CERN CH-1211 Geneva 23 Switzerland
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Date: 2013-09-23
ENGINEERING SPECIFICATION
ELENA ELECTROSTATIC QUADRUPOLES
ABSTRACT:
The extraction beamlines of ELENA will use electrostatic elements to guide and deliver
the beam to the experiments. This document describes the parameters and the design of
the electrostatic quadrupoles.
PREPARED BY:
TO BE CHECKED BY:
TO BE APPROVED BY:
Daniel Barna TE-ABT-BTP
Wolfgang Bartmann TE-ABT-BTP
Jan Borburgh TE-ABT-SE
Bruno Balhan TE-ABT-SE
John Alistair Baillie TE-EPC-MPC
Roberto Kersevan TE-VSC-IVM
Jean-Francois Poncet EN-MME-ED
+
Members of the ELENA Approval
list
Christian Carli BE-ABP-LIS
M. Meddahi
B. Goddard
DISTRIBUTION LIST:
This document is uncontrolled when printed. Check the EDMS to verify that this is the correct version before use.
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HISTORY OF CHANGES
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All
DESCRIPTIONS OF THE CHANGES
First version
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TABLE OF CONTENTS
1.
Introduction ........................................................................................................ 4
2.
Requirements ...................................................................................................... 5
3.
Diagnostics ........................................................................................................ 10
LIST OF TABLES
Table 1: Capacitance matrix of the quadrupole electrode system ....................................... 11
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1. INTRODUCTION
Figure 1: Layout of ELENA and the transfer lines
A 2-dimensional electrostatic quadrupole field can be produced by electrodes of
hyperbolic shape. In practice, however, cylindrical electrodes are most often used. An
appropriately chosen electrode-diameter (Re=1.145∙Ra, where Re is the electrode’s
radius, Ra is the radius of the aperture, i.e. the inscribed circle between the
electrodes) zeroes the contribution of the dodecapole, the first higher-order term
(Baartman, 1995). Since aberrations due to electrode shape are typically smaller than
the intrinsic aberrations of quadrupoles due to their finite length (Baartman, 1995),
cylindrical electrodes are preferred to avoid the higher electric fields in the smaller gap
between hyperbolic electrodes.
Electrostatic quadrupoles will be used in two configurations in the transfer lines:
 Single quadrupoles with a repetition period of 1.5 m will constitute the FODO
structure used to transfer the beam in the straight sections
 Quadrupole doublets will be used after the strongly focusing spherical
deflectors to re-match the beam to the FODO structure, and before the
experiments to focus the beam to the required spotsize.
Both singlets and doublets will have two types: combined with horizontal/vertical
correctors and without correctors.
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2. REQUIREMENTS




Since the total number of quadrupoles is large (???), the design should be
simple and cheap to manufacture
The device must be bakeable at 250 oC
In agreement with all other electrostatic elements of the transfer lines, the
device should have a mechanical aperture of diameter 60 mm.
Switching time: slow
3. DESIGN
The straight sections of the beamline will contain a regular FODO structure of
quadrupoles. Since these sections are sparsely populated with devices, the most
efficient solution is to have the longest possible beam pipe segments and insert and
mount the short quadrupole singlets at the CF flange of the vacuum chamber (Figure
2).
Figure 2: Quadrupole singlet
On the other hand, the longer quadrupole doublets can not be supported at only one
of their ends. They will be mounted in shorter vacuum chambers, supported at both of
their ends at the CF flanges of the vacuum chamber (Figure 3)
Figure 3: Quadrupole doublet
The design uses the same concept and building elements for both singlets and
doublets. The electrodes (A) will be constructed as extruded aluminium profiles, which
is a cheap and simple technique for large quantities. They will be cut into pieces of
length of 100 mm, and mounted on longitudinal rails (B). These rails will be fixed to
the supporting aperture disks (C) by ..... bolts, which in turn will be mounted into the
circular recess machined into the CF flange of the vacuum pipe. These aperture disks
limit the fringe fields of the electrodes and thus avoid possible variations due to the
presence of nearby elements. The intermediate aperture disks of the doublets (D)
make the structure rigid and decouple the two quadrupoles. One set of bolts (E in
Figure 3) of the doublets will allow a longitudinal expansion of the rails during baking.
