+---------+---------+---+-----+----------------+-----------------------+
|         |         |   |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| Engi    |         |   |     |                |                       |
| neering |         |   |     |                |                       |
| Specif  |         |   |     |                |                       |
| ication |         |   |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| Defi    |         |   |     |                |                       |
| nitions |         |   |     |                |                       |
| of      |         |   |     |                |                       |
| Survey  |         |   |     |                |                       |
| and     |         |   |     |                |                       |
| M       |         |   |     |                |                       |
| agnetic |         |   |     |                |                       |
| Data    |         |   |     |                |                       |
| for the |         |   |     |                |                       |
| inner   |         |   |     |                |                       |
| triplet |         |   |     |                |                       |
| systems |         |   |     |                |                       |
| at ir1, |         |   |     |                |                       |
| 2, 5    |         |   |     |                |                       |
| and 8   |         |   |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| A       |         |   |     |                |                       |
| bstract |         |   |     |                |                       |
|         |         |   |     |                |                       |
| The     |         |   |     |                |                       |
| acc     |         |   |     |                |                       |
| eptance |         |   |     |                |                       |
| of the  |         |   |     |                |                       |
| inner   |         |   |     |                |                       |
| triplet |         |   |     |                |                       |
| systems |         |   |     |                |                       |
| at      |         |   |     |                |                       |
| inte    |         |   |     |                |                       |
| raction |         |   |     |                |                       |
| regions |         |   |     |                |                       |
| 1, 2, 5 |         |   |     |                |                       |
| and 8   |         |   |     |                |                       |
| p       |         |   |     |                |                       |
| rovided |         |   |     |                |                       |
| within  |         |   |     |                |                       |
| the     |         |   |     |                |                       |
| scope   |         |   |     |                |                       |
| of the  |         |   |     |                |                       |
| US-LHC  |         |   |     |                |                       |
| collab  |         |   |     |                |                       |
| oration |         |   |     |                |                       |
| (the    |         |   |     |                |                       |
| Q1-Q3   |         |   |     |                |                       |
| optical |         |   |     |                |                       |
| el      |         |   |     |                |                       |
| ements) |         |   |     |                |                       |
| f       |         |   |     |                |                       |
| oresees |         |   |     |                |                       |
| that    |         |   |     |                |                       |
| survey  |         |   |     |                |                       |
| and     |         |   |     |                |                       |
| m       |         |   |     |                |                       |
| agnetic |         |   |     |                |                       |
| data is |         |   |     |                |                       |
| p       |         |   |     |                |                       |
| rovided |         |   |     |                |                       |
| to      |         |   |     |                |                       |
| CERN.   |         |   |     |                |                       |
| This    |         |   |     |                |                       |
| specif  |         |   |     |                |                       |
| ication |         |   |     |                |                       |
| esta    |         |   |     |                |                       |
| blishes |         |   |     |                |                       |
| the     |         |   |     |                |                       |
| conv    |         |   |     |                |                       |
| entions |         |   |     |                |                       |
| and     |         |   |     |                |                       |
| ref     |         |   |     |                |                       |
| erences |         |   |     |                |                       |
| to be   |         |   |     |                |                       |
| used to |         |   |     |                |                       |
| re      |         |   |     |                |                       |
| present |         |   |     |                |                       |
| the     |         |   |     |                |                       |
| data in |         |   |     |                |                       |
| order   |         |   |     |                |                       |
| to make |         |   |     |                |                       |
| them    |         |   |     |                |                       |
| u       |         |   |     |                |                       |
| niquely |         |   |     |                |                       |
| defined |         |   |     |                |                       |
| for     |         |   |     |                |                       |
| acce    |         |   |     |                |                       |
| ptance, |         |   |     |                |                       |
| insta   |         |   |     |                |                       |
| llation |         |   |     |                |                       |
| and LHC |         |   |     |                |                       |
| op      |         |   |     |                |                       |
| eration |         |   |     |                |                       |
| pu      |         |   |     |                |                       |
| rposes. |         |   |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| P       |         |   | C   |                | To be approved by:    |
| repared |         |   | hec |                |                       |
| by :    |         |   | ked |                | O. Bruning            |
|         |         |   | by  |                |                       |
| Luca    |         |   | :   |                | C. Hauviller          |
| BOTTURA |         |   |     |                |                       |
|         |         |   | J.  |                | S. Myers              |
| <       |         |   | Di  |                |                       |
| Luca.Bo |         |   | Ma  |                | N. Siegel             |
| ttura@c |         |   | rco |                |                       |
| ern.ch> |         |   |     |                | J. Strait             |
|         |         |   | S.  |                |                       |
| Do      |         |   | Fa  |                | T. Taylor             |
| minique |         |   | rto |                |                       |
| M       |         |   | ukh |                |                       |
| ISSIAEN |         |   |     |                |                       |
|         |         |   | M.  |                |                       |
| [Domin  |         |   | L   |                |                       |
| ique.Mi |         |   | amm |                |                       |
| ssiaen] |         |   |     |                |                       |
| {.under |         |   | R.  |                |                       |
| line}\@ |         |   | O   |                |                       |
| cern.ch |         |   | sto |                |                       |
|         |         |   | jic |                |                       |
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|         |         |   | J   |                |                       |
|         |         |   | .P. |                |                       |
|         |         |   | Qu  |                |                       |
|         |         |   | esn |                |                       |
|         |         |   | el, |                |                       |
|         |         |   |     |                |                       |
|         |         |   | W.  |                |                       |
|         |         |   | Sc  |                |                       |
|         |         |   | and |                |                       |
|         |         |   | ale |                |                       |
|         |         |   |     |                |                       |
|         |         |   | P.  |                |                       |
|         |         |   | Sch |                |                       |
|         |         |   | lab |                |                       |
|         |         |   | ach |                |                       |
|         |         |   |     |                |                       |
|         |         |   | G.  |                |                       |
|         |         |   | Ve  |                |                       |
|         |         |   | lev |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
|         |         |   |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| History |         |   |     |                |                       |
| of      |         |   |     |                |                       |
| Changes |         |   |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| Rev.    | Date    | P |     | Description of |                       |
| No.     |         | a |     | Changes        |                       |
|         |         | g |     |                |                       |
|         |         | e |     |                |                       |
|         |         | s |     |                |                       |
+---------+---------+---+-----+----------------+-----------------------+
| 0.1     | 200     | A |     | Submission for |                       |
|         | 3-01-09 | l |     | check          |                       |
| 0.2     |         | l |     |                |                       |
|         | 200     |   |     | Submission for |                       |
| 1.0     | 3-02-21 | A |     | approval       |                       |
|         |         | l |     |                |                       |
| 1.1 ?   | 200     | l |     | Released       |                       |
|         | 3-03-27 |   |     | version        |                       |
|         |         | A |     |                |                       |
|         | 200     | l |     | Changed the    |                       |
|         | 4-02-12 | l |     | position of    |                       |
|         |         |   |     | the BPMS (warm |                       |
|         |         | A |     | BPM installed  |                       |
|         |         | l |     | at CERN) in    |                       |
|         |         | l |     | schematic      |                       |
|         |         |   |     | drawings       |                       |
|         |         |   |     |                |                       |
|         |         |   |     | Changed the    |                       |
|         |         |   |     | topology of    |                       |
|         |         |   |     | the corrector  |                       |
|         |         |   |     | packages MQSX  |                       |
|         |         |   |     | and MCBXA in   |                       |
|         |         |   |     | LQXC, adding   |                       |
|         |         |   |     | MCSOX          |                       |
|         |         |   |     |                |                       |
|         |         |   |     | Removed BPMS   |                       |
|         |         |   |     | on Q1 (mounted |                       |
|         |         |   |     | at CERN        |                       |
|         |         |   |     | external to    |                       |
|         |         |   |     | the LQXA       |                       |
|         |         |   |     | assembly)      |                       |
+---------+---------+---+-----+----------------+-----------------------+

