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Tolerance studies of the Mu2e solenoid system
M. L. Lopes, G. Ambrosio, M. Buehler, R. Coleman, D. Evbota, S. Feher
M. Lamm, V. Kashikhin, G. Moretti, T. Page, M. Tartaglia, Fermilab,
J. Miller, Boston University
J. Popp, York College CUNY
R. Ostojic, CERN
Abstract—The Muon-to-electron conversion experiment
(Mu2e) at Fermilab is designed to explore charged lepton flavor
violation. It is composed by three large superconducting
solenoids: the production solenoid (PS), the transport solenoid
(TS) and the detector solenoid (DS). Each sub-system has a set of
field requirements. The tolerance sensitivity studies of the
magnet system were performed with the objective to demonstrate
that the present magnet design meets all the field requirements.
Systematic and random errors were considered on the position
and alignment of the coils. The study helps to identify the critical
sources of errors and which are translated to coil manufacturing
and mechanical supports tolerances.
Index Terms—Solenoid, Superconducting Magnets.
I. I
NTRODUCTION
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), and [3] and the detector solenoid
(DS) [4].
Fig. 1. The Mu2e experiment overview
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).
Due to the strict requirements on the field, it is necessary to
assess the robustness of the solenoid in presence of
geometrical errors in the positions of the coils. Errors can be
present both because of the manufacturing tolerances due to
the technological fabrication process and because of the
mechanical and thermal solicitations present during the
operation of the system: particularly, thermal deformations are
induced by the cryogenic system.
II. M
ETHODOLOGY
In this work we summarize the changes in the magnetic
performance due to misalignment errors in the coils. Two
types of errors are studied: systematic and random. Systematic
errors are the ones occurring when a group of coils belonging
to the same section (TS1, TS2 etc.) has known misalignment
errors. Random errors are the ones occurring when each
individual coil has an unpredictable deviation from its nominal
position. In the case of random errors, each coil is allowed to
move in one particular direction. The field is calculated for
each geometrical configuration. The process is repeated 100
times, with different individual displacement. The maximum
displacements of the coils are limited to a value specified in
each case.
For the DS a similar approach was used for the random
errors. However, given its cylindrical symmetry some errors
were suppressed from the study.
III. T
RANSPORT SOLENOID TOLERANCES
The most critical areas of the TS are the straight sections.
The magnetic field requirements are described in [1]. The
magnetic requirements on those regions are such that the
longitudinal field gradient has to be always negative. Positive
gradient could potentially trap particles. Figures 2-4 show the
longitudinal filed gradient inTS1, TS3 and TS5 (the three
straight sections of TS) when the coils are at the nominal
position. For each section, the gradient is calculated in five
different azimuthal points.
The tolerances on the position (radial, vertical and
longitudinal) and angles (yaw and pitch) were studied. Figures
5-7 show variations of the longitudinal gradient in TS1, TS3
and TS5 respectively when the TS coils have errors of
± 10 mm applied to the radial position of the coils. Random
errors of up to 10 mm are very larger compared to the typical
manufacturing tolerances. The results show that the magnetic
design is very robust because, even in the presence of large
errors, the longitudinal gradient in the TS straight sections
T
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keeps negative. In fact, Table 1 summarizes the results of all
the errors studied and the maximum achieved gradient.
TABLE I
S
UMMARY OF MAXIMUM GRADIENT IN THE STRAIGHT SECTIONS AND ERROR
TYPES
.
Error type
Maximum logitudinal gradient
(T/m)
TS1
TS3
TS5
Radial (mm)
10
-0.072
-0.072
-0.083
2
-0.114
-0.106
-0.115
Vertical (mm)
10
-0.103
-0.082
-0.104
Longitudinal
(mm)
10
-0.023
-0.009
-0.060
2
-0.111
-0.096
-0.109
Pitch (mrad)
10
-0.120
-0.092
-0.116
2
-0.124
-0.112
-0.120
Yaw (mrad)
10
-0.108
-0.086
-0.105
2
-0.121
-0.107
-0.119
Fig. 2. Nominal longitudinal gradient along TS1.
Fig. 3. Nominal longitudinal gradient along TS3.
Fig. 4. Nominal longitudinal gradient along TS5.
Fig. 5. Longitudinal field gradient along TS1 when radial errors of ± 10 mm
are present.
Fig. 6. Longitudinal field gradient along TS3 when radial errors of ± 10 mm
are present.
Fig. 7. Longitudinal field gradient along TS5 when radial errors of ± 10 mm
are present.
Systematic errors on the TS coils were also studied. These
errors, however, have a much smaller impact in the magnetic
performance. In particular, the systematic changes needed to
correct the magnetic center position [5] do not cause any
violation of the magnetic requirements. As an example of
systematic error, Figure 8 shows the field gradient in TS1
when TS1 coils are bent vertically by 1
o
.
Fig. 8. Longitudinal field gradient along TS1 when the coils are bent
vertically by 1
o
.
