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Studies on the Magnetic Center of the Mu2e Solenoid
System
M. L. Lopes, G. Ambrosio, M. Buehler, R. Coleman, D. Evbota,
V. Khalatian, M. Lamm, G. Moretti, T. Page, M. Tartaglia, Fermilab,
J. Miller, Boston University
Abstract—The definition of the magnetic center in the Mu2e
solenoid system is not trivial given the S-shaped nature of the
transport solenoid (TS). Moreover, due to the fringe field of the
larger bore adjacent magnets – production solenoid (PS) and the
detector Solenoid (DS) – the magnetic center does not coincide
with the geometric center of the system. The reference magnetic
center can be obtained by tracking a low momentum charged
particle through the whole system. This paper will discuss this
method and will evaluate the deviations from the nominal
magnetic center given the tolerances in the manufacturing and
the alignment of the coils. Methods for the correction of the
magnetic center will also be presented.
Index Terms—Solenoid, Superconducting Magnets, Beam
Transport
I. INTRODUCTION
HE Mu2e experiment [1] proposes to measure the ratio of
the rate of the neutrino-less, coherent conversion of muons
into electrons in the field of a nucleus, relative to the rate of
ordinary muon capture on the nucleus. The conversion process
is an example of charged lepton flavor violation, a process that
has never been observed experimentally. The conversion of a
muon to an electron in the field of a nucleus occurs
coherently, resulting in a monoenergetic electron (105 MeV)
near the muon rest energy that recoils off of the nucleus in a
two-body interaction. At the proposed Mu2e sensitivity there
are a number of processes that can mimic a muon-to-electron
conversion signal. Controlling these potential backgrounds
drives the overall design of Mu2e. The overview of the Mu2e
experiment can be seen in Fig 1. It is primarily formed by
three large solenoid systems: the production solenoid (PS) [2]
the transport solenoid (TS) [3] and the detector solenoid (DS)
[4]. The muon beam is created by an 8 GeV, pulsed beam of
protons striking a production target.
Fig. 1. The Mu2e experiment overview
Manuscript received July 16, 2013. Work supported in part by FRA under
DOE Contract DE-AC02-07CH11359.
M. L. Lopes - Fermi National Accelerator Laboratory, Batavia, IL 60510
USA (mllopes@fnal.gov).
The negative magnetic field gradient in the PS creates a
mirror effect on the charged particles produced after the
collision, pushing the beam downstream. This beam is then
transported by the TS. The three collimators in TS provide
charge and momentum selection. In the DS, the muon beam is
stopped in the stopping target and the 105 MeV electrons from
the conversion are detected in the tracker.
The magnetic system is formed by 3 coils for PS, 52 Coils
for TS and 11 coils for DS. Each subsystem is in a separate
cryostat module. TS is divided into two cryostats (named TSu
and TSd).
II. M
AGNETIC CENTER VERSUS GEOMETRIC CENTER
In general, straight solenoids have a well-defined magnetic
axis. The magnetic axis is important because it determines the
beam center. Very often, because of limitation in the
construction, the magnetic axis does not coincide with the
geometric center of a straight solenoid. In general, magnetic
measurements should be carried out to relate them.
The magnetic center of the Mu2e solenoid system cannot be
easily determined, given the S-shape nature of the TS. Fig. 2
shows, schematically, the difference between the geometric
center of the coils and the magnetic center. Just like in a
straight solenoid, the beam follows the magnetic center of the
TS. The knowledge of that is crucial because the experiment
relies on collimators to make both charge and momentum
selection. If the beam is off-set with respect to a collimator,
that may impact the muon transmission through the channel.
-5000 0 5000 10000 15000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
Z [mm]
X [mm]
Centroid
Nominal Traject ory
Fig. 2. Schematics of the geometric center (dashed line) vs. the magnetic
center (solid line). The differences between the two were exaggerated for
clarity.
T
FERMILAB-CONF-13-263-TD
Operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy
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III. LOW-MOMENTUM PARTICLE TECHNIQUE
The magnetic center can be determined by tracking a low-
momentum charged particle (LMCP). The vertical drift is
relatively small; therefore it stays close to the geometric
center. Fig 3 shows the track of three different momentum
particles through the magnetic system with respect to its
geometric center.
-8000 -6000 -4000 -2000 0 2000 4000
6000 8000
-30
-20
-10
0
10
20
30
Nominal tracks with respect to the centroid of the coils
1 MeV e-
Horizontal
Vertical
-8000 -6000 -4000 -2000 0
2000 4000
6000 8000
-50
0
50
100
150
Displacement (mm)
25 MeV/c
µ
-
-8000 -6000 -4000 -2000
0 2000 4000 6000 8000
-50
0
50
100
150
s (mm)
50 MeV/c
µ
-
Fig. 3. Track of particles through the TS. Top: 1MeV e
-
, Middle: 25 MeV/c
µ
-
, Bottom: 50 MeV/c µ
-
.
