LHC Project Note XXX
2009-01-01
Ezio.Todesco@cern.ch
Magnetic model of the LHC interaction region quadrupoles MQXA
N. Ohuchi and E. Todesco for the FiDeL team
CERN, Technology Department
Keywords: Superconducting Magnets, Magnetic Field Model, Harmonics, LHC.
1. Introduction
Function in the machine: the MQXA is a superconducting quadrupole with 70 mm aperture
and operational current of ~7150 A and nominal gradient of ~215 T/m [1,2]. Together with
the MQXB, it is a part of the inner triplet present on each side of the four experimental areas.
Its optical function is to focus the beam in the interaction point.
Fig. 1: MQXA cross-section, drawing (right) and picture (left).
Numbers and variants: we have 16 MQXA, two per side of each of the four experimental
areas. They are all the same. Four spares have been manufactured.
Naming convention: During construction, cold masses have been identified by progressive
numbers 1 to 20 [2]. Cold mass 20 has not been tested at 1.9 K: therefore, no measurements at
1.9 K exist. The same progressive number xx is used in the cryostat id., which is
HCMQXA_001_FL0000xx. Cold mass 2 has been disassembled due to damage of coil
insulation, repaired, and cold tested. Magnetic measurements are relative to the second
assembly (file identifier: MQXA-2b).
Expected operational cycles, range of current and operational temperature: The injection
current is 418-452 A, corresponding to a gradient of 13.2-14.3 T/m. During the ramp the
This is an internal CERN publication and does not necessarily reflect the views of the LHC project management.
current increases with the energy as for the main magnets, i.e., reaching 6811 A and a
operational gradient of 205 T/m. During the squeeze the current remains approximately stable.
Summary of manufacturing parameters, and manufacturers: the MQXA have been built as a
special contribution by Japan. All magnets have been built by Toshiba Corporation. Cold
masses 1 to 19 have been tested at KEK in vertical cryostats. The assembly into the final
cryostat has been done at FNAL.
Table I: Main parameters of MQXA. Current calculated based on measurements presented in Section 3.
Magnetic length (m)
Operational temperature (K)
Aperture (mm)
Current at injection in IP1 And IP5 (A)
Gradient at injection in IP1 and IP5 (T/m) *
Current at injection in IP2 And IP8 (A)
Gradient at injection in IP2 and IP8 (T/m) *
Current at collision (A)
Gradient at collision (T/m) *
Minimal operational current (A)
Maximal operational current (A)
* Assuming nominal magnetic lenght
6.37
1.9
70
418.2
13.2
452.9
14.3
6811
205
418.2
6811
2. Layout
Slots and positions: the 16 MQXA are allocated to 8 Q1 and 8 Q3 positions (see Fig. 2),
according to the Table II, which refers to as installed in 1-9-2008. From this list, cold masses
11 and 20 are absent. Cold masses 9 and 19 are spares.
Table II: Slot allocation, cryostat name and cold mass id for the MQXA.
Slot
MQXA.1R1
MQXA.3R1
s (m)
26.2
50.2
Cryostat
Magnet
Cold mass id
HCLQXA_001-FL000006 HCMQXA_001-FL000006
6
HCLQXC_001-FL000008 HCMQXA_001-FL000018
18
MQXA.3L2 3282.2 HCLQXC_001-FL000001 HCMQXA_001-FL000007
7
MQXA.1L2
MQXA.1R2
MQXA.3R2
MQXA.3L5
MQXA.1L5
MQXA.1R5
MQXA.3R5
3306.2
3358.5
3382.5
13279.3
13303.3
13355.6
13379.6
HCLQXA_001-FL000007
HCLQXA_001-FL000009
HCLQXC_001-FL000002
HCLQXC_001-FL000004
HCLQXA_001-FL000003
HCLQXA_001-FL000005
HCLQXC_001-FL000007
HCMQXA_001-FL000008
HCMQXA_001-FL000010
HCMQXA_001-FL000012
HCMQXA_001-FL000014
HCMQXA_001-FL000003
HCMQXA_001-FL000005
HCMQXA_001-FL000017
8
10
12
14
3
5
17
MQXA.3L8
MQXA.1L8
MQXA.1R8
MQXA.3R8
23265.2
23289.2
23341.5
23365.5
HCLQXC_001-FL000003
HCLQXA_001-FL000001
HCLQXA_001-FL000004
HCLQXC_001-FL000006
HCMQXA_001-FL000013
HCMQXA_001-FL000001
HCMQXA_001-FL000004
HCMQXA_001-FL000016
13
1
4
16
MQXA.3L1 26608.7 HCLQXC_001-FL000005 HCMQXA_001-FL000015
MQXA.1L1 26632.7 HCLQXA_001-FL000002 HCMQXA_001-FL000002
spare
HCMQXA_001-FL000009
spare
HCMQXA_001-FL000019
15
2
9
19
-2-

Circuits: the two MQXA Q1 and Q3 are powered in series with the Q2a and Q2b. An
additional power converter feeds Q2a and Q2b to make them reach the nominal
current of ~12 kA. A trim is available for Q1.