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IV. DETECTOR SOLENOID TOLERANCES
The DS is mainly divided into three sections: DS1 (gradient
region), DS2 (transition region), DS3-4 (spectrometer and
calorimeter region). The magnetic field in these three regions
can be seen in figure 9. The general requirement for the DS is
that the longitudinal field gradient has to be negative. Figure
10 shows the longitudinal field gradient in these regions.
As can be seen in figure 10, at R=0.7 m the gradient is, at
times, often positive. That happens because the bore radius of
the coils is 1.05 m, therefore at R=0.7 m is relatively close to
the coil's bore and the ripple is given, essentially, by the space
in-between the coils. In the same way, at R=0.4 m in the DS1
region, a positive gradient is present. This is due the fact that
the TS coils have a bore radius of 0.405 m and TS and DS
have an overlap as can be seen in figure 1.
Fig. 9. Longitudinal field in the DS when the coils are at the nominal
position.
Fig. 10. Longitudinal field gradient in the DS when the coils are at the
nominal position.
A. Cable thickness tolerances
The DS has 11 coils. They are winded from 2 conductors:
DS1 and DS2 types. DS1-type has a bare thickness of
5.25 mm, while DS2-type 7.0 mm (figure 11). Around each
conductor is applied 0.250 mm insulation. The coils are
winded in the hard-bend mode. Eight of the coils use the
DS1-type conductor. The three coils located on the DS3-4
region use the DS2-type conductor. Given the number of turns
in each coil, small errors on the conductor thickness will have
a direct impact on the length of each coil.
Fig. 11. Top: DS1-Type conductor cross-section. Bottom: DS2-Type
conductor cross-section.
The tolerances on the conductor were studied. The only
noticeable variation detected is on the longitudinal gradient on
the DS3-4 region. The nominal negative gradient there is
fairly weak. Figure 12 shows an example of the variation of
the longitudinal gradient on that region when the cable
thickness can vary ± 50 µm. The negative gradient should be
guaranteed up to R = 0.4 m.
As can be seen, 50 µm variation in the cable is the
acceptable limit for the thickness of the cable. The
manufacturing specification was set in 30 µm.
Fig. 12. Longitudinal field gradient in the DS3-4 region when the cables have
a variation of ± 50 µm.
B. Systematic change on the position of the
superconductor inside the Al matrix
In this study it is assumed that the superconducting part of
the cable could be displaced with respect to the Aluminum
matrix. A systematic change of the position could result in a
higher density of turns in one side or the other of the coils. It
was considered that only the 3 coils made of DS2-type
conductor: coils # 8, 9 and 10 would be affected by that. For
this study each individual turn was modeled. Each coil has 244
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turns.
Several configurations of coils densities were simulated
including linear and quadratic distributions. In all the cases,
the variation of the superconductor inside the Al matrix was
assumed to be ± 0.3 mm (according to the cable specifications
shown in figure 11).
The results have shown that, at this level of errors, no
positive gradients (up to R = 0.4 m) arise from this problem.
C. Mechanical tolerances for the coils
In this study the coils were assumed to have perfect length
and winding. The coils are positioned off their nominal values.
Given the cylindrical symmetry of the problem, changes in the
X axis are equivalent to changes in Y axis. The same way,
changes in the Pitch and Yaw of the coils are equivalent. Like
in the previous section, given the level of errors that was
assumed during the analysis, the only noticeable differences
can be seen in the longitudinal gradient of the DS3-4 region.
Figures 13 - 15 show the worst cases among all the cases
studied. It will be required that the coils be positioned better
than ± 5 mm radially and ±1 mm longitudinally. The coils
must be aligned within ±2 mrad.
Fig. 13. Longitudinal field gradient in the DS3-4 region when the coils have
errors of ± 10 mm in the radial direction.
Fig. 14. Longitudinal field gradient in the DS3-4 region when the coils have
errors of ± 1 mm in the longitudinal direction.
Fig. 15. Longitudinal field gradient in the DS3-4 region when the coils have
errors of ± 2 mrad in the pitch angle.
V. C
ONCLUSIONS
A sensitivity study was performed on the TS and the DS.
This study helped to identify the weak spots in the design and
correct them.
The most sensitive areas of the TS are the straight sections
where positive gradients could potentially trap particles, being
a source for backgrounds. The present TS magnetic design has
enough margins that make it very robust. Even in the presence
of large positioning errors the TS fulfills the magnetic
requirements. The mechanical tolerances for the TS coils are
given by other sources [5].
The most sensitive region of the DS is the spectrometer and
calorimeter regions (DS3-4). Errors on the coils in this area
could create a positive gradient that needs to be avoided for
the reasons described before. The design is robust otherwise,
even with larger errors.
The tolerances on the DS conductors are adequate. It will be
required that the DS coils to be positioned better than ± 5 mm
radially and ± 1 mm longitudinally. The coils must be aligned
within ± 2 mrad.
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] M. Lopes et al. – "Studies on the Magnetic Center of the Mu2e Solenoid
System"- this conference