Figure 4 shows the transverse cross section of the muon
beam (simulated) at the entrance of the third collimator
(located at the last TS straight section). In red is the result
before corrections in the toroidal magnetic system. As can be
seen, the center of the beam is displaced by 100 mm. Taking
into consideration that the collimator radius is 150 mm, a
significant portion of the beam will be lost. This problem is
caused by the fringe field of the adjacent solenoids (PS and
DS) extending into the curves. This can be corrected by
applying an extra yaw angle (~1 degree) on the coils that
compose the curves (Fig 5). In blue is the result after the
magnetic design was corrected.
This simulation, using G4Beamline [5], takes a relatively
long time (~1 h). The tracking of a single MeV e
-
is very fast
(<1 min) and was sufficient to correct the beam position.
Fig. 4. The transverse cross section of the muon beam (simulated) at the
entrance of the third collimator. Red: before the correction; blue: after the
correction.
This technique can be used to study the geometrical
tolerances of the magnetic system. Simulations of three
different energy particles going through the transport solenoid
were done. In each simulation a different random error was
applied to each coil. The difference between the track in the
perturbed state and at the nominal position is computed. The
process is repeated 100 times for statistics. Average and
standard deviations are calculated at each point of the track.
Fig. 5. Mu2e solenoid system. The coils in blue represent the ones with extra
yaw of ~1
o
(the picture shows an exaggerated effect for clarity).
Figures 6 shows the standard deviation as function of the
positions, when a random error of up to 5 mrad is applied to
the pitch angle of each coil. The pitch angle only affects the
vertical position of the magnetic center. Likewise, yaw errors
only affect the horizontal displacement.
As can be seen, the relative change of the beam position is
independent of the particle momentum. Therefore, the
standard deviation of a LMCP can be used as an indication of
the displacement of the magnetic center. The advantage is that
LMCPs have a very small pitch compared to the general
dimensions of the magnets and the strength of the TS
magnetic fields (from 2.5 to 2 T). Hence the errors associated
with the calculation of the magnetic center with respect to the
geometric center are minimized.
-8000
-6000
-4000 -2000
0
2000
4000 6000
8000
-10
-5
0
5
10
Displacement with respect to the nominal track
1 MeV e-
Vertical
-8000 -6000
-4000
-2000 0
2000
4000 6000
8000
-10
-5
0
5
10
Displacement (mm)
25 MeV/c
µ
-
-8000 -6000
-4000
-2000 0
2000 4000
6000
8000
-10
-5
0
5
10
s (mm)
50 MeV/c
µ
-
Fig. 6. Standard deviation as function of the trajectory of the perturbed coils
system when random errors of up to 5 mrad are present.
This technique was applied to study the sensitivity of three
translational errors of the coils (X, Y and Z) and the two
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rotation angles (yaw and pitch). The errors associated with the
rotations are more critical for the displacement of the magnetic
center. The mechanical specifications for the coils placement
were limited to be smaller than 0.2
o
. In that case the maximum
RMS deviation is limited to ~3 mm.
IV. V
ERTICAL ALIGNMENT
As mentioned before, one of the functions of TS is
momentum and charge selection. The first curve in TS will
drift the beam vertically; the drift is proportional to
momentum of the particles. Given the orientation of the field,
negative particles drift upward and positive particles drift
downward. After the first curve, the beam goes through an
asymmetric collimator (Fig 7). The second curve of TS brings
the beam back to the center.
Fig. 7. Asymmetric collimator located in between the two curves.
The vertical alignment of the magnetic center is especially
critical in the middle collimator region. Normally, 105 MeV
electrons coming from the production target are intercepted by
the middle collimator. Studies have shown that misalignments
may allow these electrons to go through the collimator and
make their way to the detector, mimicking the signal from the
conversion. In that case, the beam needs to be displaced
upwards so that the 105 MeV electrons hit the upper region of
the collimator.
A. Gravity supports
In the previous section the track of a LMCP was used to
determine the mechanical tolerances due to random errors on
the positioning of the coils. The same technique can be used to
study the mechanical tolerances that must be achieved by each
TS cryostat.
Figure 8 shows the TSu cold mass with its radial, axial and
gravity supports. The present design has three pairs of gravity
supports (TS1, TS2 and TS3). The gravity supports can be
used to intentionally misalign the coils, that is, introduce a
systematic pitch angle error to affect the vertical displacement
of the magnetic center on the middle collimator. Table I
summarizes the results of the shift in vertical magnetic center
as function of different gravity supports misalignments.
TABLE I
V
ERTICAL MAGNETIC CENTER DISPLACEMENT AS FUNCTION OF THE GRAVITY
SUPPORTS MISALIGNMENT
. UNITS ARE mm.