Fig. 2: Schematic lay-out [1] of the triplet (interaction point is 23 m on the right of Q1).
3. Measurements
3.1 ROOM TEMPERATURE MAGNETIC MEASUREMENTS
Device: Measurements are done with a rotating coil 600 mm long, in 35 positions spaced by
approximately 100 mm. First and last positions have a transfer function of approximately one
third of the body. Positive and negative currents of 10 A are used.
Available and missing measurements: we have two sets of warm magnetic measurements, one
before and one after thermal cycle. Measurements of cold masses 2 to 19 are present; cold
masses 1 and 20 are missing. After thermal cycle, measurement of cold mass 12 is missing.
Rejected or faulty measurements: there is indication that some measurements are affected by
sign problems. For instance, b4 has a wrong sign in cold mass 16 and 17 before thermal cycle,
and in cold mass 16 after thermal cycle. The magnetic length of cold mass 7 before thermal
cycle should be rejected (52 units larger than average).
Use of the measurements in FiDeL: there is no direct use in FiDeL since we have
measurements in cold conditions of all magnets. Warm measurements can be used for data
validation and cross-check, or in case of doubts on measurements at 1.9 K.
3.2 “INTEGRAL” MEASUREMENTS AT 1.9 K
Device: Harmonics coil of 599 mm length, displaced every 300 mm [3]. Measurements are
post processed, giving a central position, 5400 mm long, and two heads positions, with a
magnetic length of 340 mm and 620 mm respectively (see Fig. 3). The whole coil is covered.
Available and missing measurements: we have six sets of measurements, corresponding to
different currents, namely 392 A, 2011 A, 3207 A, 6134 A, 6677 A, 7228 A. Measurements
start 1 hour after reaching target current. Measurements of cold masses 2 to 19 are present;
cold masses 1 and 20 are missing.
Pre-cycle: Flat-top at 7350 A and reset at 50 A, with 10 A/s ramp rate (see Fig. 4).
-3-
Fig. 3: Position of measuring coils in “Integral” measurements.
Fig. 4: Pre-cycle and cycle for “Integral” measurements.
Rejected or faulty measurements: Cold mass 11 has a missing measurement at 7228 A – this
is not relevant since this current is well beyond nominal. Moreover there are two sets of
measurements, files labelled 135-139 have been retained (The other set 55-58 has TF different
up to ±5 units and b6 different up to -0.3 units). Cold mass 3 has three sets of data,
measurements, files labelled 127-132 have been retained (The second set 42-47 has TF
different up to 4 units and b6 different up to -0.05 units; the third set 133-136 has TF different
up to 6 units). Cold mass 2 has two sets of measurements, but the set relative to cold mass 2b
has been retained.
Use of the measurements in FiDeL: These are the reference measurements for the integrals in
a machine cycle. In order to have a detailed model as a function of the current the “DCloop”
set is also used. This second set has a much denser scan of the current, but is not covering the
whole magnet (see next section), so in principle should be less precise.
3.3 “DCLOOP” MEASUREMENTS AT 1.9 K
Device: 600-mm-long rotating coil. Measurements are done in three central positions, and two
coil heads [3]. This gives a sample of the straight part, and a measurement of the heads (see
Fig. 5). The two heads positions have a TF of ~55% w.r.t. the TF in the body.
-4-
0
1
2
3
4
5
6
7
8
9
Fig. 5: Measuring positions of rotating coils in “DCloop” measurements.
Available and missing measurements: For each loop we have around 20 current in the ramp
up, from 50 A to 7400 A. Current is ramped down and up two times (see Fig. 6).
Measurements are taken at ~50 currents. Measurements in the heads are missing for cold
masses 2-3-4-5-6-7. Measurements start 4 minutes after reaching target current.
8000
7000
Current (A)
6000
5000
4000
3000
2000
1000
0
19:50
21:50
23:49
1:49
Time (h)
3:49
Fig. 6: Current cycle used in “DCloop” measurements.
Pre-cycle: flat-top at 7395 A, ramp down to 50 A, with 10 A/s ramp rate. Then the
measurements are carried out for one and a half cycle, see Fig. 6.
Rejected or faulty measurements: Nothing to report.
