LHC Project Note XXX
2009-06
laurent.deniau@cern.ch
Magnetic model of Main Quadrupoles
L. Deniau
CERN, Technology Department
Keywords: Superconducting Magnets, Magnetic Field Model, Harmonics, LHC.
1. Introduction
Function in the machine: Quadrupole magnets (four poles) produce a magnetic field gradient,
i.e, a magnetic field which is zero on the reference orbit and whose magnitude linearly grows
with the radial distance from it (see Fig. 1., left). A charged particle traveling longitudinally in
a quadrupole magnet experiences a magnetic Lorenz force F = q v × B proportional to the
magnetic field that either pushes the particle toward the magnetic axis or expel the particle
away from the magnetic axis, depending on the field direction. Hence, the effect of a
quadrupole magnet on particle beam is to focus in one plane (e.g. horizontal) and defocus in
the other plane (e.g. vertical). By alternating focusing quadrupole magnets F with defocusing
quadrupole magnets D (same magnet rotated by 90 degrees), we obtain a strongly focusing
FODO cell, where O stands for Open space (no gradient field) which is filled with bending
magnets (dipole magnets).
The LHC MQs are superconducting quadrupoles designed with two 56 mm apertures (see Fig.
1, right), a magnetic length of 3.10 m and an operational current of 11870 A delivering a
nominal magnetic field gradient of 223 T/m. The measured strength is in perfect agreement at
3.10-4 with the nominal gradient.
Fig. 1: Field lines in two consecutive quadrupoles (left) and cross-section of the LHC MQ (right).
This is an internal CERN publication and does not necessarily reflect the views of the LHC project management.
Table I: Summary of MQ manufacturing parameters.
Nominal magnetic Length
Operationnal temperature
Aperture diameter
Minimal operationnal current
Maximal operationnal current
Nominal integrated strength at nominal current
Measured integrated strength at nominal current
[m]
[K]
[m]
[A]
[A]
[T]
[T]
3.1
1.9
0.056
650
11870
691.3
691.5
Numbers and variants: we have 360 MQs installed into the arc cells (2 MQs per arc cell)
constituting the main components of the 8 machine sectors, and 32 MQs installed into the
insertion regions (IR 1-8: 2 MQs in Q11, IR 3/7: 8 MQs in Q7-10) for a total of 392 MQs
installed, plus 18 MQs spares for a total of 410 MQs produced. They are all of the same type
but the two apertures of a MQ are connected on two different circuits per sector, one focusing
(RQF) and the other defocusing (RQD), including the MQs in the insertion regions. The goal
of these connections is to make the FODO cells with only one type of 2-in-1 magnet (no
rotation allowed) by swapping the aperture circuits between each pair of consecutive MQs.
With this configuration, beam 1 will see a MQF followed by a MQD while beam 2 will see a
MQD followed by a MQF when they both cross the same FODO cell. In the following of this
document, MQF refers to the focusing aperture in the horizontal plane (i.e. RQF circuit) and
MQD to the defocusing aperture in the horizontal plane (i.e. RQD circuit) of the same MQ
magnet, independently of the aperture or beam number.
Naming convention: During construction, cold masses HCMQ__S018-AC000xxx have been
identified by progressive numbers from 1 to 410. At the time of writing (2009-06) the spares
MQs are 008, 009, 070, 078, 079, 113, 124, 125, 139, 156, 167, 171, 252, 294, 306, 392, 402
and 409. The SSSs do not have the same name nor follow the same ids as MQs.
Expected operational cycles, range of current and operational temperature: The injection
current is 686 A for MQDs and 717 A for MQFs, corresponding to gradients of 12.9 T/m and
13.5 T/m respectively, see Table II. During the ramp the current increases with the beam
energy, following the main dipole magnets, and reach the nominal values of 10681 A for
MQDs and 11167 A for MQFs, corresponding to operational gradients of 200.8 T/m (11%
margin) and 209.9 T/m (6% margin) respectively. The operational temperature is 1.9 K.
Table II: MQD and MQF operational currents and gradients for injection and collision(s) energy.
E (GeV)
450
3500
5000
7000
I [A] G [T/m] I [A] G [T/m]
686
12.9
717
13.5
5534 100.4 5575 104.9
7625 143.4 7970 149.9
10681 200.8 11167 209.9
2. Layout
Slots and positions: the 360 MQs are spread over the machine arc cells (see Fig. 2). The
remaining 32 MQs are in the Q11 at all IRs, close to the arc cells in sector extremities, and in
Q7 to Q10 at IR3 (momentum cleaning) and IR7 (betatron cleaning). All MQs are allocated to
-2-
slots of type MQ.xxx in the machine layout description. A summary of the number of
quadrupoles in each sector is given in Table III.
Fig. 2: Schematic LHC arc cell with MQs marked as red boxes.
Table III: Summary of MQ per circuits.
Circuit
#MQs
RQF.A12
47
RQF.A23
51
RQF.A34
51
RQF.A45
47
RQF.A56
47
RQF.A67
51
RQF.A78
51
RQF.A81
47
Aperture
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
#MQs
24
23
26
25
25
26
24
23
24
23
25
26
26
25
24
23
Circuit
#MQs
RQD.A12
47
RQD.A23
51
RQD.A34
51
RQD.A45
47
RQD.A56
47
RQD.A67
51
RQD.A78
51
RQD.A81
47
Aperture
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
#MQs
23
24
25
26
26
25
23
24
23
24
26
25
25
26
23
24
Circuits: the 8 machine arc sectors (including MQs in IRs) from A12 to A81 have two circuits
each, one focusing RQF and one defocusing RQD, each powering half of the MQs apertures
connected in series. The LSA software is able to deal with the average of the RQF and RQD
circuits in order to power the arc cells as-if they were made of FODO cells (i.e. beam point of
view).
3. Measurements
3.1 ROOM TEMPERATURE MAGNETIC MEASUREMENTS
Device: Measurements are done with a rotating coil 750 mm long at 12.5 A (QIMM), in 5
positions spaced by approximately 750 mm to ensure no gap between measurements. First and
last positions have approximately 425 mm inside and 325 mm outside the magnet.
Available and missing measurements: All the MQs have been measured in warm condition.
Rejected or faulty measurements: none.