Table of Contents

1\. Introduction [4](#introduction)

2\. Q1 [5](#q1)

2.1 survey data for Q1 [6](#survey-data-for-q1)

2.2 Magnetic data for Q1 [7](#magnetic-data-for-q1)

2.2.1 MQXA quadrupole [7](#mqxa-quadrupole)

2.2.2 MCBX corrector [9](#mcbx-corrector)

3\. Q2 [10](#q2)

3.1 survey data for Q2 [11](#survey-data-for-q2)

3.2 Magnetic data for Q2 [12](#magnetic-data-for-q2)

3.2.1 MQXB quadrupole [12](#mqxb-quadrupole)

3.2.2 MCBX corrector [12](#mcbx-corrector-1)

4\. Q3 [13](#q3)

4.1 survey data for Q3 [13](#survey-data-for-q3)

4.2 magnetic data for Q3 [14](#magnetic-data-for-q3)

4.2.1 MQXA quadrupole [14](#mqxa-quadrupole-1)

4.2.2 MCBXA and MQSXA correctors [14](#corrector-packages)

5\. References [16](#references-and-other-applicable-documents)

6\. Appendix I -- Expansion of the magnetic field in the complex plane
[17](#appendix-i-expansion-of-the-magnetic-field-in-the-complex-plane)

7\. Appendix II -- definitions for integral and average field values
[18](#appendix-ii-definitions-for-integral-and-average-field-values)

7.1 Integrated QUADRUPOLE strength [18](#integrated-quadrupole-strength)

7.2 quadrupole Magnetic Length [18](#quadrupole-magnetic-length)

7.3 Integrated quadrupole transfer function
[18](#integrated-quadrupole-transfer-function)

7.4 average quadrupole field direction
[19](#average-quadrupole-field-direction)

7.5 average quadrupole harmonics [19](#average-quadrupole-harmonics)

7.6 integrated and average field for corrector magnets
[19](#integrated-and-average-field-for-corrector-magnets)

# Introduction

This note provides definitions for the magnetic and survey data measured
on the Q1, Q2 and Q3 inner triplet systems for the LHC interaction
regions 1, 2, 5 and 8. The aim of the definitions reported here is to
avoid any ambiguity in the magnetic and survey data, and to make them
directly usable for the qualification process, the installation and the
operation of the LHC.