Use of the measurements in FiDeL: These measurements give a fine track of transfer function
and multipoles along the loadline. The absolute value is questionable since the measure does
not cover all magnet. These data have to be used as difference with respect to a reference
value that can be established with “Integral” data. They can also be used to cross-check
“Integral”.
-5-
3.4 DECAY MEASUREMENTS
Device: Measurements are done with a rotating coil 600 mm long, in one central position [3].
Measurements are taken every ~3.7 minutes, for ~6 h.
Available and missing measurements: we have two sets of decay measurements, one close to
injection (390 A) and one close to collision (7184 A). Measurements of cold masses 2 to 19
are present; cold masses 1 and 20 are missing. Measurement of cold mass 3 at collision is
missing. Seven measurements are available for cold mass 4 at collision.
Pre-cycle: The pre-cycle is the same for the “Integral” measurements, with measurements
starting right after the reset at 50 A (see Fig. 7).
Fig. 7: Pre-cycle and cycle for “Integral” measurements.
Rejected or faulty measurements: Nothing to report.
Use of the measurements in FiDeL: Measurements are used to


Have an estimate of the stability of the transfer function at collision, of b6, and of the
low order multipoles.
Estimating the decay of transfer function and field harmonics at injection.
4. Transfer function
4.1 GEOMETRIC
The geometric has to be taken at 1500 A (see Fig. 8), average of current ramping up and
down. Saturation starts to be significant at 3000 A, and at maximum current is about 5% (500
units). Two sets of measurements are available: “Integral” and “DCloop”. The “Integral”
measurement is taken as the reference, and the geometric is evaluated from the “DCloop”
measurement, where both branches are measured.
We checked the consistency between the two sets of measurements “DCloop” and “Integral”
at two currents, namely 2010 A and 7230-40 A, ramp up. We find a systematic difference of
about 4 units with a spread of 2.5 units (one sigma) in both cases: this means that the
“DCloop” TFs are about 4 units larger than the “Integral” one (see Table III). The “Integral”
cover all the magnet, but not the “DCloop”: indeed, there should be no systematic difference
between the two; the 4 units can be considered as the absolute error (calibration) of the
measurement system. The spread of 2.5 units can be considered as the reproducibility of the
measurement system, or of the magnet itself.
-6-
MQXA19
0.00345
MQXA - Integral
0.00340
TF (T m / A)
MQXA - DCloop
0.00335
0.00330
0.00325
0.00320
0
2000
4000
Current (A)
6000
8000
Fig. 8: Integrated transfer function versus current during “DCloop” measurements for cold mass 19, and
comparison with “Integral” measurements.
Table III: Comparison of the TF at 2010 A and at 7230-40 A, DCloop and integral measurements.
Integral
Cold mass Current (A) TF (Tm /A)
2
2011.2
0.0034027
3
2011.3
0.0034046
4
2011.2
0.0034063
5
2011.3
0.0034046
6
2011.2
0.0034053
7
2011.4
0.0034048
8
2011.3
0.0034054
9
2011.3
0.0034053
10
11
2011.3
0.0034066
12
2011.3
0.0034041
13
2011.4
0.0034042
14
2011.4
0.0034037
15
2011.3
0.0034046
16
2011.3
0.0034040
17
2011.4
0.0034037
18
2011.3
0.0034041
19
2011.4
0.0034031
Dcloop
Diff.
Integral
Current (A) TF (Tm /A) (units) Current (A) TF (Tm /A)
2011.2
0.0034057
-8.8
7227.9
0.0032373
2011.2
0.0034051
-1.5
7228.0
0.0032387
2011.3
0.0034062
0.0
7227.9
0.0032393
2011.3
0.0034051
-1.4
7228.0
0.0032387
2011.3
0.0034058
-1.3
7227.9
0.0032392
2011.4
0.0034053
-1.5
7228.1
0.0032396
2011.3
0.0034067
-3.9
7228.1
0.0032399
2011.3
0.0034069
-4.5
7228.0
0.0032401
7228.1
0.0032389
2011.3
0.0034085
-5.5
2011.3
0.0034055
-4.3
7228.1
0.0032388
2011.4
0.0034046
-1.0
7228.1
0.0032383
2011.4
0.0034055
-5.3
7228.0
0.0032397
2011.3
0.0034056
-2.9
7228.1
0.0032402
2011.3
0.0034059
-5.5
7228.0
0.0032403
2011.3
0.0034060
-6.9
7228.1
0.0032404
2011.3
0.0034062
-6.0
7228.1
0.0032408
2011.4
0.0034051
-5.9
7228.1
0.0032402
Average
Stdev
(units)
(units)
-3.9
2.5
Dcloop
Diff.