Use of the measurements in FiDeL: the room temperature measurements are used to estimate
the geometric at 12.5 A for the transfer function and for the harmonics in all the MQs,
including those measured in cold conditions. Warm-to-cold offsets (one for the transfer
function and one for each harmonic) are added to the room temperature measurements to
-3-
extrapolate the geometric at 5000 A as defined in FiDeL. Hence, the spread of the geometric
component over all MQs is the spread of the warm geometric (e.g. ~12 units for the transfer
function).
3.2 MEASUREMENTS AT 1.9 K: “LOADLINE”
Device: Measurements are done with rotating shafts made of six sectors which include five
700 mm tangential coils each, and spaced with a gap of 110 mm for a total length of 4.750 m.
Only five sectors were used to measure the integral field about 3.1 m long which means that
first and last positions have approximately 280 mm inside and 420 mm outside the magnet.
The SMA system extrapolated the missing magnetic field in the gap after feed-down
correction (i.e. magnetic alignment of sectors) to deliver the integral field. Some MQs have
also been measured with the Single Stretched Wire system for axis measurements and crosscalibration of the rotating coils for field gradient measurements.
Available and missing measurements: we have 18 (17) currents plateaux (400 A, [550 A,] 600
A, 760 A, 1000 A, 1500 A, 2000 A, 2500 A, 3000 A, 4000 A, 5000 A, 6000 A, 7000 A, 8000
A, 9000 A, 10000 A, 11000 A, 11850 A, [12000 A]) with dI/dt = 0 A/s during measurements.
The ramp rate between each plateau is dI/dt = ±10 A/s. For each plateau, two measurements
were performed and averaged (no decay correction) per ramp-up and ramp-down for each
aperture, for a total of 700 measurements per loadline as shown in Fig. 3. The measurements
performed during the plateau at 550 A were systematically removed during the standard
analysis because the integrator amplifiers had wrong automatic gain adjustment at this current,
leaving only 660 magnetic field measurements available for analysis. The measurements of
the plateau at 12000 A were not systematically performed for various reasons during series
tests (e.g. time). A total of 25 MQs or equivalently 50 apertures have been measured at 1.9 K,
see Table IV, but 11 apertures have been discarded for transfer function analysis, see Table V.
Table IV: List of magnets measured with Loadline cycle.
019
040
061
068
071
074
114
120
138
142
145
151
173
248
258
263
264
364
376
383
384
394
400
405
407
Table V: List of rejected apertures during transfer function analysis.
MQ
061
114
120
151
173
258
263
264
384
Motivation for rejection
body shift in ap2 + anormal saturation in ap1 (strong contribution from heads)
anormal saturation in ap1 (strong contribution from heads)
body shift in ap2 + anormal saturation in ap1 (strong contribution from heads)
anormal saturation in ap2 (strong contribution from heads)
anormal saturation in ap1 (strong contribution from heads)
flat-low curvature in ap2 in range 5000-10000A (small effect)
anormal saturation in ap2 (strong contribution from heads)
body shift in ap2
body shift in ap2
-4-
12000
10000
current [A]
8000
6000
4000
2000
0
0
1000
2000
3000
4000
5000
6000
time [s]
Fig. 3: Current profile during “loadline” measurements.
Cross-calibration of measurements: The TF measurements performed during the loadline
cycle have been cross-calibrated with TF measurements made with the single stretch wire
(SSW) system. When SSW measurements are available, the TF geometric has been adjusted
by applying the SSW-to-rotating coils offset on the entire Loadline cycle, for each magnet and
aperture (see Table VI). This offset in many cases is close to 17 units.
Table VI: List of SSW cross-calibration for Loadline cycle.
MQ
019
040
061
068
071
074
114
120
138
142
145
151
173
V1
-6
-3
-11
15
2
14
5
15
0
1.3
-2
2
5
V2
16
16
-28
2.5
-7
0
16
-14
13.5
16
17
18.3
19
MQ
248
258
263
264
364
376
383
384
394
400
405
407
V1
-1
-2
12
16
8
15
17
23
16
18
16
16
V2
16
12
12
-4
2
-3
-3
-6
-7
-10
-7
-6
Pre-cycle: 20 minutes top energy pre-cycle with 50 A/s ramp-up and ramp-down.
Rejected or faulty measurements: the details of rejected measurements can be found in the
standard analysis reports available on the FiDeL web repository for each MQs measured cold.
Use of the measurements in FiDeL: the Loadline cycle measurements are used to extract the
static components of the model, namely the warm-to-cold correlations necessary to
extrapolate the room temperature measurements to the geometric, the DC magnetization, the
residual magnetization and the saturation.
-5-
3.2 MEASUREMENTS AT 1.9 K: “MACHINE CYCLE”
Device: measurements are done with rotating shafts made of six sectors which include five
700 mm tangential coils each, and spaced with a gap of 110 mm for a total length of 4.750 m.
Only five sectors were used to measure the integral field about 3.1 m long which means that
first and last positions have approximately 280 mm inside and 420 mm outside the magnet.
The SMA system extrapolated the missing magnetic field in the gap after feed-down
correction (i.e. magnetic alignment of sectors) to deliver the integral field.
Available and missing measurements: we have 131 measurements points versus time,
decomposed in 34 points for injection plateau with dI/dt = 0 A/s (~1000 s), 52 points for the
10 A/s ramp-up (~25 min), 9 points for flat-top plateau with dI/dt = 0 A/s (~5 min) and 36
points for decreasing -20 A/s ramp down (~20 min). For each time point, one measurement
was performed per aperture, for a total of 1310 measurements per “machine cycle”, as shown
in Fig. 4. The ramp-up of the “machine cycle” follows the so called PELP curve (i.e.
Parabolic-Exponential-Linear-Parabolic) to minimize the decay and snap-back effect. For the
model, only the 340 measurements performed during the injection plateau with I = 760 A and
dI/dt = 0 A/s have been used to compute the decay component. A total of 21 MQs or
equivalently 41 apertures (magnet 071 has only measurements for aperture 1) have been
measured in a “machine cycle” (see Table VII) and no aperture have been discarded for
transfer function analysis. The only measurement with an injection plateau of 10000 seconds
was rejected to an abnormal behaviour of the decay component.