Table 1.1 Naming of components for the superconducting inner triplet
systems

+-----------+-------------+-------------+-------------+-------------+
| Optics    | Main        | Corrector   | Magnet Code | Cr          |
| Name      | Quadrupole  | Packages    |             | yo-Assembly |
|           | Code        | Codes       |             | Code        |
+-----------+-------------+-------------+-------------+-------------+
| Q1        | MQXA        | MCBX        | LMQXA       | LQXA        |
+-----------+-------------+-------------+-------------+-------------+
| Q2        | MQXB        | MCBX        | LMQXB       | LQXB        |
+-----------+-------------+-------------+-------------+-------------+
| Q3        | MQXA        | MCBXA       | LMQXC       | LQXC        |
|           |             |             |             |             |
|           |             | MQSX        |             |             |
|           |             |             |             |             |
|           |             | MCSOX       |             |             |
+-----------+-------------+-------------+-------------+-------------+

Reference naming of the magnetic elements assembled in the cold masses
of the Q1, Q2 and Q3, and the Cryo-assembly code are reported in Tab.
1.1, reported from \[2\], \[4\]. The schematic layout of the inner
triplet system is shown in Fig. 1.1.

All definitions in this note apply to single cryomagnets, so that the
reference frames for the magnetic and survey data is independent of the
location in the LHC ring. The orientation of the magnet is defined
through reference to the lead end and the non-lead end of the assembled
cold mass, i.e. at the level of the LMQXA, LMQXB and LMQXC components.
This is the most convenient reference frame for magnet measurement. All
coordinate frame transformations from the reference frames defined in
this note to the reference frames used for installation and operation
will be performed by CERN.

![](media/image1.wmf)

Figure 1.1. Inner triplet system schematic, from \[2\], \[4\].

# Q1

The Q1 (LQXA) optical element contains a quadrupole MQXA whose cross
section is shown in Fig. 2.1. The MQXA has a nominal magnetic length (in
cold conditions) of 6.37 m. Q1 is equipped with a MCBX corrector package
(on the lead end side). The lead end of Q1 is located away from the IP.
See Fig. 2.2 for a schematic of the Q1 configuration and the location of
the lead and IP ends. The cryostat of Q1 is equipped with eight fiducial
nests for Taylor-Hobson fiducial spheres. Four of these fiducials are
used for alignment, while the other four will house instrumentation for
monitoring the position of the cryo-assembly after installation.

![](media/image2.png){width="3.8159722222222223in"
height="3.8826388888888888in"}

Figure 2.1. Cross section of the MQXA quadrupole, from \[2\].

![](media/image3.wmf)

Figure 2.2. Schematic composition and location of the lead and IP ends
for the LQXA cryo-assembly (Q1), from \[2\].

## survey data for Q1

Survey data for the Q1 optical element are obtained during the
measurement of the magnetic axis of the MQXA magnetic element in warm
conditions. The reference frame for survey data is the triad RST shown
schematically in Fig. 2.3. The plane RS is the *magnetic midplane* of
the magnet (defined later in this section). The S axis is the *average*
*magnetic axis* of the magnet (defined later in this section), pointing
in the direction from the lead end to the non-lead end. The R axis is
oriented to the right when looking at the magnet from the lead end. This
direction corresponds to the inwards direction of the LHC (pointing
towards the centre of the accelerator) for the Q1 magnets installed on
the *left* of the IP in the sense of Beam 1 (turning clockwise in the
LHC). The T axis is normal to the RS plane, and is oriented so that the
triad RST forms a right-handed reference frame.

![](media/image4.wmf)

Figure 2.3. Definition of the reference frame for magnetic and alignment
data of Q1. The fiducials positions on the cryostat are shown
schematically. The position of points Q, the origin of the RST frame, W
and W', the origins for magnetic data, is determined in warm conditions.

The *average* *magnetic axis* is the straight line such that the
weighted average of the locus of the centers of the main quadrupole
field (magnetic centers) along the MQXA quadrupole is zero. The weight
for the average is the quadrupole field strength.

The *magnetic midplane* is defined as the plane normal to the average
field direction measured in the MQXA quadrupole, and containing the
magnetic axis S. The average field direction is defined in Appendix II

The origin of the RST frame is located at the point Q that corresponds
to the beginning of the *magnetic length* of the MQXA, on the lead end,
in warm conditions. The magnetic length is defined in Appendix II.