Current (A) TF (Tm /A) (units)
7244.0
0.0032403
-9.4
7244.0
0.0032395
-2.7
7244.0
0.0032413
-6.1
7244.1
0.0032399
-4.0
7244.0
0.0032408
-4.9
7244.1
0.0032401
-1.5
7244.0
0.0032415
-5.1
7244.0
0.0032420
-6.0
7244.0
0.0032408
-5.9
7244.0
7244.2
7244.1
7244.1
7244.1
7244.1
7244.1
7244.1
0.0032407
0.0032394
0.0032404
0.0032407
0.0032410
0.0032412
0.0032414
0.0032407
-5.8
-3.3
-2.3
-1.6
-2.0
-2.5
-1.7
-1.7
Average
Stdev
(units)
(units)
-3.9
2.2
The geometric evaluated from “DCloop” measurements is given in Table IV: the average of
over the 18 magnets is 0.0034117 Tm/A, with a very low spread of 2.8 units. The model has a
different geometric for each magnet.
-7-
Table IV: Geometric values, computed as average up-down at 1500 A from “DCloop” measurements.
Cold mass
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
TF (Tm /A)
0.0034055
0.0034049
0.0034058
0.0034048
0.0034056
0.0034049
0.0034065
0.0034066
0.0034055
0.0034085
0.0034050
0.0034043
0.0034055
0.0034055
0.0034057
0.0034059
0.0034062
TF (units)
-0.6
-2.3
0.6
-2.6
-0.3
-2.1
2.5
2.8
-0.4
8.3
-1.9
-4.1
-0.4
-0.4
0.1
0.8
1.5
19
0.0034051
-1.6
Average
Stdev
0.0034057
0.0000009
0.0
2.8
4.2 STATIC COMPONENTS
The static part is computed using the “Integral” measurements, and the geometric has been
already evaluated from the “DCloop” measurements. A different fit is given for each magnet.
The example of cold mass 19 is given in Fig. 9. Here we also show the “DCloop” data that
have been used to compute the geometric, i.e., to set the zero of the vertical scale in units. The
fit is done according to the equation
𝑇𝐹 = 𝛾 𝑔𝑒𝑜 + 𝑇𝐹 𝑠𝑎𝑡 + 𝑇𝐹 𝑑𝑐_𝑚𝑎𝑔
𝑇𝐹 𝑠𝑎𝑡 = −
𝜎𝑚
𝐼 − 𝐼0
[1 + 𝑒𝑟𝑓 (𝑆
)]
2
𝐼𝑛𝑜𝑚
𝑞𝑚
𝑇𝐹
𝑑𝑐_𝑚𝑎𝑔
𝐼𝑖𝑛𝑗 2−𝑝𝑛 𝐼𝑐 − |𝐼|
= 𝜇𝑚 ( )
(
)
|𝐼|
𝐼𝑐 − 𝐼𝑖𝑛𝑗
Saturation: the fit is based on the four measurement points “Integral” at 2000, 3000, 6000 and
6700 A, all the other data being neglected. Inom is taken as 1 A. The measurement at injection
is neglected since this part is taken into account with magnetization. The measurement at 7150
is neglected since it is well beyond nominal. In this way the fit is optimized from 2000 to
6700 A: the fit error is negligible (less than 1 unit, one sigma), see Table V. An estimate of
the precision of our model, including calibration, is about 5 units in average over all magnets,
and up to 10 units for each individual magnet, as estimated in Table III. Please note that for
currents larger than 6700 A the fit underestimates the measurements: at 7228 A the fit
underestimates measurements of about 12 units.
-8-
MQXA19
TF (Units)
100
0
MQXA - integral
-100
MQXA - DCloop
-200
saturation
-300
DC magnetization
-400
-500
-600
0
2000
4000
6000
Current (A)
8000
10000
Fig. 9: Fit of the TF using “Integral” from 2000 A to 6700 A, and comparison to “DCloop” measurements.
