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Detector Alignment with Tracks
Wouter Hulsbergen (Nikhef, BFYS)
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Detector alignment
LHC silicon detectors provide
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Alignment parameters
alignment parameters: rotations and translations of solid objects in spacealignment performed at different levels of granularity, with different constraints from assembly or surveysmallest granularity: silicon wafer, driftchamber module etc
other DOFs (e.g. deformations of wafers) usually considered later
number of parameters considered in alignment of tracking detectors
see e.g. proceedings of1st LHC alignment workshop
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Tracks and residuals
* ** *
* *x
z
detector plane
hit: strip/wire/pad with fixed coordinate x
track model, e.g. x(z) = a0 + a1 z
track fitting and alignment is all about hit 'residuals'
track parameters
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Track fitting
* ** *
* *
the track fit is a 'leastsquaresestimator' minimizes track 'chisquare'→
x
z
minimization performed with (semi) analytic method
'minimize' means
NewtonRaphsonmethod for findingthe 'zero' of a nonlinear function
it is not so different from what's happening inside MINUIT
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Perfect detector: residuals are unbiased
* ** *
* *x
z
x_hit x_track
residual RMS ~ detector position resolution ⊕ multiple scattering etc
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Misaligned detector: biased residuals
x
z
note: one layer was misaligned … but next layer has biased residuals as welltypical problem in detector alignment: residuals from track fit are correlated
* * **
* *
x_hit x_track
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Alignment using residuals
simple alignment method: extract misalignments from residual histogramsnot easy to extend to detector displacements other than measurement direction
no straightforward method to deal with correlations, especially in 'segmented' detectors
tracks constrain correlated movements
alignment becomes a bookkeepingproblem: 'residuals in module A1' with respect to tracks in B1, C2, D3' etc
most popular solution: 'minimum chisquare method for alignment'consider the chisquare of a sample of tracks
minimize this chisquare simultaneously with respect to alignment parameters and track parameters
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Minimum chisquare method for alignment
the solution to this minimum chisquare problem can again be written as
'the big matrix' 'the big vector'
average residualscorrelations between elements
change in alignmentparameters
eliminating the trackparameters from this problem is actually not totally trivialneed to exploit that different tracks only correlated via alignment parameters
best known implementation of this idea: MILLIPEDE by Viktor Blobel
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Weak modes: poorly constrained common movements
special complication in alignment with tracks: some (linear) combinations of alignment parameters are unconstrained
global translation zscale shearing
more dangerous than unconstrained modes are socalled 'weak modes''statisticallyunderconstrained' common movements in a track samples with finite size
extremely sensitive to mistakes ('outliers') in track reconstruction
can lead to poor convergence of alignment procedure
weak modes are the major concern in detectors that require alignment of many elements
LHCb inner tracker: silicon tracker with O(500) ladders
LHCb outer tracker: drift chamber with 216 modules
CMS/Atlas inner detectors
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Example: weak modes in central Si tracker
(from
C. E
scob
ar, V
erte
x 20
08)
weak modes affect physics .... but
almost everything that affects track, vertex or momentum resolution can be extracted from data
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Constraining 'weak modes'
design: overlap, redundancy
overlaps constrain radial expansionand clamshell effects
different data sets: cosmics, beamhalo, magnetoff
offaxis events constrain twist andeleptical distortions
survey constrains scale,like zscale of LHCb VELO
survey measurementsmultitrack constraints: vertex,invariant mass, beam kinematics
mass constraints fix curvature biasvertex constraints fix clamshell
beforeafter
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Example of unconstrained mode in LHCb spectrometer
LHCb spectrometer measures 'kink' of particle around magnet axis: kink Q/pxzshearing of the tracking stations leads to bias in the kink momentum bias→
magnetforward tracker
veloTT
Q/p bias bias in mass as →function of asymmetry of decay
can be used to extract shearing with
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