In "as built" magnets the average magnetic axis, the magnetic midplane,
the RST reference frame and the position of Q are obtained directly
during magnetic measurements, in warm conditions.

The survey data consists in a set of coordinates of the magnet fiducials
(located on the cryostat) in the RST reference frame. The format of this
data is defined in \[9\].

The results on magnets tested at cold (a sample of the whole
production), as well as computed values, are used to establish the
correlation between warm measurements and cold (operating) conditions.
The warm-to-cold correlation is defined through the change in the
position of the fiducials in the RST frame.

## Magnetic data for Q1

### MQXA quadrupole

The reference frame for magnetic data of the MQXA quadrupole in Q1 is
the XYZ triad shown in Fig. 2.3. The Z axis is the *magnetic axis* of
the quadrupole (see the definition given later) and the X axis points
right when looking at the magnet from the lead-end. The X axis is
parallel to the RS plane, as defined for the survey data. The Y axis is
normal to the RS plane, and is oriented so that the diad XY forms a
right-handed reference frame. The origin W of the XYZ frame is located
in the longitudinal magnetic center of the cold mass, i.e. the point
along the Z axis such that the field integral from the magnet connection
end to W is equal to the field integral from W to the magnet
non-connection end. The origin W is hence determined based solely on
magnetic measurements. This definition applies both to warm and cold
measurements.

The Z axis (magnetic axis) is defined as the locus of the magnetic
centers MC measured along the magnet length. The magnetic center MC is
defined as the point in the magnet cross section where the dipole
component of the field is zero. An operative definition of the magnetic
axis is given in \[10\]. In the ideal case of perfect dimensions and
assembly, the Z axis is coincident with the average magnetic axis S. In
"as built" magnets the Z axis can deviate from this theoretical
position.

The magnetic data consists of local and integrated main quadrupole
strength and transfer function, magnetic length, local and average field
direction, local and average higher order harmonics, local and average
magnetic center coordinates. All magnetic data are reported for the
situation when the magnet current is "positive", i.e. entering terminal
A.The definition of the local value of the field and harmonics,
including the notation and conventions referred to the local XY plane,
is given in \[11\], \[10\].

The local main field for a quadrupole is indicated with *B~2~*, and
corresponds to the strength (with sign) of the second complex component
of the harmonic expansion, indicated with **C**~2~. *B~2~* is given in
units of \[T @ R~ref~\] where R~ref~ is the reference radius. The
reference radius for the LHC magnets is 17 mm, but a different value can
be used if convenient (e.g. in case of very low numerical values of the
field harmonics) provided they it is reported in the magnetic data
tables.

The local transfer function *T~2~* for the main quadrupole field is the
ratio of the main field strength to the operating current. The transfer
function *T~2~* is given in units of \[T @ Rref/A\].

The local field direction *α~2~* is given with respect to the Y axis,
i.e. with respect to the normal T to the mechanical midplane of the
magnet, in units of \[radians\]. The sign convention for the angle is
the trigonometric positive direction (counter-clockwise). A physical
rotation of the magnet by an angle -*α~2~* would align the field to the
Y axis. See \[10\] for the details on the procedure to be followed to
obtain the field direction. Note that following the definitions of
\[10\] the field direction for a quadrupole is bounded between --π/4 and
π/4.

The higher order local harmonics of order n, normal *b~n~* and skew
*a~n~* are reported in a xy reference frame with the y axis aligned with
**C**~2~, i.e. they are obtained from the measured values after a
rotation by *α~2~*, as defined in \[10\]. The local harmonics *b~n~* and
skew *a~n~* are given in relative units of 10^-4^ of the main field
strength at the reference radius.

The format for the local harmonics and local field direction tables is
specified in \[12\] and \[13\].

The local magnetic center coordinates are defined as the two components
of the local vector between the S axis and the points on the Z axis. The
magnetic center coordinates are reported in the survey frame RST.

The format for local magnetic center table is specified in \[14\].

The definition of the integral and average quantities based on local
field measurement of the field (at several locations in the magnets) is
given in Appendix II.

The integrated quadrupole strength *B~2~dl* includes the end
contributions. The integrated quadrupole strength *B~2~dl* is given in
units of \[T m @ R~ref~\].

The integrated transfer function *T~2~dl* is given in units of \[T m @
R~ref~ /A\].

The magnetic length L~m~ is given in units of \[m\].

The average quadrupole field direction \<*α~2~\>* is given with respect
to the normal to the magnetic midplane of the magnet, T, in units of
\[radians\]. Note that because of the definition of the RST reference
frame, the average field direction should be zero at the reference
conditions used for the fiducialisation.

The average higher order harmonics of order n, normal \<*b~n~\>* and
skew \<*a~n~\>* are given in relative units of 10^-4^ of the main
quadrupole field strength at the reference radius.

The format for the integrated transfer function and harmonics is
specified in \[15\] and \[16\].

