arXiv:0805.3984v1 [physics.ins-det] 26 May 2008

Preprint typeset in JINST style - HYPER VERSION

Alignment of the Pixel and SCT Modules for the2004 ATLAS Combined Test Beam

A. Ahmad35,a , A. Andreazza21 , T. Atkinson20 , J. Baines31, A.J. Barr28 , R. Beccherle11,
P.J. Bell17 , J. Bernabeu38, Z. Broklova30 , P. A. Bruckman de Renstrom28,b , D. Cauz37 ,L. Chevalier32, S. Chouridou33, M. Citterio21 , A. Clark10 , M. Cobal37 , T. Cornelissen6,
S. Correard19 , M.J. Costa38 , D. Costanzo34 , S. Cuneo11, M. Dameri11 , G. Darbo11 ,J.B. de Vivie19 , B. Di Girolamo6 , D. Dobos9 , Z. Drasal30 , J. Drohan, K. Einsweiler2,
M. Elsing6 , D. Emelyanov31, C. Escobar38 , K. Facius7 , P. Ferrari6 , D. Fergusson2,
D. Ferrere10 , T. Flick39 , D. Froidevaux6, G. Gagliardi11 , M. Gallas6 , B.J. Gallop31 ,K.K. Gan27 , C. Garcia38 , I.L. Gavrilenko22, C. Gemme11 , P. Gerlach39 , T. Golling2 ,S. Gonzalez-Sevilla38 , M.J. Goodrick5, G. Gorfine26 , T. Göttfert24 , J. Grosse-Knetter3,
P.H. Hansen7, K. Hara36 , R. Härtel24 , A. Harvey13 , R.J. Hawkings6,
F.E.W. Heinemann28, T. Henns39 , J.C. Hill5 , F. Huegging3, E. Jansen25, J. Joseph2,
M. Karagöz Ünel28∗, M. Kataoka6, S. Kersten39, A. Khomich18, R. Klingenberg9,P. Kodys30, T. Koffas6 , N. Konstantinidis16, V. Kostyukhin11, C. Lacasta38, T. Lari21 ,S. Latorre21 , C.G. Lester5 , W. Liebig26 , A. Lipniacka1, K.F. Lourerio27 ,M. Mangin-Brinet10, S. Marti i Garcia38†, M. Mathes3, C. Meroni21 , B. Mikulec10 ,B. Mindur8 , S. Moed10 , G. Moorhead20, P. Morettini11 , E.W.J. Moyse6 , K. Nakamura38 ,P. Nechaeva11, K. Nikolaev15, F. Parodi11 , S. Parzhitski, J. Pater17 , R. Petti4 ,P.W. Phillips31 , B. Pinto29 , A. Poppleton6, K. Reeves39, I. Reisinger9, P. Reznicek30 ,P. Risso11 , D. Robinson5, S. Roe6 , A. Rozanov19 , A. Salzburger14 , H. Sandaker1,
L. Santi37 , C. Schiavi11 , J. Schieck24, J. Schultes39, A. Sfyrla10 , C. Shaw12 ,F. Tegenfeldt6, C.J.W.P. Timmermans25, B. Toczek8, C. Troncon21, M. Tyndel31 ,F. Vernocchi11 , J. Virzi2 , T. Vu Anh10 , M. Warren16 , J. Weber9 , M. Weber31 ,A.R. Weidberg28, J. Weingarten3, P. S. Wells6 , A. Zhelezko23–1–

1 University of Bergen, Department for Physics and Technology, Allegaten 55, NO - 5007 Bergen,

Norway2 Lawrence Berkeley National Laboratory and University of California, Physics Division,

MS50B-6227, 1 Cyclotron Road, Berkeley, CA 94720, United States of America
3 Physikalisches Institut der Universitaet Bonn, Nussallee 12, D - 53115 Bonn, Germany
4 Brookhaven National Laboratory, Physics Department, Bldg. 510A, Upton, NY 11973, UnitedStates of America
5 Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE,

United Kingdom
6 CERN, CH - 1211 Geneva 23, Switzerland
7 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK - 2100 Kobenhavn 0,

Denmark
8 Faculty of Physics and Applied Computer Science of the AGH-University of Science andTechnology, (FPACS, AGH-UST), al. Mickiewicza 30, PL-30059 Cracow, Poland
9 Universitaet Dortmund, Experimentelle Physik IV, DE - 44221 Dortmund, Germany
10 Universite de Geneve, Section de Physique, 24 rue Ernest Ansermet, CH - 1211 Geneve 4,

Switzerland
11 INFN Genova and Università di Genova, Dipartimento di Fisica, via Dodecaneso 33, IT -16146 Genova, Italy
12 University of Glasgow, Department of Physics and Astronomy, UK - Glasgow G12 8QQ, United

Kingdom
13 Hampton University, Department of Physics, Hampton, VA 23668, United States of America
14 Institut fuer Astro- und Teilchenphysik, Technikerstrasse 25, A - 6020 Innsbruck, Austria
15 Joint Institute for Nuclear Research, JINR Dubna, RU - 141 980 Moscow Region, Russia
16 University College London, Department of Physics and Astronomy, Gower Street, London

WC1E 6BT, United Kingdom
17 School of Physics and Astronomy, University of Manchester, UK - Manchester M13 9PL, United

Kingdom18 Universitaet Mannheim, Lehrstuhl fuer Informatik V, B6, 23-29, DE - 68131 Mannheim,

Germany
CPPM, Aix-Marseille Universit, CNRS/IN2P3, Marseille, France
20 School of Physics, University of Melbourne, AU - Parkvill, Victoria 3010, Australia
21 INFN Milano and Università di Milano, Dipartimento di Fisica, via Celoria 16, IT - 20133
Milano, Italy
22 P.N. Lebedev Institute of Physics, Academy of Sciences, Leninsky pr. 53, RU - 117 924 Moscow,
Russia23 Moscow Engineering & Physics Institute (MEPhI), Kashirskoe Shosse 31, RU - 115409
Moscow, Russia
24 Max-Planck-Institut für Physik, (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805
München, Germany
25 Radboud University Nijmegen/NIKHEF, Dept. of Exp. High Energy Physics, Toernooiveld 1, NL- 6525 ED Nijmegen , Netherlands
26
Nikhef National Institute for Subatomic Physics, Kruislaan 409, P.O. Box 41882, NL - 1009 DB
Amsterdam, Netherlands
27 Ohio State University, 191 West Woodruff Ave, Columbus, OH 43210-1117, United States ofAmerica
28 Department of Physics, Oxford University, Denys Wilkinson Building, Keble Road, Oxford OX1
3RH, United Kingdom
29 Laboratorio de Instrumentacao e Fisica Experimental de Particulas - LIP, and SIM/Univ. deLisboa, Avenida Elias Garcia 14-1, PT - 1000-149, Lisboa, Portugal
19–2–