Table V: Error of the fit (units) for different current and cold masses, w.r.t. “Integral” measurements, high field
Parameters
cold mass
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
s
0.00018730
0.00017876
0.00018051
0.00017875
0.00017879
0.00017365
0.00017544
0.00018225
0.00018220
0.00017383
0.00017706
0.00017362
0.00017368
0.00017709
0.00017369
0.00017881
0.00017882
0.00017877
sig
0.000480
0.000520
0.000520
0.000520
0.000520
0.000540
0.000520
0.000500
0.000500
0.000520
0.000520
0.000540
0.000520
0.000520
0.000520
0.000500
0.000500
0.000500
Fit error
I0
5470
5450
5460
5450
5450
5390
5360
5460
5460
5340
5400
5370
5320
5400
5340
5400
5400
5420
Average
Stdev
3208 A
0
1
1
0
0
1
0
-1
-1
0
0
1
-1
1
0
-1
-1
-1
6134 A
1
1
0
0
1
0
0
1
1
0
0
-1
-1
0
0
0
0
1
6677 A
-1
0
-1
0
1
0
1
0
1
0
1
0
1
-1
1
0
-1
0
7228 A
8
13
10
13
14
16
17
10
11
-11
14
17
18
11
17
12
12
12
0.0
0.8
0.2
0.5
0.1
0.7
11.9
6.4
Magnetization: The DC magnetization component is fit on the “DCloop” measurements – this
measurement shows a magnetization component systematically larger than the integral
measurements (about 10 units). The fit is done on the two measurements at 300 A and 500 A,
i.e. around the injection current of 418 A- 450 A. The error is obviously low, having so few
points to fit. The parameter q is set to zero; the parameter p is around 0.6 (see Table VI).
-9-
Table VI: Error of the fit (units) for different current and cold masses, w.r.t. “Integral” measurements, injection
Parameters
Fit error
cold mass
2
3
4
m
0.0000100
0.0000098
0.0000094
p
0.6
0.6
0.6
300 A
-2
-2
-2
500 A
-4
-3
-3
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0.0000101
0.0000094
0.0000094
0.0000088
0.0000085
0.0000089
0.0000094
0.0000069
0.0000077
0.0000080
0.0000087
0.0000092
0.0000082
0.0000086
0.0000082
0.6
0.6
0.7
0.5
0.5
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.6
-2
-1
-2
-3
-3
-2
-2
-1
-2
-3
-3
-4
-4
-3
-1
-4
-4
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
Average
Stdev
0.0000088
0.0000008
0.6
0.1
-2
0.8
-3
0.3
4.3 DYNAMIC COMPONENTS
Decay at injection: measurements are taken at 392 A, i.e., about 25 to 60 A lower than
nominal. The transfer function has a decay ranging from 1 to -4 units after 20 minutes, and
ranging from 2 to -8 units after 200 minutes (see Fig. 10). The average has a systematic offset
of about -1.5 units after 20 minutes, and -2.0 units after 200 minutes. This effect is neglected.
6
4
db2 (units)
2
0
-2
-4
-6
-8
-10
0
25
50
75
100
125
150
175
Time (minutes)
200
225
250
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
Fig. 10: Decay of integrated transfer function at injection (392 A).
Decay at high field: measurements are taken at 7228 A, i.e., well above high field current.
As expected, there is no decay (see Fig. 11 for cold mass 19). The transfer function is stable
within one unit, multipoles within 0.1 units.
- 10 -
3
2
db2 (units)
1
0
-1
-2
-3
0
20
40
60
80
100
120
140
Time (minutes)
Fig. 11: Decay of integrated transfer function at 7228 A for cold mass 19.
5. Field errors
In a quadrupole coil, allowed harmonics are b6 and b10. In this case we compute a
geometric, a persistent and a saturation component for each magnet. Higher order as b10 are
treated as not allowed, i.e. only the geometric is given.
For the MQXA, a deformation induced by the yoke creates a systematic b4 of about 1 unit
[2]. This field error has been considered as compatible with the beam dynamics requirements,
and therefore no correction has been carried out during the production.
5.1 ALLOWED: GEOMETRIC AND STATIC COMPONENTS
Geometric: the allowed harmonics have been carefully optimized at high field: average b6
is about 0.3 units, with a spread of 0.09 units, and b10 is zero within 0.01 units (see Table VII).
For each magnet we set the geometric b6 and b10 as the values, ramp up, at 6700 A, evaluated
in the “Integral” measurements, and the saturation is set to zero. This is can be done since the
saturation is negligible, (for instance, it is less than 0.1 units for b6). This ensures the best fit at
high field, where the beam dynamics is more sensitive to these magnets.
DC Magnetization: In average, the persistent current component of b6 is about 1.1 units,
with a spread of about 0.24 units (see Table VII). This is taken into account by a generic term,
the same for every magnet:
𝑞
𝐼𝑖𝑛𝑗 2−𝑝6 𝐼𝑐 − |𝐼| 6
𝑏6 = 𝜇6 ( )
(
)
𝐼
𝐼𝑐 − 𝐼𝑖𝑛𝑗
The fit is done on the average of the “DCloop” measurements (see Fig. 12), with m6=-1.45,
p6=0.4 and q6=1.5. Modeling this contribution as a generic term, and not magnet by magnet,
one has to add for each magnet at injection an uncertainty of 0.24 units (1 sigma, see Table
VII). The b10 component of the persistent current is less than 0.1 units and is therefore
neglected.