### MCBX corrector

The MCBX corrector contains two co-axial windings generating a normal
and a skew dipole. It is assumed in this note that the two co-axial
windings are aligned longitudinally and transversally in accordance with
the manufacturing drawings, and the only error considered is a relative
angle misalignment. The reference frame for magnetic data of the MCBX
dipole corrector in Q1 is the triad X'Y'Z' shown in Fig. 2.3. The Z'
axis is the *magnetic axis* of the dipole (see the definition given
later). The X' and Y' axes are defined as for the MQXA quadrupole, i.e.
respectively parallel and normal to the RS plane.

The Z' axis (magnetic axis of MCBX) is defined as the locus of the
magnetic centers MC' measured along the magnet length. The magnetic
center MC' is defined as the point in the magnet cross section where a
non-allowed multipole of high order (typically order 10 or 12) is zero.
The actual choice of the non-allowed multipole is inessential.

The origin of the X'Y'Z' frame is located in the longitudinal magnetic
center of the dipole corrector, i.e. the point W' along the Z' axis such
that the field integral from one magnet end to W' is equal to the field
integral from W' to the other magnet end. Note that with the hypothesis
of perfect relative longitudinal alignment of the two windings the
location of the point W' can be measured indifferently on any of the two
corrector windings.

The magnetic data for the MCBX corrector consists in the integrated
field strength of the two windings B~1~dl and A~1~dl, the magnetic
length of the two windings L~mB1~ and L~mA1~, the average field
direction of the two windings \<α~1B1~\> and \<α~1A1~\>, and the
longitudinal position of the magnetic element with respect to the main
quadrupole MQXA (assumed to be identical for the two corrector
windings).

The integrated strength B~1~dl and A~1~dl, including the end
contributions, are given in units of \[T m\].

The magnetic lengths of the two windings L~mB1~ and L~mA1~ are given in
units of \[m\].

The average field directions for the two windings \<α~1B1~\> and
\<α~1A1~\> are reported in the X'Y' reference frame. As by definition
the X' and Y' axes are parallel and normal to the RS plane, the average
field directions of the MCBX dipole corrector windings are hence given
with respect to the normal of the magnetic midplane of the MQXA
quadrupole, T.

The longitudinal position of the dipole corrector is defined as the
distance between the longitudinal magnetic centers W and W', measured
along the S axis.

# Q2

The Q2 (LQXB) optical element contains two quadrupoles MQXB whose cross
section is shown in Fig. 3.1. The MQXB has a nominal magnetic length (in
cold conditions) of 5.5 m. The MQXB quadrupoles are identified as Q2a,
on the IP end, and Q2b, on the non-IP end of the assembly.

Q2 is equipped with a MCBX corrector package (placed between the MQXB
quadrupoles) and a BPM (on the IP end side). The lead ends of the two
MQXB quadrupoles are pointing in opposite directions. See Fig. 3.2 for a
schematic of the Q2 configuration and the location of the lead and IP
ends. The Q2 cold mass is mounted in a single cryostat.

The cryostats of Q2 is equipped with twelve fiducial nests for
Taylor-Hobson fiducial spheres. Six of these fiducials are used for
alignment, while the other six will house instrumentation for monitoring
the position of the cryo-assembly after installation.

![](media/image5.png){width="4.955555555555556in"
height="2.642361111111111in"}

Figure 3.1. Cross section of the Q2 quadrupole, from \[1\], as seen
facing the lead end. Note the direction pointing towards the ring center
for the magnet installed on the left of the IP (pointing right in the
figure).

![](media/image6.wmf)

Figure 3.2. Schematic composition and location of the lead and IP ends
for the LQXB cryo-assembly (Q2), from \[1\].

## survey data for Q2

The reference frame for survey data is the triad RST shown schematically
in Fig. 3.3. The plane RS is the *magnetic midplane* of the magnet
(defined later in this section). The S axis is the *average* *magnetic
axis* of the magnet (defined later in this section), pointing in the
direction from the lead end of Q2b to the lead end of Q2a. The R axis is
oriented to the right when looking at the magnet from the lead end of
Q2b. This direction corresponds to the inwards direction of the LHC
(pointing towards the centre of the accelerator) for the Q2 magnets
installed on the *left* of the IP in the sense of Beam 1 (turning
clockwise in the LHC). The T axis is normal to the RS plane, and is
oriented so that the triad RST forms a right-handed reference frame.

![](media/image7.wmf)

Figure 3.3. Definition of the reference frame for magnetic and alignment
data of Q2. The fiducials positions on the cryostat are shown
schematically. The position of points Q, the origin of the RST frame, W,
W' and W'', the origins for magnetic data, is determined in warm
conditions.

The *average* *magnetic axis* is the weighted average of the locus of
the centers of the main quadrupole field (magnetic centers) of the
assembly formed by the two MQXB quadrupoles. The weight for the average
is the local quadrupole field strength. The average magnetic axis is
hence the line along which the integrated skew and normal dipole from
MQXB (Q2a) and MQXB (Q2b) is zero.