TS3
0
10
20
30
TS1
0
0.0
6.4
12.7
19.0
10
-4.0
2.3
8.7
15.0
20
-8.0
-1.6
4.7
11.0
30
-11.9
-5.6
0.7
7.0
The maximum displacement of 19 mm is achieved when the
TS3 gravity support is lifted (or lowered) by 30 mm and the
TS1 is left at the nominal elevation. The disadvantage of this
method is that solenoids naturally align their axis. Therefore a
centering force of -68 kN will appear. This amount can be
handled by the gravity supports (in either direction).
Fig. 8. TSu cold mass with the supports.
B. Correction coils
Another method that was investigated was the use of
correction coils. A dipole winding could be placed on top of
the first straight section (Figure 9). The interesting thing about
the correction coils for solenoid systems is that for a vertical
beam displacement, one has to apply a vertical field, unlike
beam in accelerator where a vertical field displaces the beam
horizontally. A 0.1 T-m integrated field from the correction
coil will result in around 40 mm beam displacement (Figure
10). This displacement, again, is independent of the particle
momentum.
Fig. 9. A vertical correction coil model.
The disadvantage of a correction coil is the considerable
complication in the design of the magnetic system. The
needed integrated field is relatively high and its location (in
the PS fringe field) would subject the coils to high forces. The
mechanical support would be complicated. The quench
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protection will be also an issue. All result in a significant
increment in the overall cost of the magnet.
C. Trim power supplies
Another option to tune the beam position is the use of trim
power supplies (TPS). Those devices typically allow changes
of a few percent of the main current. Figure 11 shows,
schematically, the coils where a TPS is employed.
In order to allow 105 MeV electrons to be displaced
upwards, one must reduce the current on the coils. A reduction
of 10% of the current results in these electrons moving around
20 mm up. An interesting fact is that this kind of modification
is momentum-dependent; the LMCP technique does not apply
in this particular case. 105 MeV electrons were tracked
instead. Table II summarizes the results of the horizontal and
vertical displacements (with respect to the coils in nominal
conditions) of electrons with different energies.
-8000
-6000 -4000 -2000 0 2000 4000 6000
8000
-60
-40
-20
0
20
40
60
Displacement with respect to the nominal track
1 MeV e-
Vertical
-8000 -6000 -4000 -2000
0
2000
4000 6000 8000
-60
-40
-20
0
20
40
60
Displacement (mm)
25 MeV/c
µ
-
-8000 -6000 -4000 -2000 0 2000
4000 6000 8000
-60
-40
-20
0
20
40
60
s (mm)
50 MeV/c
µ
-
Fig. 10. Vertical beam displacement using a correction coil located in the first
straight section.
Fig. 11. Mu2e solenoid system. The coils in blue represent the ones powered
with trim power supplies.
TABLE
II
V
ERTICAL AND HORIZONTAL DISPLACEMENT USING TRIM POWER SUPPLIES.
UNITS ARE mm.
105 MeV
1 MeV
Vertical
19.0
-1.4
Horizontal
-1.4
3.1
V. CONCLUSION
Tracking studies have shown that the beam follows the
magnetic center. A technique to determine the magnetic center
of the Mu2e solenoid system was discussed. The technique is
useful to predict displacements and indicate corrections to the
magnetic design in a reliable and fast way. The technique was
also used to estimate the mechanical tolerances of the support
structures. The most restrictive tolerance has to do with the
angles of the coils (yaw and pitch). The mechanical tolerances
were fixed at 0.2
o
in order to keep the RMS displacements
lower than 3 mm.
The magnetic center can be measured in the same way as
presented here. The use of a beta source or even a cathode
installed in the same location as the production target in PS is
planned to be used [6]. The position could be measured with a
detector in between TSu and TSd. In the event that corrections
are needed, a controlled misalignment of TSu can provide the
necessary vertical displacement. In that case a vertical
centering force will push the magnet back to the center. The
supports are design to withstand those additional forces.
During the commissioning of the experiment, a trim power
supply (connected to the majority of the coils that compose the
first curve) can be used on-the-fly for calibration purposes.
R
EFERENCES
[1] Mu2e Collaboration, "Mu2e Conceptual Design Report",
arXiv:1211.7019, http://arxiv.org/abs/1211.7019
[2] V. V. Kashikhin et al., “Conceptual Design of the Mu2e Production
Solenoid Cold Mass,” Advances in Cryogenic Engineering, AIP Conf.
Proc., 1434, 893-900 (2012).
[3] G. Ambrosio et al. – "Challenges and Design of the Transport Solenoid
for the Mu2e experiment"- this conference;
[4] S. Feher et al. – "Reference Design of the Mu2e Detector Solenoid" -
this conference.
[5] J. Miller, R. Coleman Reference.
[6] M. Buehler et al. – "Mu2e Magnetic Measurements", this conference
(3PoAJ-12)