30 Charles University in Prague, Faculty of Mathematics and Physics, Institute of Particle andNuclear Physics, V Holesovickach 2, CZ - 18000 Praha 8, Czech Republic
31 Rutherford Appleton Laboratory, Science and Technology Facilities Council, Harwell Scienceand Innovation Campus, Didcot OX11 0QX, United Kingdom
32 CEA, DSM/DAPNIA, Centre dâĂŹEtudes de Saclay, FR - 91191 Gif-sur-Yvette, France
33 University of California Santa Cruz, Santa Cruz Institute for Particle Physics (SCIPP), Santa

Cruz, CA 95064, United States of America
34 University of Sheffield, Department of Physics & Astronomy, Hounsfield Road, UK - Sheffield S3

7RH, United Kingdom
35 Insitute of Physics, Academia Sinica, TW - Taipei 11529, Taiwan
36 University of Tsukuba, Institute of Pure and Applied Sciences, 1-1-1 Tennoudai, Tsukuba-shi,

JP - Ibaraki 305-8571, Japan
37 INFN Gruppo Collegato di Udine and Università di Udine, Dipartimento di Fisica, via delle

Scienze 208, IT - 33100 Udine; INFN Gruppo Collegato di Udine and ICTP, Strada Costiera 11,
IT - 34014 Trieste, Italy
38 Instituto de Física Corpuscular (IFIC), Centro Mixto UVEG-CSIC, Apdo. 22085, ES-46071
Valencia; Dept. Física At., Mol. y Nuclear, Univ. of Valencia and Instituto de Microelectrónica
de Barcelona (IMB-CNM-CSIC), 08193 Bellaterra, Barcelona, Spain
39
Bergische Universitaet, Fachbereich C, Physik, Postfach 100127, Gauss-Strasse 20, DE-42097
Wuppertal, Germanya Now at State University of New York at Stony Brookb also at H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland

Email: muge.karagoz.unel@cern.ch,martis@ific.uv.es

A BSTRACT: A small set of final prototypes of the ATLAS Inner Detector silicon tracker (Pixel andSCT) were used to take data during the 2004 Combined Test Beam. Data were collected from runs
with beams of different flavour (electrons, pions, muons and photons) with a momentum range of 2
to 180 GeV/c. Four independent methods were used to align the silicon modules. The corrections
obtained were validated using the known momenta of the beam particles and were shown to yieldconsistent results among the different alignment approaches. From the residual distributions, it isconcluded that the precision attained in the alignment of the silicon modules is of the order of 5 µ m
in their most precise coordinate.
K EYWORDS : Detector alignment and calibration methods, Solid state detectors, Particle tracking
detectors, Large detector systems for particle and astroparticle physics.

∗ Corresponding author† Corresponding author

Contents
1.

Introduction1

2.

Setup, Data Samples and Tracking
2.1 Simulation
2.2 Tracking and Reconstruction

2
3
4

3.

Alignment of the CTB Data
3.1 The Robust approach
3.2 The Valencia approach
3.3 The Local χ 2 approach
3.4 The Global χ 2 approach

6
7
8
11
12

4.

Results

14

5.

Summary and Conclusion

16

1. Introduction
This note reports the results of the alignment of the ATLAS Inner Detector [1] silicon tracker (Pixeland SCT) modules at the ATLAS Combined Test Beam data-taking (CTB) which took place at the
CERN H8 beam-test facility in 2004. The purpose of the CTB was to study the combined performance of ATLAS. The setup represented a full barrel slice of the Inner Detector (ID), Calorimeterand Muon Spectrometer of the complete ATLAS detector and was instrumented with final prototypes.
Once the Pixel and SCT modules had been installed in the CTB setup in addition to the alreadyoperational TRT, the Inner Detector was fully integrated into the common data acquisition system.
Data were collected with this fully integrated ID, using beams with different characteristics. Pion,
electron, muon and photon beams were used in a wide range of momenta from 2 to 180 GeV /c,and some data were taken without magnetic field (B).
The CTB setup represented an ideal framework for testing the Inner Detector software. The
offline reconstruction was tested on real data using the ATLAS software framework (ATHENA) [2]and was particularly useful for tracking [3], pre-commissioning tests, and for testing the alignment
software.
Determining the locations of the tracking detector elements is crucial for the performance of
the ID tracker. For this purpose, various alignment algorithms, based on optimization of track
hit residuals, were applied to align the CTB silicon setup. An alignment algorithm specificallydeveloped for the CTB (hereafter referred to as Valencia approach [4]) had been adapted from an–1–algorithm used in previous SCT standalone test beams [5], by the time the first data were collected.
The Valencia approach produced alignment corrections for the initial CTB data analysis. For the
final analysis of the alignment, three more algorithms were tested. These algorithms, developed
for the alignment of the entire Inner Detector silicon tracker, are: Robust [6], Local χ 2 [7, 8] andGlobal χ 2 [9, 10] approaches [11].
The resulting sets of alignment constants were used to measure the momenta of the incident
particles in electron and pion runs. A comparison with the nominal momenta was used to crosscheck the different alignment procedures. The residual distributions and reconstructed track parameters were studied for electrons and pions with and without B field. The global reference frame
was also studied by matching the alignment results via a global offset optimization.

2. Setup, Data Samples and Tracking
The Inner Detector volume in the CTB setup was divided into three containers for each subdetector: Pixel, SCT and TRT. Six Pixel and eight SCT modules were placed in their respective
containers1 . The TRT setup consisted of two barrel wedges, equivalent to 1/16 of the circumferenceof a cylinder.
The coordinate system was chosen to be right-handed, with the X -axis along the beam direction and the Y -axis pointing vertically upward as depicted in Fig. 1 [12]. The origin was located at
the entrance of the dipole magnet that produced a maximum 1.4 T field in the negative Z-direction.
The Pixel and SCT detectors were located inside the magnet whereas the TRT detector was locatedoutside due to its larger dimension.

y

x

z

6 PIX

8 SCT

Figure 1. Schematic representation of the ID components and the magnet at the CTB. The reference coordinate system is also shown. The long pixel coordinate and the SCT module strips are almost parallel to theZ-axis.

