Detector Alignment Thomas Naumann DESY Zeuthen

1
Detector AlignmentThomas NaumannDESY Zeuthen

• Detector alignment and calibration often imply
• least squares fits with many parameters
• n global parameters of calibration, n 10 ...
  10.000
• m local parameters of the data model of an
  object (e.g. tracks)
• where m k x l with
• k ... nr of objects , large , gtgt
  1000
• l ... nr of parameters , small , lt 10
• Traditional method to correct by residuals of
  local fit is
• biased since correlation between local and
  global parameters is ignored
• (wrong global constants used)
• Solution Fit local and global parameters
  simultaneously
• large nr of paras N n k x l
• time to solve linear problem by matrix inversion
  N³
• N 1.000 1 min _at_ 1 GHz with double prec.

2
Millepede
Author V.Blobel, Hamburg University 1)
The matrix equation for the simultaneous fit of
global and local parameters

• contains
• the normal equations of least squares of the
  local fits i1,...,l
• Gi ai bi
• with the local parameter vectors ai and the
  covariance matrices Gi .
• Traditionally neglect correlations Gi with
  global parameters and
• solve locally for ai inverting l matrices Gi
  of rank k.
• the sum over the covariance matrices Ci of the
  correlations
• between the vector a of the m global
  calibration parameters.
• the matrices Gi of the correlations between
  local and global parameters.

Inversion by partition Use the special shape of
this matrix and rewrite the matrix equation
as C a b with new matrices C and b
which sum over the local parameters and
explicitly contain only the global parameters
Solve the linear least square problem independent
of nr of local parameters !
1) for code and manual see http//www.desy.de/b
lobel/millepede.f and .ps
3
The H1 Backward Silicon Detector
at HERA
e
e
measure scattered electrons polar angle q
and energy momentum E and p

• BST uses r and j Si detectors
• with 640 strips of
• 96 and 75 mm pitch

• has 8 z planes along the beam axis with 16 j
  sectors each.
• Q needs alignment of the r detectors
• p needs alignment of the j detectors.
• get z correlations between BST parts 1 and 2
• and j correlations between sectors

4
The Alignment of the H1 BST
This gives a polar angle precision of 0.4 mrad.
For the r detectors a resolution of 18 mm is
reached.
Alignment of the j detectors get electron
charge and momentum up to the HERA beam energy of
Ee 27.5 GeV from BST alone !
E / p
The ratio of the energy E of the scattered
electron measured by the H1 SPACAL calorimeter to
the momentum p measured by the H1 BST versus E.
Both measurements agree well.