Title: Radio Frequency Quadrupole (RFQ)
1Radio Frequency Quadrupole(RFQ)
2Introduction
- The first linac was built in 1928 by Widröe
1MHz 25 kV
50 keV K ions
K ions
d??/2
3The 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.
4The 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.
7Principle 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.
9Modulation
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.
11The 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 -
14The 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.
18The 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. -
19Voltage 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
20The 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
21The 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.
23The actual vane tip profiles
The vertical vane
One quadrant of RFQ
The horizontal vane
?
r0
24Characteristics 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. -
25Characteristics 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.
26The 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.
28RMS
SHAPER
29Accelerator
Longitudinal profile of the vane tip in 4.5 MeV
50 mA RFQ
30The 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.
31The 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
34Resulting 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
36Eigen 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.
38The longitudinal mode spectrum of a 4 vane RFQ
cavity
39The 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.
40Vane 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
41PISL
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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