A Pixel module [13, 14] consists of a single silicon wafer with an array of 50×400 µ m2 pixels
that are read out by 16 chips [15]. The active area of each module is ∼ 60.8 × 16.4 mm2 . In the
CTB setup, six Pixel modules were distributed in three layers (0,1,2) and two sectors (0,1). The
distances along the beam axis between the different layers and the locations of modules withineach layer mimic the arrangement of the modules in ATLAS. The first Pixel layer was nominally
1 The ATLAS detector has, in total, 1744 Pixel modules and 4088 SCT modules.

–2–

located at 195.986 mm from the global coordinate center along the beamline (X -axis) and the lastlayer was located at 268.277 mm. Each module was positioned at an angle of about 20◦ with respect
to the incident beam, around the long pixel coordinate. Modules in the same layer overlapped by∼ 200 µ m.
A SCT module is built from four single-sided silicon microstrip sensors glued back to back inpairs with 40 mrad stereo angle for a 3D space-point reconstruction [16, 17]. The modules produce
two hits, one in each plane. The SCT end-cap modules have a wedge-shaped geometry which
results in variable pitch sizes (Fig. 5). In the CTB setup, one of the four shape-wise distinct SCT
end-cap module types was used (outer module). For the outer end-cap modules, the readout strip
pitch is 70.9-81.1 µ m. Each plane has a length of about 120.0 mm and bases of about 72.0 mm
and 57.0 mm. The readout is provided by a binary chip [18]. Eight SCT modules were used ineach of the four layers (0,3) of the CTB setup; distributed in two sectors (0,1) with a 4 mm overlap.
The arrangement of the modules was similar to the SCT barrel configuration in ATLAS2 , however,
the modules were not mounted at an angle with respect to the beam axis. The SCT modules were
nominally positioned from 378.198 mm to 598.218 mm along the beam axis.
The beam-line instrumentation, including trigger and veto scintillators, Cherenkov countersand readout system is documented elsewhere [19, 20]. The Inner Detector magnetic field profile
was measured [12] and its non-uniformity was taken into account during the track reconstruction.
The absolute momentum as measured by the silicon detector which was located in a very uniform
magnetic field region was certified to better than 1% by comparing the momentum reconstructedfrom silicon alone with that obtained independently using the angular measurement in the TRT.
The CTB ID data taking was divided into five different periods between September 2004 andNovember 2004 [12], where 22 million usable events were collected. In order to evaluate thematerial effects in the tracker, aluminum plates (10% X0 ) were inserted and removed between the
Pixel, SCT and TRT setups (Fig. 1) in alternate runs. The TRT was repositioned in the transverse
plane of the beam. Particle type and energy of the beam also alternated during the periods.
The algorithms provided a valid silicon detector alignment for all the CTB data-taking periods. However, this article reports on the last period (period 5) of stable data-taking when no extra
material layers were used. Table 1 lists the runs used for alignment studies in this period. Eventsfrom run 2102355, a 100 GeV pion beam run without a B-field, were used as input to all algorithms
for the production of alignment corrections. For the Local χ 2 approach, two other pion runs wereused in addition. Further event selection details are given in Section 3.
2.1 Simulation
The CTB setup was simulated with Geant 4 using the same geometry description as the event reconstruction. Detector positions and initial numbers were provided through an Oracle-based conditions
database (look-up information) which allowed the five different periods to be distinguished fromone another.
CTB specific modifications were applied to the simulation for studying the Pixel and SCTalignment, i.e. the propagation through material upstream of the ID and the inclusion of mea2 The rectangular barrel modules which have uniform 80 µ m pitch were not used due to their unavailability during

test beam data-taking.

–3–

Table 1. List of selected runs used to assess the alignment results.

Run Number
210235521024392102400
21024522102399210246321024422102365

Particle Type
π
e
e
e
e
e
π
π

Energy (GeV)
100
20
50
80
100
180
20
100

B field
Off
On
On
On
On
On
On
On

sured beam profiles. The upstream material (mainly air and triggering/monitoring scintillators)
corresponded to 13.2% radiation lengths and was taken into account to mimic the momentum distribution in the data properly. Profiles, consisting of beam incidence positions and angles, weretaken from the data and were applied during the upstream simulation to bring the simulated hit
maps and residual distributions of the silicon modules into agreement with the data.
The magnetic field map was calculated taking into account the magnet geometry, in one quadrant of the transverse plane with respect to the beam axis. The remaining field map was modeled
assuming a symmetric the field map around the main axis of the magnet. The field map calculated along these lines compares well with the actual measurement of the dipole field which were
performed before and after the CTB runs.
2.2 Tracking and Reconstruction
The default tracking algorithm in the CTB was the ‘CTBTracking’ algorithm [3]. CTBTracking
consists of a pattern recognition part, developed specially for the CTB, and a track fitting algorithm
that is in use in full ATLAS as well as in the CTB. The pattern recognition finds the tracks bylooping through combinations of space points. The track fitting algorithm is based on a global
χ 2 minimization technique, often called the ‘breakpoint’ method in the literature [21]. Multiple
scattering and energy loss enter into the algorithm as additional fit parameters at a given numberof scattering planes. The track fit has a custom description of the detector material in the test beam
setup, with one scattering plane for each layer of silicon modules. This material description was
precisely tuned to give the best possible track resolutions, down to very low energies (1 GeV orless). A number of options and features exist in the track fit that are particularly useful for thealignment algorithms, such as the possibility of setting the momentum to a fixed value in the fit,
and the ability to retrieve the fitted scattering angles and their covariances.
Fig. 2 shows typical hit maps for a Pixel and SCT module. The illumination was rather uniform
for the channels that lay within the scintillator trigger acceptance window in the central region(∼ 3 × 3 cm2 wide). More details on the tracking performance of the pixel detectors can be foundelsewhere [22]. Unmasked noisy channels can be distinguished in the SCT hitmap. Those that weremasked during data-acquisition appear as zero-entry channels. The illumination was not uniform

–4–

14

250
12

200

10

150

8

Entries / strip

Row number

16

300

6

100

4

50
0
0

250
200
150
100
50

2

20

40

60

80 100 120 140
Col number000

100 200 300 400 500 600 700
Strip number

Figure 2. Left: Pixel hit map for 100 GeV pion runs. The x-axis corresponds to Pixel η -coordinate andthe y-axis to Pixel φ -coordinate. Empty horizontal bands correspond to the ganged pixels. Lighter verticalbands are due to the 600-µ m-wide pixels. Right: SCT hit map for the same run.