- 11 -
Table VII: Average and standard deviation of allowed harmonics at ~injection, intermediate and high field, and
average and standard deviation between injection and high field.
392 A
3207 A (3.3 TeV)
6177 A (6.3 TeV)
6677 A (6.8 TeV)
6677 A - 392 A
b6
Ave -0.81
Std 0.22
Ave 0.38
Std 0.08
Ave 0.33
Std 0.08
Ave 0.33
Std 0.09
Ave 1.13
b10
0.04
0.01
-0.01
0.00
-0.01
0.01
-0.01
0.01
-0.06
Std
0.01
0.24
0.5
b6 (units)
0.0
Integral
Dcloop
Fit
-0.5
-1.0
-1.5
0
2000
4000
Current (A)
6000
8000
Fig. 12: Average of offset in b6 w.r.t. values at high field, “Integral” measurements, “DCloop” measurements,
and fit of the “DCloop” measurements.
5.2 NOT ALLOWED: GEOMETRIC
Since the MQXA dominate the dynamics after the squeeze, also for the non allowed
harmonics we propose to set the geometric on the value measured at high energy, i.e. at the
squeeze, ramp up, and to have no persistent or saturation component. In this way the highest
precision is reached at high field, which is the most relevant current.
7 TeV correspond to 6811 A. Here we use the measurement “integral” taken at the closest
value of the current, i.e. 6677 A. An overview of average and standard deviations are given in
Table VIII. All harmonics have an average close to zero (within a fraction of unit) with the
exception of b4 and a3. For b4 one has a systematic component of about 1.3 units with a low
spread (0.11 units). This is believed to be due to the mechanical impact of the horizontally
split yoke. The other multipole that has a non zero systematic component is a3; in the
hypothesis of a Gaussian distribution with stdev 0.37, the sigma of the average of a
distribution of 18 magnets is 0.09, and the measured average is 0.21, i.e. more than two sigma.
This origin of this slight systematic asymmetry is not known.
A set of measurements at 6177 A, corresponding to 6.5 TeV, and at 3207 A (3.3 TeV)
shows a negligible dependence of the multipoles on the energy in this range (see Table VIII).
Therefore one should not expect variations from 4 to 7 TeV runs.
Our model assumes to have at injection the same values as at high field. The validity of this
assumption is checked in the last two rows of Table VIII. The average of each multipole
- 12 -
change between injection and high field, taken over the set of magnets is zero within a
fraction of unit with the exception of a4, which is a rather large negative systematic
component (1.3 units) at injection. The spread at injection is always larger than the spread at
high field, as expected. The last line of the Table gives that spread to be added to MAD
simulations to correctly model the injection values. The model is given individually for each
magnet (see Table IX).
Table VIII: Not allowed multipoles, average and spread over the 18 magnets, at four different currents
(“Integral” data).
392 A
3207 A (3.3 TeV)
6177 A (6.3 TeV)
6677 A (6.8 TeV)
6677 A - 392 A
b3
Ave -0.12
Std 1.05
Ave 0.01
Std 0.28
Ave 0.03
Std 0.30
Ave 0.04
Std 0.31
Ave 0.17
b4
1.28
0.15
1.24
0.11
1.28
0.11
1.30
0.11
0.02
b5
0.04
0.31
0.00
0.04
-0.01
0.04
0.00
0.04
-0.05
b7
0.00
0.03
0.00
0.01
0.00
0.01
0.00
0.01
0.00
b8
0.03
0.02
0.02
0.00
0.02
0.00
0.02
0.00
0.00
b9
-0.01
0.02
0.00
0.00
0.00
0.01
0.00
0.01
0.00
a3
0.35
1.01
0.21
0.35
0.21
0.36
0.21
0.37
-0.14
a4
-1.32
1.27
-0.06
0.26
-0.02
0.27
-0.02
0.28
1.30
a5
0.09
0.27
0.02
0.04
0.01
0.04
0.01
0.04
-0.08
a6
-0.01
0.04
-0.03
0.02
-0.03
0.02
-0.03
0.02
-0.02
a7
-0.01
0.03
0.00
0.01
0.00
0.01
0.00
0.01
0.01
a8
0.00
0.02
0.00
0.01
0.00
0.01
0.00
0.01
0.00
Std
0.14
0.29
0.03 0.02
0.02
1.08
1.26
0.29
0.04
0.03
0.02 0.01
1.08
a9
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Table IX: Geometric used in FiDeL for MQXA magnets (values at 6677 A, ramp up, “Integral” data).
