RFQ Basics July 7

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RFQ BasicsJuly 7
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Some RFQ General References

• I.M.Kapchinskiy and V.A.Tepliakov,
  Prib.Tekh.Eksp.2,19-22(1970).
• I.M.Kapchinskiy and V.A.Tepliakov,
  Prib.Tekh.Eksp.4,17-19(1970).
• M.Weiss, Radio-Frequency Quadrupole, presented at
  CERN Accelerator School, Aarhus, 15-26 September,
  1986, CERN-PS/87-51(LI) 29 May 1987.
• A.Schempp, CERN Accelerator School, CERN 92-03,
  Vol.II,p.522.
• J.Staples, Radio-Frequency Quadrupole, in
  Handbook of Accelerator Physics and Engineering,
  Eds. Alex Chao and Maury Tigner, (1999) pp.36-40.
• T.P.Wangler, RF Linear Accelerators, second
  edition, Wiley-VCH (2008), Chapter 8

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RFQ Overview
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Shown below are the two main RFQ structures,
4-vane (left) and 4-rod (right)
Four Vane RFQ
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4-vane RFQ (left) and 4-rod RFQ (right)

• Modulation of the vane tips produces a
  longitudinal electric-field to accelerate the
  beam.
• The RFQ was invented by Kapchinsky and Tepliakov
  and published in 1970. I.M.Kapchinskiy and
  V.A.Tepliakov, Prib.Tekh.Eksp.2,19-22(1970).
  I.M.Kapchinskiy and V.A.Tepliakov, Prib.Tekh.
• Eksp.4,17-19(1970).

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Radiofrequency quadrupole (RFQ) is used for ions
with approximately blt0.1

• The RFQ can provide bunching, acceleration, and
  transverse focusing all from rf electric fields.
• The desired field near the axis is determined by
  machining of vanetips.
• The RFQ provides strong electric focusing for
  high current, low emittance ion beams.

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Four-vane Radiofrequency-Quadrupole (RFQ) Cross
Section

• The four vanetips (poletips) are excited in a
  TE210-like mode with rf electric quadrupole
  voltages that focus the beam transversely.
• This produces time-dependent alternating polarity
  FD focusing. (Other accelerators provide
  spatially-dependent alternating polarity
  focusing)

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Two views of a 4-vane RFQ showing the unit cell
which is the equivalent of one accelerating gap
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Views showing the vanes and a four-vane RFQ unit
cell
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Electric-field vectors in x-z plane over half
period of vanetip modulations (a unit cell)
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The RFQ has produced a major improvement in the
architecture and performance of ion linacs
Pre-RFQ linac architecture used an ion source in
a large Cockcroft-Walton, an inefficient buncher
(poor capture efficiency) and weak magnetic
focusing at low velocities, resulting in poor
beam quality which typically led to beam loss.
RFQ provides stronger (electric) focusing,,lowers
injection velocity, eliminates the large
Cockcroft-Walton. Raises injection energy into
the DTL (stronger magnetic focusing. As option
eliminates external buncher system. Beam quality
improvement.
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Some RFQ Examples
CERN- LINAC II SNS LANL-LEDA TRIUMF- ISAC
Ion H H- H Q/Agt1/30
Freq (MHz) 202.56 402.5 350 35
Win(keV) 90 65 75 2 keV/u
Wout(MeV) 0.75 2.5 6.7 0.150 MeV/u
Structure 4-vane 4-vane 4-vane Variant of 4-rod Uses split ring, CW
Length(m) 1.8 3.76 8.0 8.0
E(Kilpatrick) 2.5 1.85 1.8 --
I(mA) 165 to 200 38 100 lt1 mA
erms,n(mm-mrad) 1.2 0.21 0.2 0.026
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Proof-of-Principle RFQ 1980- Los Alamos30-mA
proton beam, 100 keV to 640 keV, 425 MHz
15 cm
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ATS RFQ 1985100 mA protons, 100 kev to 2 MeV,
425 MHz
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Beam Aboard A Rocket (BEAR) RFQ 198930 mA H-,
425 MHz, 30 keV to 1 MeV
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1-MeV Bear RFQ which is the first and only
accelerator that has operated in space is shown
with Cockcroft-Walton that it could replace.
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LEDA CW RFQ 1999100-mA protons CW, 75 keV to 6.7
MeV, 350-MHz, 8-m long
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SNS H- RFQ designed and built by LBNL65 keV to
2.5 MeV, 402.5 MHz