12000
10000
current [A]
8000
6000
4000
2000
0
0
500
1000
1500
2000
2500
3000
3500
4000
time [s]
Fig. 4: Current profile during “machine cycle” measurements.
Table VII: List of magnets with “machine cycle” measurements.
019
040
061
071 (1)
074
120
138
142
145
151
173
248
258
264
364
376
383
384
394
400
405
Pre-cycle: 20 minutes top energy pre-cycle with 50 A/s ramp-up and ramp-down.
-6-
Rejected or faulty measurements: the details of rejected measurements can be found in the
standard analysis reports available on the FiDeL web repository for each MQs measured cold.
Use of the measurements in FiDeL: the measurements are used to evaluate the dynamic
components of the model. The decay is implemented for b6, and is neglected for all other
components.
4. Transfer function
4.1 GEOMETRIC
The geometric is defined as the average ramp-up and ramp-down transfer function at 5000 A
during the “loadline” measurement. The persistent currents and DC magnetization
contribution start to be significant below 4500 A and the saturation contribution starts to be
significant above 6000 A with ~13 units at nominal current (11850 A), see Fig. 5 for a typical
case. The geometric component is estimated by adding the warm-to-cold offset of 28 units to
the room temperature measurements. This procedure is used also for the magnets measured at
1.9 K.
0.0009926
0.0009924
0.0009922
0.0009920
TF [T.m/A]
0.0009918
0.0009916
0.0009914
0.0009912
0.0009910
0.0009908
0.0009906
0.0009904
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 5: “Loadline” measurements of quadrupole MQ138 aperture 1.
Room temperature measurements
The room temperature measurements of all quadrupoles during the production show a rather
uniform behaviour, without trends (see Fig. 6). Average transfer function is 9.8910-4
Tm/A, and the standard deviation is about 12 units.
-7-
9.94E-04
aperture 1
9.93E-04
aperture 2
transfer function [T.m/A]
9.92E-04
9.91E-04
9.90E-04
9.89E-04
9.88E-04
9.87E-04
9.86E-04
9.85E-04
magnet number
Fig. 6: Transfer function measured at room temperature during the production with running average
Warm to cold extrapolation
The standard deviation of the room temperature and 1.9 K geometrics are respectively 10.9
units and 14.4 units (see Table VIII) for the population measured at cold: this means that the
standard deviation of the TF is dominated by the 1.9 K measurements, and there is a poor
warm-to-cold correlation, as shown in Fig. 7. Once MQ 061 (upper right point in Fig. 8) with
a warm-to-cold offset > 3σ of the rest of the population measured at cold has been removed,
the warm-to-cold offset becomes 28 units with σ of 11.8 units (table VIII, last row). This
warm-to-cold offset is used to extrapolate all the warm geometrics to 1.9 K for the
REFPARM.
Table VIII: Geometric warm-to-cold offset, including per aperture offset and spread for MQs measured at 1.9K.
GEO ALL
µ [T.m/A] σ [units]
ΔWC 2.96E-06 13.61
Warm 9.89E-04 10.92
Cold 9.92E-04 14.42
Covar.
8.42
ΔWC 28.14 units 11.81
GEO V1
µ [T.m/A] σ [units]
2.67E-06 11.23
9.89E-04
9.41
9.91E-04 13.36
8.39
-8-
GEO V2
µ [T.m/A] σ [units]
3.25E-06 15.31
9.89E-04 12.42
9.92E-04 15.31
8.78
0.000995
0.000994
TF Cold [T.m/A]
0.000993
0.000992
0.000991
0.000990
aperture 1
aperture 2
0.000989
0.000988
0.000987
0.000988
0.000989
0.000990
0.000991
TF Warm [T.m/A]
Fig. 7: 1.9 K TF measurements versus room temperature TF measurements.
0.000008
W/C Offsets Aperture 2 [T.m/A]
0.000007
0.000006
0.000005
0.000004
0.000003
0.000002
0.000001
0.000000
0.000000 0.000001 0.000002 0.000003 0.000004 0.000005 0.000006 0.000007
W/C Offsets Aperture 1 [T.m/A]
Fig. 8: Warm-to-cold transfer function offsets per aperture.
Geometric per circuit
The geometric is provided per aperture for each circuit (see Table IX) and for each magnet.
The geometric of magnets is given by the procedure previously described in this section and
the geometric of circuits is given by the average of their magnets geometric. The σ in the
following table includes only the magnets statistics, but not the uncertainty on the warm-tocold offset of the Table VIII. The σ of the RQ circuits over the all machine is about 3.9 units,
and all the TF relative to the different circuits are within ±10 units. The σ of all the magnets is
11.9 units which correspond to the standard deviation of the warm geometric.
-9-
Table IX: Geometric of transfer function, including per aperture offset and spread.
Circuit
ALL
RQD.A12
RQD.A12
RQD.A23
RQD.A23
RQD.A34
RQD.A34
RQD.A45
RQD.A45
RQD.A56
RQD.A56
RQD.A67
RQD.A67
RQD.A78
RQD.A78
RQD.A81
RQD.A81
RQF.A12
RQF.A12
RQF.A23
RQF.A23
RQF.A34
RQF.A34
RQF.A45
RQF.A45
RQF.A56
RQF.A56
RQF.A67
RQF.A67
RQF.A78
RQF.A78
RQF.A81
RQF.A81
Aperture
0
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
#MQs
784
23
24
25
26
26
25
23
24
23
24
26
25
25
26
23
24
24
23
26
25
25
26
24
23
24
23
25
26
26
25
24
23
µ [T.m/A]
0.000992
0.000992
0.000991
0.000992
0.000992
0.000992
0.000991
0.000991
0.000992
0.000992
0.000991
0.000991
0.000992
0.000992
0.000992
0.000992
0.000992
0.000991
0.000992
0.000992
0.000992
0.000992
0.000991
0.000992
0.000991
0.000991
0.000992
0.000992
0.000991
0.000992
0.000992
0.000992
0.000992
µ [units]
0.0
2.7
-3.2
2.1
0.4
0.8
-2.0
-7.0
1.7
0.3
-8.6
-2.3
1.8
5.8
3.9
0.8
3.7
-4.9
0.4
2.7
1.8
0.9
-1.2
3.5
-4.7
-9.9
-0.2
2.0
-3.9
3.7
4.3
3.9
-0.5
σ [units]
11.9
10.0
11.4
10.6
16.2
11.2
9.9
10.2
8.7
14.2
7.9
11.8
11.7
9.8
9.0
12.4
12.8
11.7
14.2
14.7
11.5
8.9
14.1
9.4
11.8
9.6
13.4
11.3
13.7
12.8
8.1
10.3
10.6
4.2 STATIC COMPONENT
The static part of the FiDeL model is computed using the “loadline” measurements. 11
apertures over the 50 apertures available have been discarded (22%) during the analysis (see
Table V). Results are shown in Figs. 9 and 10 for aperture 1 and 2 respectively, were the
geometric is subtracted and the TF is expressed in units w.r.t. the geometric.