The *magnetic midplane* is defined as the plane normal to the average
field direction of the assembly formed by the two MQXB quadrupoles, and
containing the magnetic axis S. The average field direction is defined
in Appendix II

The origin of the RST frame is located at the point Q that corresponds
to the beginning of the *magnetic length* of the MQXB (Q2b) element, on
the lead end, in warm conditions. The magnetic length is defined in
Appendix II.

The survey data consists in a set of coordinates of the Q2 magnet
fiducials (located on the cryostat) in the RST reference frame. The
format of this data is defined in \[9\].

Warm-to-cold correlation for the survey data is defined as for Q1
through the change in the position of the fiducials in the RST frame. It
is planned to measure the warm-to-cold geometry change for all the
magnets.

## Magnetic data for Q2

### MQXB quadrupole

The reference frame for magnetic data of the two MQXB quadrupoles in Q2
is defined using the same convention as for Q1. The definition is
reported schematically in Fig. 3.3. The magnetic data consists of local
and integrated main quadrupole strength and transfer function, magnetic
length, local and average field direction, local and average higher
order harmonics, local and average magnetic center coordinates and the
longitudinal distance between the quadrupoles, defined as the distance
between the longitudinal magnetic centers W and W', and measured along
the S axis.

### MCBX corrector

The reference frame for magnetic data of the MCBX corrector package in
Q2 is defined as for Q1, and is shown schematically in Fig. 3.3. The
magnetic data for the MCBX corrector consists in the integrated field
strength of the two windings B~1~dl and A~1~dl, the magnetic length of
the two windings L~mB1~ and L~mA1~, the average field direction of the
two windings \<α~1B1~\> and \<α~1A1~\>, and the longitudinal position of
the magnetic element with respect to the main quadrupole MQXB (Q2b),
i.e. the distance between the longitudinal magnetic centers W and W'',
measured along the S axis. (assumed to be identical for the two
corrector windings).

# Q3

The Q3 (LQXC) optical element contains a quadrupole MQXA (the same as in
Q1) whose cross section is shown in Fig. 2.1.

Q3 is equipped with a MQSX corrector package (on the lead-end side) and
a MCBXA corrector package (on the non-lead end side). A combined,
non-linear correction coil package MCSOX is mounted adjacent to the
MCBXA. The lead end of Q3 is located on the IP side of the
cryo-assembly. See Fig. 4.1 for a schematic of the Q3 configuration and
the location of the lead and IP ends.

Similar to Q1, the cryostat of Q3 is equipped with eight fiducial nests
for Taylor-Hobson fiducial spheres. Four of these fiducials are used for
alignment, while the other four will house instrumentation for
monitoring the position of the cryo-assembly after installation.

![](media/image8.wmf)

Figure 4.1. Schematic composition and location of the lead and IP ends
for the LQXC cryo-assembly (Q3), from \[2\].

## survey data for Q3

The definition of the survey frame for the fiducials on Q3 is the same
as for Q1. The definition is reported schematically in Fig. 4.2. As for
Q1, the R axis is oriented to the right when looking at the magnet from
the lead end. This direction however corresponds to the inwards
direction of the LHC (pointing towards the centre of the accelerator)
for the Q3 magnets installed on the *right* of the IP in the sense of
Beam 1 (turning clockwise in the LHC).

The survey data consists in a set of coordinates of the Q3 magnet
fiducials (located on the cryostat) in the RST reference frame. The
format of this data is defined in \[9\].

The warm-to-cold correlation is defined as for Q1 and Q2, as the change
in the position of the fiducials in the RST frame. Similar to Q1, cold
tests are planned only on a sample of the Q3 production, and the
correlation will hence be based on measured and computed values.

## magnetic data for Q3

### MQXA quadrupole

The reference frame for magnetic data of the MQXA quadrupole in Q3 is
defined using the same convention as for Q1. The definition is reported
schematically in Fig. 4.2. The magnetic data consists of local and
integrated main quadrupole strength and transfer function, magnetic
length, local and average field direction, local and average higher
order harmonics, local and average magnetic center coordinates. The same
definitions as for Q1, as reported in Appendix II, apply also to Q3.

![](media/image9.wmf)

Figure 4.2. Definition of the reference frame for magnetic and alignment
data of the Q3 quadrupoles. The fiducials positions on the cryostat are
shown schematically. The position of point Q, the origin of the RST
frame, and of the points W, W', W'' and W''', the origins of the
magnetic frames, is determined in warm conditions.

### corrector PACKAGES

The MCBXA corrector package contains two dipole windings (as for the
MCBX) and additional nested corrector windings (b3 and b6). The co-axial
windings are assumed to be aligned longitudinally and transversally in
accordance with the manufacturing drawings, and the only error
considered is a relative angle misalignment among the windings. The
reference frame for magnetic data of the MCBXA corrector package is the
triad X'Y'Z' shown in Fig. 4.2. The Z' axis is the *magnetic axis* (the
locus of the magnetic centers MC') of the lowest order corrector winding
(b3) inserted in the MCBXA. The X' and Y' axes are respectively parallel
and normal to the RS plane.

