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Radio Frequency Quadrupole (RFQ)

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February 5, 2004. S. A. Pande - CAT-KEK School on SNS. 1. Radio Frequency Quadrupole ... The issues related to the electrodynamics are distinct from those associated ... – PowerPoint PPT presentation

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Title: Radio Frequency Quadrupole (RFQ)


1
Radio Frequency Quadrupole(RFQ)
  • S. A. Pande

2
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.

4
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.

5
  • 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.

6
  • 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.

7
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
8
  • 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.

9
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

10
  • 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.

11
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?.

12
  • 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.

13
  • 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

14
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.

15
  • 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

16
  • 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)

17
  • 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.

18
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.

19
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

20
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

21
The ideal vane tip profile
The hyperbolic vane tip profiles
22
  • 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.

23
The actual vane tip profiles
The vertical vane
One quadrant of RFQ
The horizontal vane
?
r0
24
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.

25
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.

26
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.

27
  • 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.

28
RMS
SHAPER
29
Accelerator
Longitudinal profile of the vane tip in 4.5 MeV
50 mA RFQ
30
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.

31
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
32
  • On Quadrant of the RFQ showing electric field
    lines of quadrupole mode

33
  • 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

34
  • 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
35
  • 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
36
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
37
  • 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.

38
The longitudinal mode spectrum of a 4 vane RFQ
cavity
39
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.

40
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

41
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
42
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