- 10 -
25
20
15
TF - GEO [units]
10
5
0
-5
-10
-15
-20
-25
-30
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 9: Integrated TF versus current during “Loadline” measurements (21 apertures 1)
20
15
10
TF - GEO [units]
5
0
-5
-10
-15
-20
-25
-30
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 10: Integrated TF versus current during “Loadline” measurements (18 apertures 2).
The extraction of the static component of the FiDeL model has been done following the
standard procedure. The maximum model error in the range of operation is about 1 units at
2000 A, which is well below the requirement (see Table X).
Table X: transfer function static model versus Loadline measurements during ramp-up
- 11 -
Measure
I [A] TF-GEO [units]
400
-2.8
600
6.0
760
1.8
1000
0.5
1500
1.0
2000
1.4
2500
0.8
3000
0.4
4000
0.1
5000
-0.3
6000
-0.7
7000
-1.4
8000
-2.4
9000
-3.8
10000
-5.9
11000
-9.1
11850
-13.0
Model
TF - GEO [units] Error [units]
3.9
-6.8
2.4
3.6
1.8
0.0
1.2
-0.7
0.6
0.4
0.4
1.1
0.2
0.6
0.1
0.3
0.0
0.1
-0.2
-0.2
-0.5
-0.3
-1.0
-0.4
-2.0
-0.4
-3.5
-0.4
-5.7
-0.2
-9.0
-0.1
-12.7
-0.3
A single model (i.e. REFPARM generic) is provided for all the MQs with the parameters set
summarized in Table XI. The fit is optimized in the range of 1000 A to 11000 A: the transfer
function below 1000 A have a large random spread which makes the model unstable and the
last point at 11850 A has an asymmetric ramp direction (i.e. ramp-up but no ramp-down).
This asymmetry results in a strong shift of the saturation comparing to previous currents
where the ramp-up and ramp-down are averaged. To consider the 11850 A measurement
point, one would need two measurements respectively before and after a short transition to the
top current 12000 A. The average static component (taken on the available sample of
measurements), the FiDeL fit, and the error is shown in Fig. 11.
Penetration field: this component is not considered in the actual MQs model.
DC magnetization: The DC magnetization component is computed from the averaged rampup and ramp-down of the “loadline”. It is pretty weak, i.e. a few units at injection. The point at
760 A is discarded since the model is not able to fit the double curvature of this component. If
the error is considered significant for the operation, the field penetration component could be
used to fit lower currents. The maximum contribution of this component to the model is about
-1.2 units at 760 A with an error of 1.6 units which will be compensated by the residual
magnetization contribution.
- 12 -
4
2.0
measure
2
1.5
model
0
error
TF Model [units]
-2
1.0
0.5
-4
0.0
-6
-0.5
-8
-1.0
-10
-12
-1.5
-14
-2.0
current [A]
Fig. 11: TF model versus measurements during the ramp-up, and error (units).
Residual magnetization: The residual magnetization component is computed from the rampup of the “loadline” up to the geometric at 5000 A. As for the DC magnetization, the point at
760 A is discarded for the same reason. The maximum contribution of this component to the
model is about 4 units at 760 A with an error of -2.5 units which will be compensated by the
DC magnetization contribution.
Saturation: The saturation is computed from the ramp up of the “loadline”, starting from the
geometric at 5000 A. It is by far the most relevant component of the static model. It gives
about 13 units at 11850 A. This component is well reproducible, and therefore the error is
pretty low (0.3 units).
Table XI: model parameters for the transfer function
Iinj
μ -1.00E-07
760
Inom 11850
p
1.9
Ic
13000
q
3.6
Tc0
9.5
h
2
Tmeas
1.9
ρ 4.00E-07
r
1.5
σ 2.01E-05
s 1.648
I0 19596
- 13 -
4.3 DYNAMIC COMPONENTS
The transfer function decay measurements were performed at 760 A during a 1000 s plateau,
as part of the “machine cycle”. Comparing to the 450 GeV injection during machine
operation, the current used was about +40 A (MQF) and +80 A (MQD) above. The decay of
the transfer function shows a pure random contribution similar in both apertures as shown in
Figs. 12 and 13. The pre-cycle was a 50 A/s ramp-up and ramp-down with a 30 min flat top,
leading to a larger decay contribution than expected in the machine, where one has 10 A/s
ramp.
6
5
4
TF decay [units]
3
2
1
0
-1
-2
-3
-4
-5
0
100
200
300
400
500
600
700
800
900
1000
time [s]
Fig. 12: Decay of integrated transfer function at injection (21 apertures 1).
8
TF decay [units]
6
4
2
0
-2
-4
-6
0
200
400
600
800
time [s]
Fig. 13: Decay of integrated transfer function at injection (20 apertures 2).
- 14 -
1000
The average decay of the transfer function of the all population is about μ = 0.78 units with a
spread of σ = 2.85 units (see Table XII). In Table XIII we give the detail of the decay
amplitude in units for each MQ aperture after a 20 minutes injection. This component has not
been included in the field model.
Table XII: Decay of the transfer function after 1000 s.
Decay
μ
0.78
σ
2.85
Min
-4.14
Max
6.16
Table XIII: Decay of the transfer function per magnet for each aperture after 1000 s.