The origin of the X'Y'Z' frame is located in the longitudinal magnetic
center W' of the dipole corrector windings (assumed to be longitudinally
centered by construction).

The MQSX corrector package contains a skew quadrupole corrector. The
reference frame for magnetic data of the MQSXA corrector packages is the
triad X''Y''Z'' defined analogously to X'Y'Z' and also shown in Fig.
4.2. The Z'' axis is the *magnetic axis* of the skew quadrupole
corrector winding. The X'' and Y'' axes are respectively parallel and
normal to the RS plane.

The origin of the X''Y''Z'' frame is located in the longitudinal
magnetic center W'' of the skew quadrupole corrector.

The reference frame X'''Y'''Z''' of the MCSOX corrector package is
defined by analogy with the above definitions, and is shown in Fig. 4.2.
The Z''' axis is the *magnetic axis* of the skew sextupole corrector
winding. The X''' and Y''' axes are respectively parallel and normal to
the RS plane. The origin of the X'''Y'''Z''' frame is located in the
longitudinal magnetic center W''' of the skew sextupole corrector.

The magnetic data for the corrector packages consist in the integrated
field strength B~n~dl and A~n~dl of each corrector of order n, the
magnetic length, the average field direction \<α~nBn~\> and \<α~nAn~\>
of each corrector winding, the average magnetic center coordinates ∆x,
∆y and the longitudinal position of the magnetic element with respect to
the main quadrupole MQXA.

The average field direction for each corrector winding, \<α~nBn~\> and
\<α~nAn~\> for normal and skew correctors, is reported in the X'Y'
reference frame for the MCBXA, in the X''Y'' reference frame for the
MQSX, and in the X'''Y''' reference frame for the MCSOX, i.e. with
respect to the normal T to the magnetic midplane of the MQXA quadrupole.

The average magnetic center coordinates are given in the RST frame, i.e.
with respect to the axis of the MQXA quadrupole.

The longitudinal position of the MCBXA, MQSX and MCSOX corrector
packages is defined as the distance between the longitudinal magnetic
centers W', W'' and W''', and the magnetic center of the main quadrupole
MQXA measured along the S axis.

# References and other applicable documents

\[1\] J. Kerby, Inner Triplet Systems at IR1, 2, 5 and 8,
LHC-LQX-ES-0001.00 rev. 2.3, US TD/FNAL/USA, EDMS Document No. 108981,
2001-05-04.

\[2\] A. Yamamoto, T. Ogitsu, N. Ohuchi, T. Nakamoto, Inner Triplet
Quadrupole MQXA -- Functional Specification, LHC-MQXA-ES-0001 rev. 0.1,
TD/KEK/JP, EDMS Document No. 313715, 2001-05-15.

\[3\] T. Ogitsu, M. Lamm, J. Kerby, Inner Triplet Quadrupole MQXA --
Interface Specification, LHC-MQXA-ES-0003 rev. 0.1, KEK, EDMS Document
No. 345952, 2002-05-27.

\[4\] J. Kerby, Inner Triplet Quadrupole Cryo-Assembly LQXB --
Functional Specification, LHC-LQX-ES-0006 rev. 1.1, TD/FNAL/USA, EDMS
Document No. 340762, 2002-06-04.

\[5\] J. Kerby, M. Lamm, Inner Triplet Quadrupole MQXB -- Functional
Specification, LHC-LQX-ES-0002 rev. 1.1, TD/FNAL/USA, EDMS Document No.
256806, 2001-04-23.

\[6\] J. Kerby, Inner Triplet Quadrupole MQXB -- Interface
Specification, LHC-LQX-ES-0003 rev. 1.2, TD/FNAL/USA, EDMS Document No.
311585, 2001-07-11.

\[7\] J. Kerby, Inner Triplet Corrector MCBX/MCBXA, LHC-LQX-ES-0005 rev.
1.0, US LHC MS 3-1.1.1-11, EDMS Document No. 342357, 2002-08-28.

\[8\] J. Kerby, Inner Triplet Corrector MQSXA, LHC-LQX-ES-0004 rev. 1.0,
US LHC MS 3-1.1.1-11, EDMS Document No. 342142, 2002-08-28.

\[9\] D. Missiaen, Survey Data Format for Beam Separation Dipoles and
Interaction Region Quadrupoles, LHC-G-ES-0036.

\[10\] L. Bottura, Standard Analysis Procedures for Field Quality
Measurement of the LHC Magnets. Part I: Harmonics, LHC-M-ES-0007 rev.
1.0, EDMS Document No. 313621, 2001-11-15.

\[11\] R. Wolf, Field Error Naming Conventions for the LHC Magnets,
LHC-M-ES-0001.00 rev. 2, EDMS Document No. 90250, 2001-09-19.