• 4-vane RFQ with p-mode stabilizers for dipole
  mode supression
• 4 modules with 3.72-m total length
• 402.5 MHz resonant frequency
• 640 kW pulsed power needed to achieve nominal
  gradientwithout beam
• 8 power couplers
• 80 fixed tuners
• Dynamic tuning implemented by adjusting cooling
  water
• 2.5 min. needed to reach stable operation from
  cold start

SNS RFQ seen from the LEBT (Low-Energy Beam
Transport) side.
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The MSU reaccelerator RFQ will look similar to
this CW 4-rod for SARAF (Soreq applied research
accelerator facility) in Israel.4 mA D, CW,
3MeV, 176 MHz, 3.8 m , 220 kW, 39 cells.Now
operating at full power.
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The ISIS Four-Rod RFQ at Rutherford Appleton
Laboratory35 keV to 665 keV H- beam, 202.5 MHz,
V90 kV
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SARAF 4-ROD RFQ 4-mA D to 3 MeV, 176 MHz, 3.8
m long, 220 kW power 39 cells
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3D Solution for Electric Potential and Fields in
the RFQ
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Quasistatic approximation results in 3D analytic
solution for the fields in time- dependent
complex 3D RFQ geometry

• For some RF cavity problems involving
  time-dependent fields, the fields can be obtained
  from a time-dependent scalar potential that
  satisfies Laplaces equation.
• This solution is called the quasistatic
  approximation and is valid when the field
  variations are determined by cavity structural
  elements whose geometrical sizes are small
  compared with the RF wavelength.
• In particular the quasistatic approximation
  applies near the beam aperture when the beam
  aperture is small compared with the wavelength.

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Kapchinsky used quasistatic approx to obtain the
general solution to Laplaces equation for a
spatially periodic RFQ geometry
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Two-Term Potential Function

• At first impression the RFQ looks like a
  complicated 3D structure for which an analytic
  solution for the fields would be difficult to
  obtain.
• We have a general solution to a potential
  function valid near the beam axis, but it has an
  infinite number of terms.
• If we can choose the vane geometry so that near
  the beam axis only a few terms are large, the
  potential description can be put to practical
  use.
• The approach used by K-T was to select only the
  two lowest terms from the general solution and
  configure vanetips that conform to the resulting
  equipotential surfaces of that potential
  function.
• This is called the two-term potential function
  description and is starting point for RFQ
  design.

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Two-term potential function can be expressed in
either cylindrical-coordinates and Cartesian
coordinates
Cylindrical coordinates
Use
Cartesian coordinates
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Vanetip Geometry Parameters
a minimum radius, ma is maximum radius, r0 is
radius at point of pure quadrupole symmetry,
called the characteristic radius.
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How to determine the two constants X and A from
the vanetip geometry a and m
Write the two term potential function as
k2p/bl
Call A and X (Greek letter chi), the acceleration
efficiency and focusing efficiency, respectively.
Evaluate potential at z0, q0, rxa where
UV0/2.
Then evaluate potential at z0, qp/2, ryma,
where U-V0/2.
We now have two equations in two unknowns, X and
A. Subtract them and solve for A, and then
substitute A into either equation to get X.
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Next relate the accelerating and focusing
efficiency parameters A and X to the cell
geometry parameters a and m.
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Electric field components
Potential function
Differentiate and obtain electric field
components (Cartesian coordinates).
The quadrupole terms contain X. The RF terms
contain A.All three components are multiplied by
sin (wtf) for time dependence.
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Modified Bessel Functions I0(x) and I1(x)
I0
x
,
I1
x
2
1.75
1.5
I0(x)
1.25
1
I1(x)
0.75
0.5
0.25
x
0.5
1
1.5
2
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Applicability of the two-term potential function

• If we could build the RFQ vanes everywhere to the
  exact shape of the equipotential surfaces of the
  potential function we would have the exact
  electric fields as calculated from the potential
  function.
• But we cant do that. For one reason, the
  equipotentials asymptotically approach each other
  at large radii and the peak fields would be too
  high resulting in arcing.
• But in practice we can still construct the
  vanetips to lie on an equipotential along z.
• It will still be a good approximation near the
  beam axis to use the electric fields from the
  two-term potential function.

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Two-term RFQ potential function provides an
approximate analytic solution for the electric
fields
Potential
Focusing efficiency
Acceleration efficiency
Electric Field Components are derivedfrom
potential.
Quadrupole focusingplus rf defocusing.
Acceleration
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The two-term potential function is a good
approximation for the basic physics. But for more
precision in beam-dynamics calculations the
PARMTEQ RFQ code now uses up to an 8-term
potential function.