MQ
019
040
061
071
074
120
138
142
145
151
173
248
258
264
364
376
383
384
394
400
405
µ
σ
V1
3.97
-1.13
0.06
3.06
-3.26
1.56
-2.64
2.19
-1.75
3.52
1.90
4.20
0.62
-0.18
-3.18
-0.16
4.47
-1.92
-2.29
1.94
2.91
0.66
2.55
V2
2.52
1.41
3.52
-4.14
4.55
-2.45
4.44
-1.31
2.47
2.25
6.16
1.63
4.27
-3.04
-0.57
1.49
-4.02
-4.08
-0.38
4.28
0.91
3.18
5. Field errors
5.1 GEOMETRIC
In quadrupole coils, allowed harmonics are b6 and b10 while other harmonics are not allowed
by the coils geometry and should have small systematic values.
- 15 -
Room temperature data
The systematic value of b6 has dropped from 5.5 units to about 3 units with the introduction of
cross-section 2 (Fig. 14) as a corrective action during the production steering, leading to an
average b6 of 3.65 units which fulfils the beam dynamic target (red solid lines).
10
b6 integral with nominal µr (units)
cross-section 1
cross-section 2
8
6
systematic
upper target systematic
4
2
lower target systematic
0
-2
0
100
200
300
400
500
600
700
Aperture progressive number
AT-MAS
Fig. 14: b6 along the production measured at room temperature with the running average (solid line)
Warm to cold extrapolation
The room temperature measurements are extrapolated to the geometric 1.9 K using the
following procedure: (i) for the non-allowed harmonics we assume the geometric equal to the
room temperature measurement (ii) for the allowed harmonics we evaluate the warm-cold
correlation and we apply the offset. The spread of the warm and cold geometrics are
respectively 1.2 units and 1.0 unit for b6 and shows a good warm-to-cold correlation (see Fig.
15). The warm-to-cold offset is -0.24 units (σ = 0.45 units) for b6 and negligible for b10.
6
5
b 6 cold [units]
4
3
2
aperture 1
aperture 2
1
0
0
1
2
3
4
b 6 warm [units]
5
Fig. 15: Room temperature versus 1.9 K measurements for b6.
- 16 -
6
7
Geometric per circuit
The geometric is provided for each circuit and for each magnet (see Tables XIV and XV). The
geometric of magnets is given by the procedure previously mentioned and the geometric of
circuits is given by the average of their magnets geometric.
For the allowed harmonics, b6 has a systematic component of about 3.4 units, which is needed
to compensate the DC magnetization so that at injection the value is close to zero. The spread
over all magnets is about 1 unit. b10 has a systematic component of about -0.2 units.
For the not allowed harmonics, both a4 and a6 have a non negligible systematic of about 0.40.5 units, all the other systematic multipoles being close to zero. A standard deviation of the
order of 1.5 to 2 units is observed in b3 and a3 and a4.
Table XIV: Estimates of the geometric component of normal harmonics.
b3
b4
b5
b6
b7
b8
b9
b10
b11
Circuit Aperture #MQs
µ
σ
µ
σ
µ
σ
µ
σ
µ
σ
µ
σ
µ
σ
µ
σ
µ
σ
ALL
0
784 -0.02 1.50 -0.03 0.32 0.02 0.48 3.42 1.20 0.00 0.14 -0.01 0.07 0.00 0.05 -0.19 0.06 0.00 0.02
RQD.A12 1
23
-0.32 1.65 -0.04 0.33 0.24 0.56 2.94 1.09 0.04 0.11 0.00 0.06 0.01 0.05 -0.19 0.07 0.00 0.02
RQD.A12 2
24
-0.08 1.33 -0.01 0.38 -0.05 0.44 2.72 0.99 0.05 0.13 0.01 0.08 0.00 0.04 -0.21 0.04 0.00 0.02
RQD.A23 1
25
0.52 1.45 -0.09 0.31 -0.12 0.46 3.38 1.04 -0.01 0.17 -0.04 0.07 0.00 0.06 -0.20 0.05 -0.01 0.02
RQD.A23 2
26
-0.14 1.93 0.10 0.40 0.13 0.74 3.31 1.42 0.06 0.16 0.02 0.09 0.00 0.05 -0.18 0.07 0.00 0.02
RQD.A34 1
26
-0.07 1.78 -0.08 0.26 0.17 0.35 2.90 1.06 0.01 0.13 -0.03 0.07 0.01 0.04 -0.19 0.06 0.00 0.02
RQD.A34 2
25
0.24 1.40 -0.09 0.26 -0.02 0.46 3.10 0.73 0.06 0.14 0.00 0.08 0.00 0.05 -0.22 0.06 0.00 0.02
RQD.A45 1
23
0.36 1.42 0.06 0.29 -0.07 0.44 2.92 0.67 -0.02 0.16 0.00 0.06 0.00 0.05 -0.20 0.04 -0.01 0.02
RQD.A45 2
24
-0.01 1.33 -0.01 0.25 -0.05 0.41 3.08 0.78 0.01 0.11 0.01 0.06 0.00 0.04 -0.19 0.05 0.00 0.01
RQD.A56 1
23