\[12\] L. Deniau, L. Bottura, Table F Local Harm, available at the URL:
http://www.cern.ch/m/mtauser/www/html/DAP/f_local_harm.html.

\[13\] L. Deniau, L. Bottura, Table F Local Angle, available at the URL:
http://www.cern.ch/m/mtauser/www/html/DAP/f_local_angle.html.

\[14\] L. Deniau, L. Bottura, Table F Local Center, available at the
URL: http://www.cern.ch/m/mtauser/www/html/DAP/f_local_center.html

\[15\] L. Deniau, L. Bottura, Table F T Function, available at the URL:
http://www.cern.ch/m/mtauser/www/html/DAP/f_t_function.html.

\[16\] L. Deniau, L. Bottura, Table F Int Harm, available at the URL:
http://www.cern.ch/m/mtauser/www/html/DAP/f_int_harm.html.

# Appendix I -- Expansion of the magnetic field in the complex plane

We recall the definition \[10\], \[11\] of the harmonic expansion of the
magnetic field ***B*** in the 2-D imaginary plane (*x*,*y*) in terms of
the complex variable **z=***x*+*i* *y*:

$B(z) = B_{y} + {iB}_{x} = \sum_{n = 1}^{\infty}{C_{n}\left( \frac{z}{R_{\text{ref}}} \right)}^{n - 1}$

where the coefficients **C***~n~* appearing above are the complex
harmonic coefficients, and *R~ref~* is the reference radius (*R~ref~* is
17 mm for LHC).

# Appendix II -- definitions for integral and average field values

## Integrated QUADRUPOLE strength

The integrated quadrupole strength *B~2~dl* is defined as the integral
of the local quadrupole strength along the length of the magnet,
including the full contribution of the magnet ends:

$$B_{2}\text{dl} = \int_{L_{\text{magnet}}}^{}{B_{2}(Z)\text{dZ}}$$

This integral generally translates in a sum of measured values over
adjacent lengths, e.g. obtained with short rotating coils, but can also
be obtained directly from an integral measurement, e.g. with a stretched
wire or a long rotating coil. The integration formula does not take into
account the possible variations in the quadrupole field direction along
the magnet length. This effect is negligible for variations of the local
field direction below 5 mrad and was not taken into account for
simplicity.

## quadrupole Magnetic Length

The magnetic length is defined as the ratio of the integrated quadrupole
strength *B~2~dl* to the average quadrupole strength in the body of the
magnet (i.e. excluding end effects):

$$L_{m} = \frac{B_{2}\text{dl}}{\frac{\int_{L_{\text{body}}}^{}{B_{2}(Z)\text{dZ}}}{L_{\text{body}}}}$$

The magnetic length corresponds to the length of an ideal magnet with
the same integrated strength as the real magnet considered and a
*window-frame* field profile, with field strength identical to the body
average strength.

## Integrated quadrupole transfer function

The integrated transfer function *T~2~dl* is defined as the ratio of the
the integrated quadrupole strength to the operating current:

$$T_{2}\text{dl} = \frac{B_{2}\text{dl}}{I}$$

The integrated transfer function is given in units of \[T m @
R~ref~/A\].

## average quadrupole field direction

The average quadrupole field direction \<*α~2~*\> is given by the
weighted integral of the local field direction along the magnet length.
The weight is the quadrupole field strength:

$$\alpha_{2}\text{>=}\frac{\int_{L_{\text{magnet}}}^{}{\alpha_{2}(Z)B_{2}(Z)\text{dZ}}}{\int_{L_{\text{magnet}}}^{}{B_{2}(Z)\text{dZ}}}$$

The average quadrupole field direction \<*α~2~*\> is given in units of
\[radians\].

## average quadrupole harmonics

The average higher order harmonics of order n, normal \<*b~n~*\> and
skew \<*a~n~*\> are computed similarly to the average quadrupole field
direction:

$$b_{n}\text{>=}\frac{\int_{L_{\text{magnet}}}^{}{b_{n}(Z)B_{2}(Z)\text{dZ}}}{\int_{L_{\text{magnet}}}^{}{B_{2}(Z)\text{dZ}}}$$

$$a_{n}\text{>=}\frac{\int_{L_{\text{magnet}}}^{}{a_{n}(Z)B_{2}(Z)\text{dZ}}}{\int_{L_{\text{magnet}}}^{}{B_{2}(Z)\text{dZ}}}$$

The average harmonics *b~n~* and skew *a~n~* are given in relative units
of 10^-4^ of the main field strength at the reference radius.

## integrated and average field for corrector magnets

The definition of the integrated and average field quantities for the
corrector magnets follows by analogy the definitions above, substituting
the main field of the corrector B~n~ or A~n~ and the local field
direction of the corrector α~nBn~ or α~nAn~ for the main quadrupole
field B~2~ and the main quadrupole field direction α~2~ respectively.