Opposite electrodes of the same quadrupole are biased to the same voltage. In order
to minimize the number of feedthroughs, these electrodes will be electrically
connected under vacuum (F). Spring-loaded pins mounted on a 2-pin high-voltage
feedthrough (DN35CF) will provide the required voltage. Electrodes of subsequent
quadrupoles in the FODO structure will not be inter-connected under vacuum due to
the large distance between them (1.5 m). Although the two quadrupoles of the
doublets are close, they must be independently powered so their electrodes cannot be
inter-connected either. There will be therefore one two-pin high-voltage feedthrough
for each quadrupole.
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4. PERFORMANCE OF THE QUADRUPOLES
4.1 FOCUSING POWER AND TRANSFER MATRIX
The quadrupole was modelled in COMSOL to simulate its electrostatic field and trace
particles through it. Figure 4 shows this model with simulated trajectories. Particles
were traced between -120 mm and +120 mm with respect to the center of the device.
Figure 4: The COMSOL model and simulated particle trajectories ±3.5 kV
The transfer matrix of the device was calculated from the input and output coordinates
of the simulated particles, as described in (Barna). The top panel of Figure 5 shows
the focusing/defocusing power (inverse of the focal length) of the quadrupole as the
function of applied voltage. The solid lines show the fitted hard-edge models :
𝐹 = sin(𝜔𝑧 𝐿) 𝜔𝑧
(1)
𝐷 = sinh(𝜔𝑧 𝐿) 𝜔𝑧
(2)
𝜔𝑧 = √
𝑞𝑈
2
𝑟0 𝐸𝑘𝑖𝑛
(3)
where L is the length of the quadrupole, ±U voltage is applied to the electrodes, q is
the particle’s charge, Ekin is its kinetic energy and r0 is the radius of the aperture
between the electrodes. The quadrupole can be very well approximated by a hardedge model of length 108 mm.
The cell length of the FODO structure is 3 m, which consists of a quadrupole in a
focusing configuration, and another quadrupole at a distance of 1.5 m in a defocusing
configuration. The transfer matrix of a complete cell was constructed as M=O*D*O*F,
where F and D are the transfer matrices of the device in the focusing and defocusing
directions, respectively, estimated from the simulation, and O is the transfer matrix of
a drift space of 1260 mm. The bottom panel of Figure 5 shows the cosine of the phase
advance per cell: cos(Δφ)=(M11+M22)/2 as a function of voltage. The required 90o
phase advance is realized with U=811 V.
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Table 1 shows the transfer matrix of the FODO quadrupoles at 811 V.
Figure 5: Top panel: focusing/defocusing power of the quadrupoles as a function
of voltage. The solid lines show a fitted hard-edge model. Bottom panel: cosine of
the phase advance per FODO cell. The inset shows a zoom and the stable region in
yellow.
xout
x’out
yout
y’out
xin
1.117
0.984
0
0
x’in
0.253
1.118
0
0
yin
0
0
0.886
-0.949
y’in
0
0
0.227
0.885
Table 1: Transfer matrix of the quadrupole at 811 V
Figure 6 shows the relative change of the focusing/defocusing power of the
quadrupole (hard-edge model approximation with L=108 mm) as the function of the
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kinetic energy of the particles. Within the range dEkin/Ekin=2·dp/p=5e-3 the focusing
powers change by less than 5 per mill.
Figure 6: Relative change of the focusing/defocusing
power of the quadrupole as a function of particle kinetic
energy (hard-edge model with L=108 mm)
4.2 FIELD PERTURBATION DUE TO THE ELECTRICAL INTERCONNECTIONS
The presence of the electrical connections between diagonally opposite electrodes (F
in Figure 2 and Figure 3) breaks the quadrupole symmetry of the system. To estimate
the effect of the field of these connections leaking through the large gap between
neighbouring quadrupole electrodes, a simple 2D model was simulated in Ansys
Maxwell:
 At the point where both, oppositely biased connections pass nearby (point G in
Figure 2), their field was assumed to cancel
 Therefore only two diagonally opposite segments of these connections were
implemented (Figure 7)
 Due to the superposition principle, the additional effect of these electrodes over
the quadrupole field can be simulated by keeping the quadrupole electrodes at
zero potential, and biasing these perturbing electrodes appropriately (±900 V)
Figure 7 shows the model and the electric field along the indicated sampling line.