0.07 1.07 0.04 0.33 0.05 0.47 3.73 1.12 -0.06 0.16 -0.01 0.07 0.01 0.05 -0.17 0.08 -0.01 0.02
RQD.A56 2
24
-0.65 1.98 0.10 0.30 0.14 0.59 2.95 1.05 0.02 0.15 -0.02 0.07 0.01 0.05 -0.19 0.04 0.00 0.03
RQD.A67 1
26
0.39 1.44 -0.03 0.38 -0.12 0.52 2.51 0.76 -0.04 0.17 -0.01 0.07 0.01 0.04 -0.20 0.06 -0.01 0.02
RQD.A67 2
25
-0.14 1.23 -0.07 0.21 0.01 0.44 3.18 0.99 0.05 0.14 0.00 0.06 0.00 0.05 -0.20 0.06 -0.01 0.02
RQD.A78 1
25
-0.01 1.44 0.12 0.34 -0.07 0.50 4.86 0.73 -0.03 0.12 -0.04 0.05 0.00 0.05 -0.17 0.04 0.00 0.02
RQD.A78 2
26
-0.14 1.52 0.04 0.42 0.03 0.45 4.46 1.16 0.00 0.11 -0.01 0.06 0.00 0.03 -0.15 0.05 0.01 0.03
RQD.A81 1
23
0.05 1.77 -0.05 0.26 0.03 0.50 4.10 1.26 0.01 0.13 0.01 0.06 0.02 0.04 -0.17 0.08 -0.01 0.02
RQD.A81 2
24
-0.22 1.33 -0.17 0.34 0.06 0.42 4.25 1.02 0.05 0.15 -0.02 0.08 -0.01 0.04 -0.17 0.06 0.00 0.02
RQF.A12
1
24
0.37 1.64 -0.07 0.34 -0.05 0.58 2.84 0.97 -0.07 0.12 -0.02 0.06 0.01 0.05 -0.22 0.04 -0.01 0.01
RQF.A12
2
23
-0.38 1.27 -0.03 0.36 0.11 0.45 3.26 1.18 0.03 0.13 0.00 0.08 0.00 0.04 -0.17 0.08 0.01 0.02
RQF.A23
1
26
0.41 1.73 -0.10 0.32 -0.14 0.43 3.41 1.23 -0.02 0.15 -0.01 0.07 0.00 0.05 -0.20 0.06 -0.01 0.01
RQF.A23
2
25
0.00 1.47 -0.07 0.37 -0.05 0.37 3.16 1.09 0.00 0.15 -0.01 0.07 -0.01 0.05 -0.20 0.06 -0.01 0.02
RQF.A34
1
25
0.04 1.52 -0.13 0.32 -0.02 0.44 2.77 0.60 -0.05 0.15 -0.02 0.06 0.01 0.05 -0.23 0.05 -0.01 0.02
RQF.A34
2
26
-0.77 1.71 -0.10 0.30 0.20 0.48 3.06 1.02 0.03 0.16 -0.01 0.06 0.01 0.04 -0.18 0.08 0.00 0.02
RQF.A45
1
24
0.09 1.26 -0.10 0.27 0.04 0.43 2.97 0.84 -0.03 0.12 0.00 0.07 0.00 0.04 -0.22 0.04 -0.01 0.02
RQF.A45
2
23
-0.01 1.19 0.03 0.21 -0.06 0.44 2.89 0.66 0.00 0.13 0.02 0.05 0.01 0.05 -0.21 0.05 -0.01 0.02
RQF.A56
1
24
0.34 1.36 -0.06 0.38 -0.01 0.44 2.98 0.97 -0.03 0.15 -0.02 0.05 0.00 0.05 -0.18 0.03 -0.01 0.02
RQF.A56
2
23
-0.46 1.18 -0.01 0.25 0.27 0.43 3.84 1.25 0.03 0.14 0.01 0.08 0.03 0.04 -0.16 0.09 0.00 0.02
RQF.A67
1
25
0.09 1.74 -0.08 0.26 0.01 0.53 3.31 1.15 -0.03 0.14 0.00 0.05 0.00 0.05 -0.20 0.05 -0.01 0.02
RQF.A67
2
26
-0.15 1.57 -0.10 0.28 -0.03 0.51 2.77 0.96 0.01 0.20 0.00 0.09 0.00 0.06 -0.19 0.04 0.00 0.02
RQF.A78
1
26
-0.09 1.45 -0.06 0.38 0.10 0.52 4.48 1.25 -0.03 0.13 -0.01 0.06 0.03 0.05 -0.16 0.06 0.00 0.02
RQF.A78
2
25
-0.07 1.36 0.14 0.36 0.02 0.41 4.83 0.92 0.02 0.11 -0.02 0.06 0.00 0.03 -0.16 0.05 0.00 0.03
RQF.A81
1
24
0.30 1.23 -0.05 0.25 -0.20 0.45 4.37 0.88 -0.10 0.12 -0.01 0.07 -0.01 0.05 -0.17 0.06 -0.01 0.02
RQF.A81
2
24
-0.29 1.42 -0.02 0.30 0.13 0.47 4.03 1.21 0.03 0.15 -0.01 0.09 0.00 0.05 -0.16 0.08 0.00 0.02
- 17 -
Table XV: Estimates of the geometric component of skew harmonics.
Circuit Aperture
ALL
0
RQD.A12
1
RQD.A12
2
RQD.A23
1
RQD.A23
2
RQD.A34
1
RQD.A34
2
RQD.A45
1
RQD.A45
2
RQD.A56
1
RQD.A56
2
RQD.A67
1
RQD.A67
2
RQD.A78
1
RQD.A78
2
RQD.A81
1
RQD.A81
2
RQF.A12
1
RQF.A12
2
RQF.A23
1
RQF.A23
2
RQF.A34
1
RQF.A34
2
RQF.A45
1
RQF.A45
2
RQF.A56
1
RQF.A56
2
RQF.A67
1
RQF.A67
2
RQF.A78
1
RQF.A78
2
RQF.A81
1
RQF.A81
2
#MQs
784
23
24
25
26
26
25
23
24
23
24
26
25
25
26
23
24
24
23
26
25
25
26
24
23
24
23
25
26
26
25
24
24
a3
µ
σ
0.16 1.84
1.45 1.90
-0.66 1.45
0.95 1.44
-0.81 1.51
0.51 1.41
-0.55 1.50
0.93 1.52
-0.72 1.37
0.95 1.13
-0.83 2.06
0.72 2.06
-0.74 1.74
1.45 1.72
-1.44 1.80
1.03 1.04
-0.73 1.35
1.52 1.90
-1.31 1.63
1.26 2.01
-0.76 1.21
1.01 1.10
-0.58 1.74
1.27 1.61
-0.16 1.61
0.71 1.46
-0.64 1.91
1.05 1.79
-0.64 1.73
1.31 1.55
-1.00 1.10
1.11 1.67
-0.55 1.14
a4
µ
σ
0.41 1.72
0.84 2.14
1.22 1.44
0.12 1.32
0.18 1.83
1.07 2.03
0.32 1.77
0.34 1.83
0.30 1.53
0.82 1.71
0.39 1.48
0.67 1.95
0.00 1.91
0.12 1.13
-0.42 1.41
-0.36 1.64
0.67 1.43
0.66 1.90
0.52 1.47
0.70 2.21
0.33 1.78
0.58 1.58
0.63 1.67
0.32 1.37
-0.31 1.57
0.55 2.33
0.47 1.84
0.48 1.61
0.57 2.25
0.23 1.76
0.62 1.41
0.39 1.56