and limited along the strip length but only in the central region, where the sensor planes overlappedcompletely with the trigger scintillator.
The limited illumination of the sensors had direct consequences on some of the alignment
degrees of freedom (DoF) due to insufficient constraints and reduced sensitivity. The problem
was more severe for SCT modules, because the SCT modules were not tilted with respect to thebeamline. As the beam incidence was almost perpendicular to the module planes, the alignment
procedures were not very sensitive towards misalignments along the beam axis.
The pixel sensors require free space in order to bond the readout chips on the surface of thesensor. In the precise φ coordinate, unbonded pixels are physically connected to nearby pixels(ganged pixels) and share a readout logic channel. Due to this connection, whenever a hit was
registered by a logic channel, there was an ambiguity as to which pixel fired. In the long coordinate
wider pixels (600-µ m instead of 400-µ m) are used. The wider pixels collect more hits. The impact
of both effects is clearly seen in the pixel module hit map (Fig. 2). The ambiguity in the ganged
pixels was also found to effect the alignment. In a highly misaligned environment, tracking may
make too many wrong decisions between ganged pixels. It was found that, in the presence of a hightrack quality cut, the ganged pixel hits were favoured, degrading the quality of the alignment [6].
The fact that the modules were exposed to almost perpendicular beams resulted in discrete
Pixel η -residual distributions. Due to the large dimension in this direction (400 µ m compared to
300 µ m of the thickness of the silicon bulk) the drift of the charge carriers along that direction is
negligible. Therefore, almost all of the clusters consist of a single pixel in the η -coordinate. As
the cluster position is located in its geometrical center, the outcome is a discrete positioning ofclusters (Fig. 3). With only three pixel layers providing three precision points, a discrete residual
distribution was obtained. The use of SCT clusters in the tracking partially removed this undesired effect [4]. Effectively the pixel η -residuals of the first and last pixel layers were somewhat
broadened by overlaps of Gaussian distributions, while the middle layer η -residuals remained dis-–5–Pixel module [layer 1, sector 1]

1800

Data

500

Simulation

Entries

Entries

Pixel module [layer 0, sector 1]

Data

1600

Simulation

1400
400

1200
1000

300

800
200

600
400

100

200
0
-0.6

-0.4

-0.2

0

0
-0.6

0.2
0.4
0.6
η-residual (mm)

-0.4

-0.2

0

0.2
0.4
0.6
η-residual (mm)Figure 3. Pixel η -residuals for tracks reconstructed with Pixels and SCT. Left: first Pixel layer. Right:
middle Pixel layer.

crete. This peculiarity of the CTB setup made the alignment along the pixel η -coordinate difficult.
ATLAS collision data will not present such difficulties.

3. Alignment of the CTB Data
The goal of alignment is to determine the corrections to the parameters that describe the positionand orientation of the module in space. Each module is treated as a flat rigid body with 6 DoFs,i.e., three translations along the local coordinate axes (x, y, z) and three rotations (α , β , γ ) aroundthe local coordinate axes, in a right-handed orthogonal frame where the origin is at the centerof-gravity of each module and the local x-coordinate is along the most precise coordinate. The
translations correspond to the shift of the module with respect to its nominal position. For the axes
orientation, the Cardano representation of angular rotation with respect to the cartesian axes wasused. The alignment corrections were stored in the conditions database.
The alignment corrections are given in terms of CLHEP [23] transform objects H, made of arotation matrix R and a translation vector ~T . The rotation matrix is defined as:
R = Rx (α ) · Ry (β ) · Rz (γ )(3.1)

with α , β and γ being the rotation angles around the x, y and z-axes. Rz (γ ) is the first rotation
applied and Rx (α ) the last. The representation of a point in the local reference frame (~P) of amodule is H · ~P = ~T + R · ~P in the global frame. Lets consider H0 as the transformation specifying
the nominal position of a given module. If δ H is a shift of the module, the new transformation of
the points measured by it becomes H = H0 · δ H. Therefore, the task of the alignment is to determine
the 6 DoFs that define δ H for each module. In the case of poorly constrained movements, some
DoFs may not be considered.
The technique to align each silicon module consists of minimizing its two residuals (pixel
modules measure two coordinates and each SCT module has two sensor planes). The i-th residual–6–

(defined by coordinate, plane or module) is thus ri ≡ (~mi −~ei (a, π )) · k̂i , where m~ i represents the
position of the hit recorded in the sensor plane, ~ei is the intersection point of the extrapolated track
to the detector that depends on the alignment parameters (a) and the vector of track parameters
π = (d0 , φ0 , z0 , θ0 , q/p). k̂i denotes the unit vector of the measurement direction [9].
All alignment algorithms were run iteratively. Initially, nominal detector and hit positions
were used for track reconstruction. After the track fit, residuals and their derivatives with respect
to alignment and/or track parameters were calculated to determine the alignment corrections. For
each module, the best fit estimates for alignment parameters were derived and its position was
updated. A new reconstruction with updated module positions was performed and the alignment
was reiterated. This procedure is expected to converge to final alignment corrections for each
module and the residual resolution is expected to improve.
The alignment was performed using two different classes of approach. The Robust approachis based on iterative minimization of the residual means of overlapping and non-overlapping modules. The approach is “robust" because the output is stable against changes in the input tracking
information.
The Valencia , Local χ 2 and Global χ 2 approaches are based on the linear least squares minimization defined for a set of reconstructed tracks as:χ 2 (a, π 1 , . . . , π t ) =∑ri T Vi−1 ri(3.2)

i ∈ tracks

where ri = ri (a, π i ) is the vector of residuals measured for the fitted track i. Vi is the covariancematrix of the residual measurements of track i. The generic solution for alignment corrections (δ a)
is:
 T

!−1
 T
 T
dri
dri
dri
∂
r
i
−1
−1
−1
δa = −
V ri = −A
V −1 ri ,∑ da V
∑
∑
∂
a
da
da
i ∈ tracksi ∈ tracksi ∈ tracks(3.3)−1
where A is the covariance matrix for δ a. The size and contents of the matrix A depend on thedetails of the alignment method which are explained in the following sections.
3.1 The Robust approach
The Robust alignment approach [6] is an iterative method to align a silicon detector with overlapping modules. In each iteration alignment corrections are calculated from measurements of mean
residuals, res, and mean overlap residuals, ovres, in the x and y coordinates. Overlap residuals aredefined as the difference between two residuals from two overlapping modules. y SCT residuals
are constructed using both hits from each side in a module. The algorithm only corrects for shifts
in the plane of the module. The alignment corrections are given by:
3
sj
1
/
.
∑
2
2
j=1 (δ s j )
j=1 (δ s j )3ax/y = − ∑(3.4)

s1 to s3 are defined as: s1 = res; s2 = ∑ ovresx ; s3 = ∑ ovresy , where δ s j are the measurement
uncertainties. The range of the sum depends on the geometry of the detector. Given the simple
CTB geometry, a straightforward implementation of Eqn. 3.4 was used.

