The orientation of the pixel pattern corresponding to a hit is also calculated. Shown in Figure 14 is a single-hit cluster pattern which is skewed due to large track crossing angles and . The angle of the pattern with respect to the padrow (indicated by the overlaid line in the figure) is correlated with these crossing angles. This angle is stored in TPHIT.PHI. What should be relevant to tracking algorithms is an estimate of the crossing angles and . As intuitively expected, the angles , , and are observed to be correlated, so that knowledge of two of them gives the third. For tracks originating from the center of the TPC, = Tan, where r is the radial distance of the hit and z is the seperation along the TPC axis between the hit and the central E-field membrane. TPH stores this value into TPHIT.LAMBDA.
Figure 14: The position and orientation of a hit in a single-hit cluster are shown. The large skew (non-zero orientation) of the pattern arises from large track crossing angles.
Given and , we should be able to estimate the crossing angle . If we use local coordinates such that x is along the padrow (in the ``pad direction''), y is perpendicular to the padrows, and z is along the drift direction, then the two crossing angles are given by
whereas our calculated ``orientation angle'' might be expected to be written
which would lead one to conclude that alpha could be calculated by
However, the orientation angle is more complicated, and we find that an empirical relation is needed.
In the simulations we find that there are strong correlations of with our assumed and our calculated . Two branches in the relation of the true vs (as given by Equation 19) are seen, and neither one is linear.
Experimenting with different parametrizations leads to a relation
The resolution for reconstructing the crossing angle is only fair (see Section 6.4 below). However, this hit-by-hit estimate of the track crossing angle, which intrinsically involves the third dimension (perpendicular to the padrow), based on the two-dimensional pad-tdc space, may still be useful to the tracking software in situations in which it is not clear to which track a hit belongs.