0.17 1.19
a5
µ
σ
-0.09 0.49
-0.11 0.60
0.07 0.52
-0.20 0.48
-0.10 0.63
-0.23 0.43
-0.03 0.48
-0.14 0.53
-0.04 0.39
-0.21 0.34
-0.11 0.44
-0.14 0.66
0.02 0.50
-0.07 0.58
-0.31 0.50
-0.14 0.41
0.02 0.39
-0.01 0.44
-0.18 0.54
-0.10 0.52
-0.08 0.40
-0.16 0.37
-0.11 0.54
-0.10 0.42
0.19 0.59
-0.37 0.54
0.09 0.48
-0.12 0.47
0.03 0.49
-0.06 0.51
-0.10 0.34
-0.15 0.40
-0.05 0.43
a6
µ
σ
0.45 0.30
0.39 0.24
0.31 0.22
0.53 0.32
0.47 0.29
0.34 0.34
0.48 0.27
0.56 0.29
0.48 0.23
0.40 0.31
0.37 0.33
0.25 0.31
0.53 0.29
0.31 0.24
0.42 0.25
0.59 0.22
0.45 0.21
0.41 0.27
0.25 0.31
0.50 0.30
0.51 0.30
0.58 0.24
0.32 0.30
0.38 0.35
0.50 0.29
0.40 0.31
0.44 0.28
0.63 0.27
0.35 0.32
0.43 0.23
0.50 0.28
0.57 0.25
0.64 0.30
a7
µ
σ
-0.01 0.16
0.06 0.13
-0.06 0.18
0.04 0.13
-0.06 0.16
0.04 0.13
-0.02 0.12
0.03 0.14
-0.02 0.12
0.08 0.12
-0.07 0.16
0.00 0.18
-0.10 0.18
0.08 0.13
-0.08 0.14
0.09 0.16
-0.09 0.15
0.02 0.19
-0.09 0.12
0.04 0.14
-0.11 0.12
0.04 0.12
-0.07 0.11
0.02 0.14
-0.04 0.16
0.09 0.14
-0.04 0.14
0.04 0.17
-0.10 0.14
0.06 0.15
-0.06 0.13
0.07 0.14
-0.03 0.15
a8
µ
σ
0.04 0.20
0.04 0.21
0.16 0.20
0.07 0.18
0.08 0.21
0.05 0.16
-0.01 0.20
0.08 0.21
0.02 0.18
0.06 0.21
0.04 0.25
0.03 0.22
-0.01 0.25
0.02 0.14
-0.02 0.16
-0.07 0.18
0.03 0.18
0.02 0.25
0.07 0.21
0.01 0.23
0.05 0.18
0.03 0.19
0.03 0.26
0.07 0.21
0.02 0.24
0.00 0.18
0.05 0.22
0.05 0.19
0.04 0.25
0.05 0.20
0.05 0.16
0.02 0.18
0.01 0.21
a9
µ
σ
0.00 0.05
0.01 0.04
0.01 0.05
0.00 0.04
-0.01 0.05
0.00 0.04
0.02 0.05
0.00 0.05
0.00 0.05
0.01 0.04
0.00 0.05
0.00 0.05
-0.01 0.04
0.00 0.05
-0.02 0.04
0.01 0.04
0.01 0.05
0.00 0.04
0.00 0.04
0.00 0.05
-0.01 0.04
0.01 0.04
-0.01 0.04
0.00 0.04
0.01 0.04
0.00 0.06
0.01 0.06
-0.02 0.06
0.01 0.04
0.01 0.05
0.00 0.04
0.00 0.04
0.02 0.05
a10
µ
σ
0.07 0.04
0.07 0.04
0.05 0.04
0.09 0.04
0.07 0.05
0.06 0.05
0.07 0.03
0.07 0.05
0.08 0.04
0.07 0.04
0.06 0.05
0.05 0.05
0.06 0.04
0.05 0.03
0.07 0.03
0.08 0.04
0.08 0.04
0.06 0.04
0.06 0.05
0.07 0.03
0.07 0.04
0.08 0.04
0.04 0.05
0.07 0.04
0.06 0.03
0.07 0.05
0.07 0.05
0.08 0.05
0.05 0.05
0.08 0.04
0.06 0.04
0.07 0.03
0.06 0.06
a11
µ
σ
0.00 0.03
0.01 0.02
-0.01 0.02
0.00 0.01
-0.01 0.03
0.00 0.02
-0.01 0.02
0.00 0.02
-0.01 0.02
0.00 0.02
0.00 0.03
0.00 0.02
-0.02 0.02
0.03 0.05
0.01 0.05
0.01 0.02
-0.01 0.02
0.00 0.02
-0.01 0.02
0.00 0.02
-0.02 0.02
0.00 0.02
-0.01 0.02
0.00 0.02
-0.01 0.01
0.01 0.02
-0.01 0.02
0.00 0.02
-0.01 0.02
0.02 0.04
0.03 0.05
0.00 0.02
-0.01 0.02
5.2 STATIC COMPONENTS
The static part of the FiDeL model is computed for the allowed harmonics b6 and b10 using the
“loadline” measurements. All the 50 apertures available have been used during the analysis
(no data rejected). The extraction of the static component of the FiDeL model has been done
following the standard procedure [1]. One fit (i.e. REFPARM generic) is provided for each
allowed harmonics (i.e. b6 and b10). The fit is the same for all the MQs; parameters are
summarized in Table XVI.
For b6, the maximum model error in the range of operation is about 0.015 units, which is well
below the requirement (see Figs. 16 for b6). The measurements of all available magnets are
shown in Fig. 17 and 18: one clearly sees that the DC magnetization is the dominant
component, and saturation is negligible. Therefore FiDeL will only implement DC Mag.
For b10, the systematic dependence on excitation current is very small, i.e. within 0.02 units
(see Fig. 19). On the other hand, a relevant spread of ±0.5 units is observed at injection. So
even though a model can be worked out, it is not implemented in FiDeL.
A rather wide spread at injection is observed also for b3 and a4 (see Figs. 20-23). Also in this
case this effect is not in FiDeL.
Penetration field: this component is not considered in the actual MQs model.