–7–

The alignment corrections were obtained as follows: The fractions of non-overlap and overlap
hits in the sample were controlled by coefficients A for overlap hits and B for non-overlap hits, toadjust the influence of each information on the x and y correction. The corrections were weighted
with the ratio of the total number of overlap hits A · nohx/y and the number of hits B · nhx/y to thetotal number of hits, Nx/y . The total residual weight rwx/y and the total overlap residual weight
orwx/y obtained this way corresponded to 1/δ s j 2 :rwx/y =

B · nhx/y,Nx/yorwx/y =A · nohX/y
Nx/y

(3.5)

There was one overlap for each two modules in a layer. Thus, this information could be used for
only one sector which was arbitrarily chosen to be sector 1. The alignment corrections for modules
in Sector 1 is given by Eqn. 3.6 and in Sector 0 is given by Eqn. 3.7, as:ax/y = −orwx/Y · ovresx/y − rwx/y · resx/y(3.6)

ax/y = −rx/y(3.7)

The CTB alignment was carried out using “unbiased" residuals, i.e., the hit of the aligned wafer
on the side of the module was removed from the track fit. About 72,000 events from run 2102355
were used for the alignment. This run contained about 10 to 50 times more hits than overlap hits.
Information from residual distributions and overlap residual distributions were weighted so thatoverlap residuals had almost similar influence: setting A to 10 and B to 1 was found optimal.
Further tests showed that other values affected the speed of convergence rather than the final result.
There were two major limitations in the application of the Robust algorithm to the CTB data.
First, significant tilts arose from the hand-mounted modules in the setup. In contrast with the otheralgorithms, the Robust algorithm does not correct for rotations. Therefore, after alignment, theresiduals still had a global Y dependence, in agreement with the tilts observed around the Pixel yaxis (see Sec. 3.4). The dependence vanished when the modules were rotated accordingly. This is
the main reason why the residual resolution after the Robust alignment were not as good as the ones
achieved by other algorithms. The modules with the largest residuals after the Robust Alignment
correspond to the modules with the largest rotations. Second, discrete Pixel y (η ) residuals resulted
in less stable mean of the residuals with respect to any small shifts.
The Robust algorithm converged on a solution without a tight track selection. Although 30
iterations were performed to align the detector, stable results were achieved after 15 iterations. The
residuals improved significantly and the track quality stabilized after a small number of iterations.
After 30 iterations, about 1 µ m global shifts of module positions in the negative x direction wereobserved (Fig. 4). The Robust algorithm had the advantage of requiring minimal computing resources. The CPU time used by the algorithm were shown to be negligible compared to that of thepreceding track reconstruction.
3.2 The Valencia approach
The Valencia alignment algorithm [4] is based on the numerical minimization of the χ 2 function
defined in Eqn. 3.2 using “biased" residuals (i.e., the hit of the module being aligned is included in
the track fit). The covariance matrix is assumed to be diagonal and the diagonal elements are filled–8–Shifts in 30th iteration [mm]

0.005
0.004

PIXEL Local X

0.003

SCT Local X

0.002
0.001
0.000
-0.001-0.002-0.003-0.004-0.005
0-1

1-1

2-1

3-1

Layer - Sector

Figure 4. Shifts in the x direction for the 30th iteration of the Robust alignment. Statistical uncertainties
on the measurements are represented by the error bars. The quality of the reconstructed track parameters isinsensitive to this negligible systematic effect.

with the measurement uncertainties, σri , for residuals, ri , both of which are calculated numerically.
The only fit parameters are the alignment corrections, ignoring the correlation between track andalignment parameters. The algorithm is therefore executed iteratively, alternating between track
and alignment fits.
The SCT endcap outer module strips follow a fan-out geometry and thus have a variable pitch
along the vertical direction (Sec. 2). Therefore, instead of using the standard “linear” residual(perpendicular distance from the track prediction to the strip), “angular” residuals (δ γ ) were used
(Fig. 5). These represent the difference between the angular separation of the signal channel anda “fictitious” strip passing through the extrapolated point. The strip-pitch dependence was thus
avoided, and uniform angular residuals were obtained.
The outlier hit rejection was applied by defining an acceptance region determined by a criticalvalue of the χ 2 (outlier rejection). This value was taken as three standard deviations with respect
to the mean value of the reduced residual distribution (ri /σri ) calculated before the minimization.
The fraction of measurements lying out of the acceptance region was ∼3%, and reduced to below
1% if five standard deviations were used.
The Valencia algorithm was intended for runs without magnetic field yielding straight tracks.
After reconstruction, each track was extrapolated to the silicon modules. If the extrapolation lay
outside the module geometrical acceptance, the track prediction was discarded. The module intersection point of the accepted tracks was transformed into the local frame and residuals were calculated. For Pixels, only measurements in the φ (x) direction were considered. The η -coordinate
was ignored due to the non-Gaussian residual distributions (Fig. 3). For the SCT modules, angular residuals and measurements from both SCT sides were used3 . Although an analytical residual
linearisation as a function of the alignment parameters was not computed, the dependence of theχ 2 on the alignment parameters remained linear. Fig. 6 shows the contour regions for two fitted
variables and three different confidence level intervals (68%, 90% and 95%) for one Pixel module.
3 Except for module [layer 2, phi 1] with a single working plane.

–9–commonintersection point

artificial strip
DOCA residual

linearresidual

prediction

signaldg
angularresidual

g1
particle trackg2

γ (mrad)Figure 5. Residual definitions in a SCT end-cap module used by the different alignment algorithms: theangular and linear residuals and distance of closest approach (DOCA).

0.4
0.2

68% CL
90% CL
95% CL

-0
-0.2
-0.4
-0.6
-0.8
-1
-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2
T y ( µm)

Figure 6. Contour ellipses of constant probability content for three different confidence levels and twoalignment parameters of a Pixel module; translation along the most sensitive coordinate (horizontal axis)
and tilt angle (vertical axis), as obtained by the Valencia alignment algorithm.