Cold mass
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
b3
0.65
-0.14
-0.54
-0.17
-0.39
0.06
0.55
0.04
-0.04
0.21
-0.01
-0.25
0.29
0.17
0.15
0.07
0.37
-0.29
b4
1.13
1.27
1.23
1.21
1.26
1.45
1.54
1.23
1.34
1.17
1.27
1.22
1.30
1.37
1.48
1.39
1.38
1.23
b5
-0.04
-0.02
0.00
-0.04
-0.02
0.08
-0.02
-0.01
0.01
-0.01
-0.07
0.01
0.00
-0.08
0.00
0.05
0.03
0.02
b7
0.00
-0.01
-0.01
-0.01
-0.02
0.00
0.01
0.00
0.00
0.01
-0.01
0.01
0.01
0.01
0.00
0.01
0.02
-0.01
b8
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.03
0.02
0.02
0.02
0.02
0.03
0.02
0.02
0.02
b9
0.00
0.00
0.01
0.01
0.00
0.00
-0.02
-0.01
-0.01
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
a3
-0.05
0.18
0.76
0.46
0.32
0.08
-0.12
0.02
0.87
0.57
0.06
-0.11
0.17
-0.22
0.78
0.42
0.01
-0.45
a4
0.23
-0.34
-0.42
-0.02
-0.08
0.16
-0.44
0.67
-0.25
0.12
-0.14
0.04
0.18
0.13
-0.05
-0.31
0.13
0.03
a5
0.04
-0.03
-0.01
0.05
0.00
0.02
-0.08
-0.04
0.07
-0.02
0.05
-0.01
0.00
0.01
0.04
0.04
-0.03
0.03
a6
-0.03
-0.03
0.00
0.02
-0.05
-0.03
-0.03
-0.06
-0.05
-0.02
0.00
-0.01
-0.05
-0.01
-0.07
-0.03
-0.01
-0.03
a7
0.00
0.01
0.00
-0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
0.00
-0.01
-0.02
a8
0.00
-0.01
-0.01
0.00
0.00
0.01
-0.01
0.03
0.00
0.02
-0.02
-0.01
0.02
-0.01
-0.01
-0.01
0.01
-0.01
5.3 DYNAMIC COMPONENTS
Decay at injection: measurements are taken at 392 A, i.e., about 25 to 60 A lower than
nominal. The b6 decay after 20 minutes ranges from 0.2 to 0.9 units (see Fig. 13), and up to
1.3 units after 250 minutes. The b4 has decay after 20 minutes ranging from -0.1 to 0.1 units
(see Fig. 14). We make a fit of the b6 component according to the equation
𝑏6 (𝑡) = 𝑐(𝑑𝑒 −𝑡/𝜏 +(1 − 𝑑)𝑒 −𝑡/9𝜏 )
The time constant is of the order of 500 s, the amplitude c between 0.5 and 1.3, and the error
of the fit of the order of 0.01 units (see Table XI). An example of the fit for cold mass 19 is
given in Fig. 15. All the other field harmonics are neglected. In average, all not allowed
- 13 -
harmonics have a zero decay: the only exception is a4, with -0.6 units (see Fig. 16). This is
consistent with the behaviour at injection, where an (unexplained) persistent current
component of about 1.3 units was observed. Half of this component decays during the
injection plateau. Systematic persistent current components on not allowed multipoles have
been observed also in the dipoles.
After 200
minutes
After 20
minutes
Table X: Decay at injection (392 A) after 20 minutes and after 200 minutes.
b3
a3
b4
a4
b5
a5
b6
a6
b7
a7
b8
a8
b9
a9
b 10
-0.02
Ave
-0.01
-0.02
-0.02
-0.57
0.02
0.00
0.58
0.01
0.00
0.00
0.00
0.00
0.00
0.00
Stdev
0.40
0.32
0.09
0.53
0.13
0.11
0.13
0.03
0.01
0.01
0.01
0.01
0.00
0.00
0.00
Min
-0.79
-0.74
-0.23
-1.61
-0.26
-0.24
0.26
-0.07
-0.02
-0.02
-0.02
-0.02
-0.01
-0.01
-0.03
Max
0.87
0.49
0.10
0.50
0.24
0.23
0.79
0.05
0.04
0.02
0.01
0.01
0.01
0.01
-0.01
Ave
0.12
-0.08
0.01
-0.66
0.01
0.00
0.88
0.01
0.01
-0.01
0.00
-0.01
0.00
0.00
-0.04
Stdev
0.64
0.44
0.15
0.81
0.20
0.20
0.23
0.04
0.02
0.03
0.01
0.02
0.01
0.01
0.01
Min
-1.26
-0.85
-0.33
-1.85
-0.47
-0.34
0.45
-0.04
-0.02
-0.06
-0.02
-0.04
-0.02
-0.01
-0.05
Max
1.52
0.75
0.30
1.18
0.29
0.47
1.27
0.07
0.06
0.05
0.02
0.03
0.01
0.01
-0.02
1.4
1.2
db6 (units)
1.0
0.8
0.6
0.4
0.2
0.0
0
25
50
75
100
125
150
175
Time (minutes)
Fig. 13: Decay of b6 at injection (392 A).