- 18 -
DC magnetization: The DC magnetization component is computed from the averaged rampup and ramp-down of the “loadline”. The point at 760 A is discarded since the model is not
able to fit the double curvature of this component (see figures 15 & 16 for b6 and 17 & 18 for
b10). If the error is considered significant for the operation, the field penetration component
could be used to fit lower currents. The maximum contribution of this component to the
model is about -3.4 units for b6 and 0.09 units for b10 at 760 A with negligible error for both
harmonics.
Residual magnetization: The residual magnetization component is computed from the rampup of the “loadline” up to the geometric at 5000 A. As for the DC magnetization, the point at
760 A is discarded for the same reason. The maximum contribution of this component to the
model is about -0.08 units for b6 and -0.11 units for b10 at 760 A with a negligible error for
both harmonics. This component is fully dominated by the DC magnetization for b6 and
almost compensated by the DC magnetization for b10.
Saturation: The saturation is computed from the ramp up of the “loadline”, starting from the
geometric at 5000 A. The maximum contribution of this component to the model is about 0.04 units for b6 (negligible but with double curvature). This component is systematic and
therefore the model has a small error for both harmonics.
Table XVI: Static model parameters for b6 and b10 and decay for b6.
Only b6 DC magnetization is implemented in the FiDeL model (i.e. μ, p, q, h for b6).
b6
Iinj
760
b10
μ -3.356 0.088
Inom 11850
p 0.522 0.543
Ic
q 0.352 2.458
13000
Tc0
9.5
h
2
2
Tmeas
1.9
ρ
r
σ
s
-0.083
1.254
4.259
5.908
-0.108
2.247
-0.013
3.561
I0 7117 10915
Δ 0.097
τ
29
- 19 -
0.0
0.020
-0.5
measure
0.015
model
-1.0
error
0.010
b 6 [units]
-1.5
-2.0
0.005
-2.5
0.000
-3.0
-0.005
-3.5
-4.0
-0.010
current [A]
Fig. 16: b6 model versus measurements during the ramp-up. The right axis shows the model error.
15
b6 [units]
10
5
0
-5
-10
0
2000
4000
6000
8000
10000
current [A]
Fig. 17: Integrated b6 versus current during “loadline” measurements (21 apertures 1)
- 20 -
12000
15
b6 [units]
10
5
0
-5
-10
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 18: Integrated b6 versus current during “loadline” measurements (20 apertures 2)
0.020
0.040
measure
model
error
0.015
0.010
0.035
0.030
0.025
0.005
0.020
b 10 [units]
0.000
0.015
-0.005
0.010
-0.010
0.005
-0.015
0.000
-0.020
-0.005
-0.025
-0.010
current [A]
Fig. 19: b10 model versus measurements during the ramp-up. The right axis shows the model error.
- 21 -
0.4
0.3
0.2
b10 [units]
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 20: Integrated b10 versus current during “loadline” measurements (21 apertures 1)
0.3
0.2
0.1
b10 [units]
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0
2000
4000
6000
8000
10000
current [A]
Fig. 21: Integrated b10 versus current during “loadline” measurements (20 apertures 2)
- 22 -
12000
10
8
6
b3 [units]
4
2
0
-2
-4
-6
-8
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 22: Integrated b3 (not allowed) versus current during “loadline” measurements (21 apertures 1)
12
10
8
6
b3 [units]
4
2
0
-2
-4
-6
-8
-10
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 23: Integrated b3 (not allowed) versus current during “loadline” measurements (20 apertures 2)
- 23 -
1.5
1.0
a4 [units]
0.5
0.0
-0.5
-1.0
-1.5
-2.0
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 24: Integrated a4 (not allowed) versus current during “loadline” measurements (21 apertures 1)
2.5
2.0
1.5
a4 [units]
1.0
0.5
0.0
-0.5
-1.0
-1.5
0
2000
4000
6000
8000
10000
12000
current [A]
Fig. 25: Integrated a4 (not allowed) versus current during “loadline” measurements (20 apertures 2)
- 24 -
5.2 DYNAMIC COMPONENTS
The transfer function decay measurements were performed at 760 A during a 1000 s plateau,
as part of the “machine cycle”. The pre-cycle was a 50 A/s ramp-up and ramp-down with a 30
min flat top, leading to a larger decay contribution than expected in the machine. The decay of
the b6 shows a systematic contribution similar in both apertures as shown in Figs. 26 and 27.
The model parameters are given in Table XVI. The average decay of the b6 after 20 minutes
for the all population is about μ = 0.56 units with a spread of σ = 0.32 units (see Table XVII).
1.8
1.6
b 6 decay [units]
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
100
200
300
400
500
600
700
800
900
1000
900
1000
time [s]
Fig. 23: Decay of integrated b6 at injection (21 apertures 1).
1.6
1.4
b 6 decay [units]
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
100
200
300
400
500
600
700
800
time [s]
Fig. 24: Decay of integrated b6 at injection (20 apertures 2).
Table XVII: Decay of the b6 after 1000 s
- 25 -
Decay
μ
0.56
σ
0.32
Min
0.02
Max
1.53
7. Summary and critical issues




Knowledge of the TF: the cross-calibration of rotating coils versus SSW is under
investigation. Warm data show that the spread of the TF is of the order of 10 units.
The TF used in the different circuits differ by less than 10 units. At high field one
has a saturation component that decreases the TF of about 13 units.
The geometric b6 is well optimized at injection. A generic model for the DC
magnetization, accounting of about 3 units at injection, is implemented.
Systematic values of non allowed multipoles are close to zero, with the exception
of 0.5 units of a4 and a6. Some low order multipoles. Such as b3 and a4 have a
large random DC magnetization component, which is neglected in the model, but
that could affect the stability at injection.
The decay of the TF at injection is negligible. One observes a small decay of b6,
which has been included in the model. This is an overestimate, since it is based on
measurements with a pre-cycle ramp rate of about 50 A/s.
8. References
[1] FiDeL web site: http://cern.ch/fidel
[2] Field error naming conventions for LHC magnets (EDMS 90250)
[3] Model specifications (EDMS 908232)
[4] Magnet specification (EDMS, CAD, report)
[5] Machine requirements (http://cern.ch/fidel/files/ABP-requirements-v2.xls)
[6] Excel/Mathcad/etc sheet used for data analysis (EDMS)
Other references cited.
- 26 -