The alignment was performed in three consecutive steps, with variable number of iterations
in each step: (st-1) internal alignment of the Pixel modules, (st-2) broad alignment of the SCT
modules with respect to the Pixel system, and (st-3) fine alignment of all silicon modules. In st-1(∼ 6 iterations), tracks lying in the overlap region between Pixel modules in the same layer wereselected to enhance the number of overlap hits and to produce a pixel alignment. In st-2 (∼ 2
iterations), tracks reconstructed only with the pixel hits were extrapolated to the SCT planes. In
this manner, it was possible to compute SCT residuals (unbiased only in this case) which served as– 10 –

input for an initial alignment of the SCT modules with respect to the Pixel modules. The required
correction of the SCT modules was several hundreds of microns. In st-3 (∼ 8 iterations), all silicon
modules were included in the track fit and all were aligned simultaneously. In this last stage thealignment corrections per module were of few micrometers.
During alignment, the first Pixel module [layer 0, phi 0] was kept as an anchor; fixed to its
nominal position to fix the global degrees of freedom. DoFs to which the sensitivity was very small
were excluded from the set of fitted alignment parameters. Module positions along the beam axis
were not considered. For Pixels, only the displacements along the sensitive coordinate were fitted.
The tilt angle (γ ) was excluded in (st-1 and st-2), but fitted in step (st-3). For the SCT modules, theparameters for displacements along and across the sensitive coordinate together with the in-planerotation were fitted in all steps. The inclusion of one additional angle (β ) during the last iterations
was found to marginally help to improve the results for both sub-detectors.
3.3 The Local χ 2 approach
The Local χ 2 approach [7, 8] derives from Eqn. 3.2. The χ 2 -function uses unbiased residuals,
which are defined as the 3D distance of closest approach (Fig. 5). The algorithm uses a diagonal covariance matrix, V that is simlar to that of the Valencia approach. The residual errors arecalculated using hit errors and the extrapolated tracking errors.
The Local χ 2 algorithm produces alignment constants for each module separately, neglecting
correlations between the modules during an iteration. Thus, the solution reduces to inverting as
many N × N matrices as there are modules, where N corresponds to the DoFs of each module (upto 6). Track parameters with a better fit quality gradually bring correlations into play after every
iteration.
The fact that CTB was found to be a degenerate setup for track-based alignment required
inclusion of external constraints to resolve some of the degeneracies. These were a momentum
constraint to the reconstructed tracks and an additional stabilization term to the diagonal elementsof the matrix A in Eqn. 3.3. The stabilization term acts like an additional measurement with azero residual, full sensitivity in the corresponding degree of freedom (the derivative in Eqn. 3.3 isequal to one) and an uncertainty σstability . The uncertainty σstability corresponds to the inverse of thesquare root of the added term. These additional stability terms constrain the movement to be withinσstability . The values for σstability are 10, 10, 100 µ m for the Pixel x,y,z coordinates and 100 µ m for
the SCT x,y,z coordinates. For the module rotations the value for σstability is set to one mrad.
The momentum of the incident particles from SPS is known more precisely than the intrinsic
momentum resolution of the CTB ID setup. Consequently, this information can be used to constrain
the track curvature. Tracks with different beam energies were used as input, using 10,000 eventsfrom each pion run listed in Table 1. The alignement procedure was parallelized where multiple
jobs with different momentum constraint settings were executed simultaneously. When jobs were
finished the alignment information was collected and merged. Subsequently a new iteration with a
new set of parallel jobs was started.
The usage of overlap hits, a hit lying in the overlap region of two modules on the same layer,
has a profound impact on alignment by constraining relative positions of both sectors, thus avoiding
divergences due to lack of external constraints. Residual calculation is also more precise for overlap
hits than non-overlap hits. Non-overlap hits were rejected for alignment once a defined limit was– 11 –

Alignment Parameter x

Alignment Parameter y

Alignment Parameter z

300

200

1000

150

200

100

0

-50

µm

µm

µm

500

100

50

0

-100

0

-500

-100
-200

-150
-200
10

20

30
40
Iterations

50

60

Layer 0, Sector 0
Layer 0, Sector 1
Layer 1, Sector 0
Layer 1, Sector 1
Layer 2, Sector 0
Layer 2, Sector 1

20

30
40
Iterations

50

60

0

10

20

30
40
Iterations

50

60

50

60

Alignment Parameter γ

4
3

40

2
20
0

1
0
-1

-20

-2

-5
-40
-10
0

0

60

mrad

mrad

5

10

Alignment Parameter β

Alignment Parameter α

10

-300
0

mrad

0

10

20

30
40
Iterations

50

60

-60
0

-3
-4
10

20

30
40
Iterations

50

60

0

10

20

30
40
Iterations

Figure 7. Flow of alignment parameters of the 6 Pixel modules through the iterations of the Local χ 2alignment algorithm.

reached. In this way the number of overlap hits was enriched with respect to the number of nonoverlap hits. The maximum number of non-overlap hits was set to 400.
For the alignment the iteration chain was performed 60 times. The flow of the 6 alignment
parameters of each Pixel module through the iterations is shown in Fig. 7. After 10 iterations, nearly
all degrees of freedom of all modules converged on stable values. Slower convergence of someparameters was due to the imposed stability term. The procedure was stopped after 60 iterations,when no significant improvement of track parameters was observed and alignment corrections forthe sensitive coordinates were at the submicrometer level.
3.4 The Global χ 2 approach
The Global χ 2 algorithm [9, 10] is based on the minimization of the χ 2 defined as Eqn. 3.2 withrespect to the alignment parameters. The residuals are defined within the module plane and are
biased (i.e., the hit of the module being aligned is included in the track fit). They depend on the
track parameters (π ) as well as on the subset of alignment parameters related to the intersected
module (a):dr∂ r ∂ r dπ
=
+
.
(3.8)
da ∂ a ∂ π da
The method has the advantage of properly treating all correlations between residuals arisingfrom common track parameters and Multiple Coulomb Scattering (MCS). Solution 3.3 requires
inverting a symmetric matrix of size N × N, where N is the number of DoFs of the problem. For– 12 –

Mode Eigenvalues

Mode Eigenvalues

Eigenvalue

Eigenvalue

6

10

6

10

105

105

104

104

103

103

102

102

10

10

1

1

10-1

10-1

10-2

10-2

10-3

10-3

10-4

10-4

-5

10 0

-5

10

20

30

40

50

60

70
80
Mode Number

10 0

5

10

15

20

25

30

35
40
45
Mode Number

Figure 8. Left: original eigenspectrum of the unconstrained solution (84 DoFs - open squares) overlaid
with that of the actual solution (42 DoFs - solid triangles) of Global χ 2 algorithm. Six solid circles on theleftmost are the singular modes of the asymptotic freedom of the system. Their actual value is zero up to thenumerical precision and were fixed to 10−5 for clarity only. Right: actual solution eigenspectrum (42 DoFs
- solid triangles) compared to the analogous one without the anchor modules requirement (open squares).
The five solid circles are the new near-singularities.