- 14 -
200
225
250
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
0.4
0.3
0.2
db4 (units)
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0
25
50
75
100
125
150
175
Time (minutes)
200
225
Fig. 14: Decay of b4 at injection (392 A).
Table XI: Fit parameters for decay at injection (392 A) for b6.
t
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
480
420
900
480
300
420
420
480
480
420
480
360
480
480
540
540
540
480
d
0.71
0.69
0.68
0.85
0.55
0.68
0.65
0.55
0.50
0.53
0.56
0.37
0.65
0.71
0.60
0.60
0.86
0.69
- 15 -
c
0.82
0.86
0.48
0.69
0.88
0.91
0.98
1.30
1.29
1.20
1.09
1.11
0.88
0.66
0.62
0.90
0.69
0.87
err
0.009
0.010
0.004
0.010
0.006
0.009
0.015
0.013
0.015
0.010
0.008
0.017
0.007
0.009
0.007
0.011
0.010
0.010
250
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1.00
0.90
0.80
db6 (units)
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
50
100
150
200
250
300
Time (minutes)
Fig. 15: Decay of b6 at injection in cold mass 19, and fit (392 A).
3
2
da4 (units)
1
0
-1
-2
-3
-4
0
25
50
75
100
125
150
175
Time (minutes)
Fig. 16: Decay of a4 at injection (392 A).
- 16 -
200
225
250
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
6. Summary and open issues






The 16 MQXA have the same transfer function within ±7 units (see Table IV). We
model them individually. The relative precision (i.e. from magnet to magnet) of the
model at 1-7 TeV (i.e. well above injection) is in this range.
Decay of the transfer function at injection is within [-10,+4] units (see Fig. 10). It is
not modelled. Therefore the relative precision (i.e. from magnet to magnet) at injection
can be ±15 units.
The large saturation component (5%) becomes visible after 3000 A – it is modelled
through the usual FiDeL fit, with a very low error (one unit).
The magnetization component is of the order of 30 units at injection. It is modelled
individually. Here the error is probably higher: the two sets of measurements
“integral” and “dcloop” have a discrepancy of about 10 units, which is the precision of
the model.
Data of cold mass 1 are not available on the database, and have been replaced in
FiDeL with data of cold mass 2.
The origin of the systematic component (about 1.3 units) of persistent current in a4,
and its decay, are unknown. They should be not critical, since present at injection only.
Ackowledgements
We wish to thank R. Ostoijc for useful remarks and comments.
References
[1] O. Bruning, et al., CERN Report 2004-003 (2004).
[2] Y. Ajima, et al., Nucl. Instrum. Meth. A 550 (2005) 499.
[3] N. Ohuchi, presentation given at the CERN-FNAL-KEK Joint meeting, March 2003.
Appendix A. Field quality in the heads
After the squeeze, the triplet dominate the optics and the dynamics, and the variation of the
beat function in the triplet is large. For this reason, the head contribution, if strongly different
from that one of the straight part, can be weighted differently and the simple use of the
harmonics integral to model these magnets can be not appropriate. For this reason, we account
of the field quality in the heads, as measured in the “Integrals” measurements.
Table XII: Field harmonics in the heads, CS: connection side, with 0.34 m magnetic length - NCS: non
connection side, with 0.62 mm magnetic length.
CS, ml=0.34 m
NCS, ml=0.62 m
ave
stdev
ave
stdev
b3
-0.26
1.20
0.21
1.20
b4
1.17
0.14
2.07
0.14
b5
0.01
0.17
0.05
0.17
b6
-0.54
0.10
2.59
0.10
- 17 -
b7
0.00
0.02
-0.01
0.02
b10
-0.08
0.01
-0.06
0.17
a3
0.24
1.20
0.59
0.01
a4
0.08
0.26
0.04
0.02
a5
0.04
0.14
-0.06
0.01
a6
-0.06
0.04
0.09
0.01
a7
0.00
0.01
0.03
0.00
a8
0.00
0.01
0.00
1.20