Download Radio Frequency Quadrupole (RFQ)

Radio Frequency Quadrupole
(RFQ)
S. A. Pande
February 5, 2004
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Introduction

The first linac was built in 1928 by
1MHz
Widröe
~
25 kV
50 keV
K+
ions
K+ ions
d=/2
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The Sloan Lawrence Structure



E. O. Lawrence in association with Sloan built
an improved version of Widröe’s 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
-1/2Vcos(t)
conducting electrodes
with alternating
polarity as we move
from one electrode to
1/2Vcos(t)
1/2Vcos(t) the next forms the
electric quadrupole.
-1/2Vcos(t)
The electric Quadrupole
February 5, 2004
Voltage 1/2V0cos(t)
is applied in quadrupolar symmetry
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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
a
ma
a
ma
Beam axis
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 r<<.
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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
2
2
1


U
1

U

U


 2U (r , , z ) 
r


0

 2
2
2
r r  r  r 
z
Where U(r,,z) electric field potential.
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Solving the above equation by the method of
separation of variables, we obtain

U (r , , z )   As r 2( 2 s 1) cos[ 2(2s  1) ]
s 0


   Ans I 2 s (nkr) cos( 2s ) sin( nkz)
n 1 s  0
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 s=0 from the first and s=0, n=1
terms from the second summation, we have
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The two term potential
Retaining only s=0 from the first and s=0, n=1 terms
from the second summation, we have
U (r , , z )  A0 r 2 cos 2  A10 I 0 (kr) cos kz
where k=2/; =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
V0 I 0 (ka)  I 0 (mka)
A0  2 2
2a m I 0 (ka)  I 0 (mka)
V0
m2  1
A10 
2 m 2 I 0 (ka)  I 0 (mka)
We define two dimensionless quantities
I 0 (ka)  I 0 (mka)
X 2
m I 0 (ka)  I 0 (mka)
m2  1
A 2
m I 0 (ka)  I 0 (mka)
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With these two dimensionless quantities, A0=XV0/2a2
and A10=AV0/2, the two term time dependent
potential is written as
V
U (r , , z )  0 [ X (r / a) 2 cos 2  AI 0 (kr) cos kz] sin( t   )
2
---------------- --------------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 m=1 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 = -V0/2[2(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
Lc
Vcell   Ez dz  AV
0
where we have used Ez as defined earlier and
Lc=/2
The energy gain is given by
W=qeAVTcos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 x=rcos and y=rsin, we
have
U(x,y,z,t)=V0/2[X/a2(x2-y2)+AI0(kr)coskz]
with U=V/2, we have for the geometry of the vane
surface
1=X/a2(x2-y2)+AI0(kr)coskz
Or
x2-y2=a2/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 .
r02=a2/X
r0=aX-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.
W=~0 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 m=1, 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
A=0 and s=-90 to the gentle buncher where A>0
and s>-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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Vane Tip profile for first 50 cm
vane tip distance from axis
(cm)
0
-0.1
RMS
-0.2
SHAPER
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
0
5
10
15
20
25
30
35
40
45
50
axial distance (cm)
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Vane Tip profile for last 50 cm
vane tip distance from axis
(cm)
0
-0.1
Accelerator
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
495
505
515
525
535
545
axial distance (cm)
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 electromagnetically.
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 Etangential=0
• 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
Desired
Field due to TE210 mode
zero last subscript denotes
that there is no variation in
the longitudinal direction.
Resulting
Field due to TE211 mode
the last subscript denotes
the no. of half wavelength
variations in z 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
Cross section through RFQ
at an end
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Side view
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A
N
E
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
Quadrupole
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x
Dipole-1
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x
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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Dipole
Quad
375
Frequency (MHz)
370
365
360
355
350
345
340
335
0
1
2
3
4
5
6
7
Longitudinal mode no. n
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 12 m in length
Difficult to mount and
cooling is a problem
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PISL
Dipole 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.
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Quad
mode
x
41
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