large systems (for instance, the entire ATLAS ID), the solution with accurate numerical precisionand in a reasonable CPU time could be a challenge [24]. In the CTB case, however, the system
consisted of just 14 silicon modules. Therefore it was free from such numerical limitations. Intrinsic alignment of an unconstrained system always leads to a singular matrix and consequently
an ill-defined solution. This is best solved by diagonalization of the matrix. The singular modes
can subsequently be ignored in the solution. The procedure can be further extended to remove all“weak modes” which either represent unphysical deformations or have an associated error exceeding expected misalignments.
In order to solve the CTB alignment, the following approach was adopted: two anchor modules were chosen (the first Pixel and the last SCT) which removed the exact singularities fromthe solution. All considered tracks were nearly parallel to one another and orthogonal to the SCT
module planes. Also the y tilt angles of the Pixel modules were considered to be very accurately
known from the survey. Consequently the following DoFs were removed from the fit: out of theplane translation and the rotations with respect to x and y-axes. This choice resulted in 3 DoFs
per module (36 in total). However, results indicated a substantial residual misalignment related tothe uncorrected y rotation of the Pixel modules. The largest misalignment was found for the upper
module in layer 2 with a value of 25.2 ± 0.5 mrad. The y rotations of the Pixel modules were eventually included in the alignment fit which efficiently eliminated the corresponding misalignments.
The final alignment was concerned with 42 DoFs. Fig. 8 shows the comparison of eigenspectra(obtained by DSPEV routine from the LAPACK library [25]) of the unconstrained CTB geometry
and the one used for the final alignment. Elimination of unphysical parameters efficiently removed
the lowest part of the eigenspectrum. Fig. 8 gives also the comparison of the final alignment to theone without anchor modules. The five weak modes correspond to the approximate4 freedom of two
global translations and three rotations of the entire setup.
4 Axes of local reference systems in different modules are not parallel which lifts the perfect translational degeneracy.

– 13 –

10

Before alignment
After alignment

3

Arbitrary units

Arbitrary units

104

104

<µ> = -0.7 µm
σ = 7 µm

102

10

3

<µ> = -0.7 µm
σ = 20 µm

102

10

10

1
-100 -80

Before alignment
After alignment

1
-60

-40

-20

0

20

40

60
80 100Residual (µm)

-200

-150

-100

-50

0

50

100
150
200
Residual (µm)

Figure 9. Residuals of all Pixels (left) and all SCT (right) modules before and after alignment. The width ofthe distribution is consistent with the intrinsic resolution of the modules.

The method required four iterations for convergence, however a total of seven iterations wasused on about 50,000 events at each iteration. Translations of some SCT modules in y-direction
were found to be as large as 1.5 mm; x translations never exceeded 0.4 mm.

4. Results
In order to assess the quality of the alignment, one must check the track reconstruction quality andphysics observables. For this purpose, the alignment corrections were applied to the data detailedin Table 1.
After aligning the modules, the track finding efficiency increased. For example, for the Robustalignment approach, the number of tracks per event was found to stabilize at around 0.95. As
expected, an average of three hits in the pixels and eight in the SCT (two per module) were found.
All four alignment approaches produced similar performances, consistent with the simulation.
A well-aligned setup returns residuals with a mean of zero and a width consistent with the
intrinsic resolution of the detector and the track fit errors. Fig. 9 shows the biased x-residualsof all the Pixel and SCT modules for the 100 GeV pion run, for those tracks which had at least
three pixel and six SCT hits. The width of the distribution after alignment is consistent with the
intrinsic resolution of Pixels and SCT modules. Fig. 10 shows the mean of Pixel module residuals
for an example run (20 GeV/c pion run). While simulation residual means are centered aroundzero for all modules, the aligned detector data show fluctuations. From the size of the fluctuations,
we conclude that the Pixel residuals of all alignment methods agree within 5 µ m over the whole
momentum range. Fig. 10 also shows a good agreement between the χ 2 minimization methods andthe simulation on the residual resolutions. The Robust method results in a worse residual resolution
since this method only corrects for alignment shifts in the module plane. Fig. 10 also reveals adependence of the σ of the pixel residuals on the module number. This indicates contributions to
the resolution from the geometry of the setup in addition to the intrinsic detector resolution. The
residuals also vary because the track error varies along the track due to MCS, for example.
Similarly, rotations are free only approximately.

– 14 –

Sigma (µ m)Mean (µ m)20
Valencia
Global15

Robust
Local
Simulation

10

30

25

20

5
0

15

-5

10

-10
5
-15
-20

00

01

10

0

11
20
21
Pixel module [layer, phi]

00

01

10

11
20
21
Pixel module [layer, phi]20

Sigma (µ m)Mean (µ m)

Figure 10. Pixel biased φ -residuals mean (left) and σ (right) distributions for the aligned data and thesimulation, for the 20 GeV pion run with B=1.4 T.

Valencia
Global15

Robust
Local
Simulation

40

35

10

30

5

25

0

20

-5

15

-10 000 001 010 011 100 101 110 111 200 201 210 211 300 301 310 311

10 000 001 010 011 100 101 110 111 200 201 210 211 300 301 310 311

SCT module [layer, phi, side]

SCT module [layer, phi, side]

Figure 11. SCT biased residuals mean (left) and σ (right) distributions for the aligned data and the simulation, for a 20 GeV pion run with B=1.4 T. A single SCT module has two entries in each plot (e.g. 000 and
001 correspond, respectively, to the module front and back plane at layer 0, sector 0). The dip at index 210
in the right plot corresponds to the plane which was not working.