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Figure 7: Simulation of the perturbation caused by the connections between the
electrodes. Left panel: simulated geometry and the sampling line (red). Right
panel: electric field along the sampling line
The transverse electric field necessary to deflect a particle of charge q, kinetic energy
Ekin, over a length L by an angle ϑ is E=2ϑEkin/qL. Requiring that this angle is less than
the minimum beam divergence, 𝜗 ≪ 𝜗𝑚𝑖𝑛 = 0.9 mrad and using the full length
L=100 mm of the quadrupoles (note that this largely overestimates the effect of the
inter-connections having a smaller width), we get 𝐸 ≪ 1800 V/m. As is visible in the
right panel of Figure 7, the perturbation is much less than this limit within the beam’s
extension, i.e. these perturbations can be safely neglected. This holds also for the
quadrupoles of the matching section with higher (up to 5 kV) voltages.
4.3 TOLERANCES
5. POWER SUPPLY
For longer sections where several quadrupoles will be powered from the same power
supply, we must ensure that the total capacitance and the power supply’s current
rating are compatible
5.1 VOLTAGE STABILITY
The proposed power supplies (FuG ......) have a typical voltage stability of 10-4·Umax.
In case higher-than-needed units (Umax=6.5 kV) are chosen, this corresponds to
δU=0.65 V. Figure 8 shows the dependence of the focusing power of the quadrupoles
on voltage deviations for the FODO quadrupoles (U=±811 V). These changes to not
exceed 2 per mill in the given range.
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Figure 8: Dependence of focusing power on voltage
at U=±811 V
6. DIAGNOSTICS
Although the device has been designed to be safe against thermal cycling, diagnostic
methods are still needed to check whether all electrical connections within the vacuum
chamber are in order – without breaking the vacuum. One way to implement this is
adding diagnostic feedthroughs. Since two diagonally opposite electrodes of the
quadrupoles are electrically interconnected within the vacuum chamber, a highvoltage feedthrough connected to one of these electrodes could provide the bias
voltage, and another high-voltage feedthrough connected to the other electrode could
be used for monitoring the voltage, as indicated in Figure 9.
Figure 9: Using diagnostic feedthroughs to monitor the electrode voltages
The device is practically an ideal capacitor with negligible leakage currents, which can
be monitored at the power supply. No resistive voltage drop of the electrodes is
therefore expected. In order to keep the design simple and the price low, the
quadrupoles will contain no extra diagnostic feedthroughs. In case of suspected
misbehaviour of the quadrupole the electrical connections within the vacuum chamber
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can be checked by measuring the capacitances at the high-voltage feedthroughs, as
described below.
The 3D model of the quadrupole was simulated in COMSOL to estimate the
capacitance matrix between the two high-voltage feedthroughs. Figure 10 shows the
model when one of the electrodes (green) had a broken connection.
Figure 10: The COMSOL model used to estimate the capacitance matrix of the
quadrupole electrode system with one electrode disconnected. Red and blue
indicate the two “terminals”, the green disconnected electrode (X- electrode) had
a “floating potential” boundary condition. The insulators with εr=9 are shown in
yellow.
Table 2 shows the values of the capacitance matrix between the two terminals
associated with the X-electrodes and Y-electrodes for different cases. Figure 11 shows
the equivalent circuit diagram of the quadrupole electrode system. Note that
Cxx=Cxg+Cxy and Cyy=Cyg+Cxy where for example Cxx is the capacitance that can be
measured at the feedthrough connected to the X electrodes with the feedthrough of
the Y-electrodes being grounded, and Cxg is the stray capacitance of the X electrodes
towards ground, and Cxy is the mutual capacitance between the two sets of electrodes
which couples the two high-voltage feedthroughs.
X electrodes
connected
disconnected
disconnected
Y electrodes
connected
connected
disconnected
Cxx [pF]
104
70.5
70.4
Cyy [pF]
104
104
70.4
Cxy [pF]
8.4
5.6
4.2
Table 2: Capacitance matrix of the quadrupole electrode system
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Figure 11: Equivalent circuit diagram of the
quadrupole
The values Cxx and Cyy clearly dominate over the mutual capacitance C xy and are
therefore easier to measure. A broken electrical connection can be detected by an
approximately 32% decrease of the measured capacitance at the corresponding highvoltage feedthrough, the other feedthrough being kept at ground.
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7. WORKS CITED
Barna, D. (n.d.). Calculating the transfer matrix from a simulated beam. Retrieved from
http://barna.web.cern.ch/barna/ELENA/transfer-matrix.pdf
Shockley, W. (1938). Currents to Conductors Induced by a Moving Point Charge. Journal of
Applied Physics, 635.