V/2sin(?t?) and
  V/2sin(?t?) and expressing the two term
  potential in cartesian coordinates by
  substituting xrcos? and yrsin?, we have
• U(x,y,z,t)V0/2X/a2(x2-y2)AI0(kr)coskz
• with UV/2, we have for the geometry of the vane
  surface
• 1X/a2(x2-y2)AI0(kr)coskz
• Or x2-y2a2/X(1-AI0(kr)coskz)
• ?The transverse cross sections are hyperbolas

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The ideal vane tip profile
The hyperbolic vane tip profiles
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• But for the ease of machining, and also to
  control the peak surface electric field, the
  electrode contours deviate from the ideal
  hyperbolas.
• A combination of circular arcs and straight
  lines is used
• At the cell centre, i.e. at z??/4
• The RFQ has exact quadrupolar symmetry
• The x and y tips of the electrode are
  equidistant from the axis (or have radius r0)
  given by .
• r02a2/X
• r0aX-1/2
• This is known as the average radius of the RFQ.
• The focusing strength of a modulated structure
  is equivalent to that of an unmodulated structure
  with radius r0.

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The actual vane tip profiles
The vertical vane
One quadrant of RFQ
The horizontal vane
?
r0
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Characteristics of RFQ

• Adiabatic Capture and Bunching
• Ion source provides a DC beam and thus is
  injected uniformly from -? to ? over one period.
• ?W0 and ??360?
• The RFQ can capture almost all the beam injected
  and bunch it slowly.
• In the initial part of RFQ there is no
  acceleration.The longitudinal electric field
  which is proportional to AV, is slowly increased
  by increasing m the modulation parameter. This
  provides bunching.

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Characteristics of RFQ (Contd.)

• Many cells are devoted to this part in an RFQ.
  This will not be economical in other linac
  structures. In RFQ, the cells are very short and
  many cells can be accommodated in a relatively
  shorter length. Thus RFQ provides adiabatic
  capture and bunching.
• The synchronous phase is kept initially at -90?
  where we have maximum longitudinal focusing and
  no acceleration (i.e. the synchronous particle
  will have no acceleration).
• Once some rough bunching is achieved, the
  synchronous phase (?s) and m are slowly increased
  further to impart energy and the bunch slowly
  becomes well defined.

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The complete RFQ

• The first RFQ was built at LANL. They divided
  the whole RFQ in 4 parts.
• 1. Radial Matching Section (RMS)
• 2. Shaper (Sh)
• 3. Gentle Buncher (GB) and
• 4. Accelerator (Acc)
• 1. Radial Matching Section (RMS)
• Matches the input DC beam to the strong
  transverse focusing structure of the RF
  quadrupole. In this section m1, no Ez no
  acceleration, few cells 5.

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• 2. Shaper (Sh)
• This is a short section which starts the
  bunching process. This section smoothly joins the
  RMS where A0 and ?s-90? to the gentle buncher
  where Agt0 and ?sgt-90?. This initiates the
  bunching process.
• 3. Gentle Buncher (GB)
• The GB adiabatically bunches the beam and also
  slowly accelerates to some intermediate energy.
  Being adiabatic, it forms the major part of the
  RFQ structure. ?s and m are increased ultimately
  to match those in the accelerator part.
• 4. Accelerator (Acc)
• In this part the major emphasis is on the
  acceleration at a faster rate. ?s and m reach
  their ultimate values. ?s -30? and m 1.5
  2.5.

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RMS
SHAPER
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Accelerator
Longitudinal profile of the vane tip in 4.5 MeV
50 mA RFQ
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The RFQ Cavity or Resonator

• Whatever we discussed was the story in the
  vicinity of the axis where the beam passes
  through.
• Let us see how we can generate these fields
  electro-magnetically.
• Two types of structures are most commonly used
• 1. The four rod structure and
• 2. The four vane structure
• 3. Split Co-axial cavity is used at few places
  for heavy ion acceleration.
• We will study the first two.

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The four vane structure
TE21 mode in circular cylindrical waveguide
We introduce the vanes
The quadrupole field concentrates near the vane
tips
Vanes divide the waveguide in 4 quadrants
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• On Quadrant of the RFQ showing electric field
  lines of quadrupole mode

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• The vanes concentrate the electric field near the
  axis providing strong quadrupole focusing field
• Magnetic field which is longitudinal is localized
  in outer part of the quadrant
• The vane to vane capacitance reduces the cutoff
  frequency of the waveguide or the resonant
  frequency of the cavity. To compensate this the
  waveguide diameter can be reduced
• The four vane cavity is obtained by shorting the
  two ends by conducting plates
• The boundary condition on each conducting end
  plate is Etangential0
• This shows a true TE210 mode cannot exist in
  cylindrical cavity with metallic end walls.
  Instead the mode will be TE211

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• TE211 and TE210 Modes

Resulting Field due to TE211 mode the last
subscript denotes the no. of half wavelength
variations in z direction
Desired Field due to TE210 mode zero last
subscript denotes that there is no variation in
the longitudinal direction.
E
Z
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• Therefore gaps are provided between the end wall
  and the vane ends
• This produces longitudinally uniform field
  throughout the interior of the cavity
• Etransverse is localized near the vane tips
• Hlongitudinal is localized in outer part of the
  quadrants

VANE
vane
Side view
Cross section through RFQ at an end
Top view
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Eigen modes of a 4 vane cavity

• There is one more important mode in the 4 vane
  cavity slightly below in frequency of the quad
  mode.
• This is the dipole mode denoted by TE11n
• The field pattern for quad and dipole modes are
  shown below

x
x
x
x
Quadrupole
Dipole-1
Dipole-2
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• Dipole modes are degenerate modes
• When a dipole mode is excited, a small potential
  difference appears across the the opposite vanes
  where as for the quad mode the opposite modes are
  exactly at the same potential.
• If these modes are close to the quadrupole mode,
  the transverse as well as longitudinal field will
  be perturbed and the performance will be
  affected. Therefore the dipole modes should be
  tuned away from the quadrupole mode.
• It may happen that the frequency of a higher
  order dipole mode may fall very close to the quad
  mode.

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The longitudinal mode spectrum of a 4 vane RFQ
cavity
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The field stabilization

• The perturbation caused due to the dipole or othe
  modes can result in unflat field distribution
  along the RFQ structure.
• Due to highly sensitive nature of the RFQ cavity,
  the machining and tuning errors can also result
  in dipole mode excitation.
• Many proposals have been made at many places.
  Most successful are the Vane Coupling Rings
  introduced at LBNL and Pi mode Stabilizing Loops
  (PISL) proposed at KEK.

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Vane Coupling Rings (VCR)

• The opposite vanes are shorted together forcing
  them to the same potential.
• The dipole modes are shifted away.
• 3 pairs of VCRs are used in structures of 1-2 m
  in length
• Difficult to mount and cooling is a problem

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PISL
Dipole mode
Quad mode

• Principle
• The total magnetic flux normal to the surface
  surrounded by a closed conducting loop is zero.
• The dipole mode fields will be perturbed more
  and thus shifted away.

x
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