The residual mean distributions for each SCT plane are shown in Fig. 11. Systematic correlations in the signs of the means is observed among the alignment algorithm results. Fig. 11 also
shows that the residual resolution of the SCT modules in aligned data (except those reconstructedusing Robust method alignment corrections) are around 20 µ m, which is in good agreement withthe simulation.
All track parameters at the perigee (d0 , z0 , φ0 and θ0 and the momentum) were examined whentracks were reconstructed with the alignment corrections from the four algorithms. The values ofspatial track parameters were not exactly similar for tracks reconstructed with different constants,
however, they followed consistent trends for the runs studied. The difference can be attributed
to the insufficiently constrained global degrees of freedom. The residuals and curvature, hence– 15 –

the track fit χ 2 and the pT , are invariant under rigid body translations and rotations of the wholesystem.
The reconstructed φ0 and θ0 values depend on the beam properties as well as the module
locations provided by the algorithms. Therefore, the measured φ0 and θ0 in data were used to tune
the beam spread in the simulation and to evaluate the interaction length, X0 , upstream of the CTB
setup. The track φ0 and θ0 resolutions improved with increasing momentum, as expected with areduced MCS for more energetic particles.
The momentum reconstruction provides a very powerful test of the alignment performance.
Fig. 12 shows the recovery of the momentum resolution of the 100 GeV pion run after alignment,from a highly degraded initial measurement. The momentum measurement does not depend on
global transformations. Therefore the momenta of the tracks reconstructed with different alignment
constants ought to agree. Fig. 13 is used to compare the electron and pion momenta resolution
as a function of the reconstructed momentum obtained from the four alignment methods to thesimulation. The momenta reconstructed using all algorithms, in particular χ 2 minimization ones,
are consistent with the simulation. The Robust method returns slightly worse results since thealignment does not take rotations of the modules into account.
The reconstructed electron momentum is significantly less than the nominal (set by the beamline), for both data and simulation. The presence of several layers of upstream material can accountfor this effect, because the radiated energy of electrons before they enter the tracking volume was
not recovered. As pions do not suffer as much from energy loss, their reconstructed momenta agree
much better with the nominal set by the SPS.
The convergence of alignment corrections per iteration and the improved residual distributions
presented are mandatory but not sufficient to ensure the success of the alignment. Unfortunately,
survey data of the CTB detector setup does not exist, therefore, a comparison with the derived
alignment sets was not possible. However, a comparison of the position and orientation of eachdetector element derived with the four algorithms served as a means of validation.
When the alignment constants for the four algorithms were compared, the algorithms wereobserved to provided large corrections (several hundred microns). The chosen alignment strategy(fixing one or several modules as opposed to leaving the whole system free floating or constraining
or removing some DoFs from the alignment) has an impact on the solution of the global DoF of
the system. In order to compare the results of the different algorithms, they need to be globallymatched. Allowing a global offset for each alignment set was chosen to be the method of finding a best match of the alignment results. After having subtracted the global offsets between thegeometries, it was observed that a good agreement between the algorithms for the most sensitive
coordinates x, y and γ was reached. Given the low sensitivity of the alignment procedures to thealpha and beta rotations, the agreement between the algorithms for these coordinates was only
marginally improved.

5. Summary and Conclusion
Four independent algorithms were used to successfully align the setup formed by the silicon modules of the ATLAS Inner Detector tracker, using data collected during the 2004 Combined Test
Beam. The reconstructed track parameters and hit residual distributions were studied. The per-– 16 –

Arbitrary units

Before alignment
After alignment

400
350

q
<p> = 0.0101 (GeV-1)qσ(p) = 0.0019 (GeV-1)

300
250
200
150
100
50

0
0 0.010.02 0.03 0.04 0.05 0.06 0.07 0.08q-1
p (GeV )Figure 12. Momentum resolution of Pixel and SCT detectors for a 100 GeV pion run with non-zero B-fieldwith and without alignment corrections.

Momentum resolution pion runs with B field

0.5

σ(p)/pσ(p)/pMomentum resolution electron runs with B field

0.45

0.5
Valencia

0.45

GlobalRobust

0.4

0.4

0.35

0.35

Local
Simulation

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0
0

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40

60

80

100

120

140

0
0

160 180p (GeV/c)

20

40

60

80

100
120
p (GeV/c)

Figure 13. Momentum resolution as a function of the reconstructed momentum.

formance of the alignment algorithms was assessed by comparing with a simulation, in which all
modules were at their nominal positions. The simulation can be taken as a benchmark where all
errors were regarded as only being due to the intrinsic resolution of the modules.
All alignment approaches yielded results for the reconstructed momentum of electrons andpions that agreed with the simulation. Slightly worse momentum resolution was observed using
the Robust algorithm. This was understood and explained by the fact that the algorithm was lim-

– 17 –

ited to re-alignment of the two in-plane translations only. The unresolved residual misalignments(e.g. in-plane rotations) unavoidably led to reduced track fit quality and consequently increased uncertainties on the reconstructed curvature. For the remaining track perigee parameters, consistent
results were obtained with each method.
All four methods agree well on the residuals for all modules and planes, and with the simulation. The resolution of individual pixel modules is around 10 µ m and the SCT around 20 µ m.
Observed differences for the residual mean values remain below 5 µ m. We conclude that the silicon modules of the ATLAS ID were aligned at the CTB with a precision of 5 µ m in their mostprecise coordinate.
The data collected at the ATLAS Combined Test Beam in 2004 served as an invaluable test bed
for the Inner Detector alignment algorithms. For the first time ever, the readiness of the alignment
algorithms was assessed with experimental data. All algorithms performed satisfactorily given thelimitations inherent to the CTB geometry and the beamline arrangement. The narrow tower ofmodules and almost parallel particle beams gave rise to undetermined degrees of freedom. These
were successfully dealt with by the four algorithms, each in its own way, providing consistent andhigh quality measurements of the test beam track parameters.

Acknowledgments
We are greatly indebted to the CERN accelerator departments and to the H8 beam-test facility for
their efforts in building and operating the beamline, to our technical collaborators for their support
in constructing and operating the Combined Test Beam components and to the IT department for
their support for the computing infrastructure. We are grateful to all the funding agencies which
supported generously the construction and the commissioning of the ATLAS detector components
and the computing infrastructure.
We acknowledge the support of ARC and DEST, Australia; Bundesministerium fürWissenschaft und Forschung, Austria; CERN; Ministry of Education, Youth and Sports of the
Czech Republic, Ministry of Industry and Trade of the Czech Republic, and Committee forCollaboration of the Czech Republic with CERN; Danish Natural Science Research Council;
IN2P3-CNRS and Dapnia-CEA, France; BMBF, DESY, and MPG, Germany; INFN, Italy;
MEXT, Japan; FOM and NWO, Netherlands; The Research Council of Norway; Ministry of
Science and Higher Education, Poland; Ministry of Education and Science of the RussianFederation, Russian Federal Agency of Science and Innovations, and Russian Federal Agency ofAtomic Energy; JINR; Ministerio de Educación y Ciencia, Spain; State Secretariat for Educationand Science, Swiss National Science Foundation, and Cantons of Bern and Geneva, Switzerland;
National Science Council, Taiwan; The Science and Technology Facilities Council, United
Kingdom; DOE and NSF, United States of America.

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