In order to reconstruct a ionizing particle trajectory, e.g. in High Energy Physics Experiments, it is needed to detect the positions of this particles within the space. The technique used for this purpose consists on the interposition along the trajectory of several sensors planes which are capable to detect the positions where the particles pass through; from the interpolation of all these points can be reconstructed the trajectories followed by the particles. In these environments one of the most important merit figure of the sensors is the spatial resolution, that is the capability to reconstruct the crossing point of the particle.
Fig. 1
Illustration of a tracking
system with three sensors
aligned one in front of each
other.
Basically, the evaluation of the spatial resolution of a particle sensor consists on the irradiation of the sensor under test with particles beam at high energy and on the measurement of the differences between the measured impact points with the real ones. It is clear that it is necessary to know the real impact points of the incoming particles, a solution of this problem is to use a known tracking system (usually called telescope) with which it is possible to measure this positions. However a solution that uses only the sensors under test can be used. We can use the same sensors for which we want to measure the resolution to track the particle. The Figure 1 illustrates schematically this system. For simplicity considering a onedimensional system we can use at least three sensors to detect the coordinate x1, x2 and x3 of the particle trajectory at three different position d1, d2 and d3, from the interpolation of two of these points we can reconstruct the particle path and use them to estimate the position on the third sensor. From simple geometry considerations, if a and b indicate the two sensors used to estimate the position on the third one, that is c, such predicted position will be:
and the difference whit the measured one (in the following called residue also) mathematically is
If we assume that the uncertainty in the position measurement has a Gaussian probability density function whit a standard deviation σ, e has a Gaussian probability density function in turn with a deviation of:
So, from the
standard deviation of e and
knowing the geometry of the
system it is possible to
retrieve σwhich
is the spatial resolution of the
sensors.
The previous example however represents an ideal case; in real cases there are several sources of uncertainty. Each sensor has a non negligible thickness which produces a deflection of the incoming particle due to the phenomenon of multiple scattering. The effect of multiple scattering is to add another aleatory contribution to the coordinates of the measured position. The effect on the residues is to enlarge its distribution; calling σs the deviation introduced by the multiple scattering we can make the approximation:
where σr is
the real resolution.
Other sources
of uncertainty come from a non
perfect alignment of the
different elements. Each sensor,
as a solid body, has six
different degree of freedom,
namely three translations and
three rotations. The two
translations perpendicular to
the particles direction adds an
offset to the coordinates of the
impact point. The effect on the
distribution of the residue is
only a shift of the Gaussian
peak. The translation along the
particle direction adds an
uncertainty to the coordinate d,
however if the distance of a
sensor to another is big
compared to the position
uncertainty this component can
be neglected.
Fig. 2
Effect of the multiple
scattering and
misalignments in the detection
of a particle trajectory.
The rotations (Figure 3) are more difficult to compensate. Referring to the Figure 4 a rotation around thex or y axis has the effect shown in Figure 4(a), mathematically:
if the angles
are little (few degrees) the
correction can be neglected. For
the θd tilt
the situation is little more
complex because each coordinate x or y of
a sensor is related to both the
coordinates of another sensor
(see Figure 4(b)).
Fig. 3 The
three tilt angles among the
sensors.
Mathematically the tilt around the axis parallel to the particles beam direction can be modelled as:
The effect of all the rotation on the residue is to widen its distribution, but from the analysis of the coordinate x2 and y2 detected on a sensor as a function of the coordinate x1 and y2 detected to another sensor the tilt can be estimated and corrected.
Fig. 4 Effect
of a non parallelism among two
different sensors.
Summarizing
the relation among two different
sensors we can consider:
From the
measured points, using a
multiple linear regression
algorithm, it is possible to
retrieve the coefficients m and q.
Another
source of uncertainty comes from
the algorithm used to define the
crossing point of the particles
with each single detector. In
order to improve the resolution
in the position reconstruction,
it is possible to exploit the
charge sharing effect among
adjacent pixels. Usually the
barycentre algorithm is used (calledCOG,
Center Of Gravity).
Mainly due to the finite nature
of the detector and the
dimensions of the cluster the
COG, even if it allows to reach
lower resolution limit (i.e.
better resolution) than the
pixel dimension, it adds a
systematic error also.
References

D. Passeri et al., Characterization
of CMOS Active Pixel Sensors for particle detection: beam test of the four
sensors RAPS03 stacked system, Nucl.
Instr. and Meth. A 617 (2010) 573–575

D.Passeri,et al. Tilted
CMOS Active Pixel Sensors for Particle Track Reconstruction, IEEE
Nucl. Sci. Symp. Conf. Rec. NSS09 (2009) 1678. July 2006.

L. Servoli et al. . Use
of a standard CMOS imager as position detector for charged particles ,
Nucl. Instr. and Meth. A 215 (2011) 228231, 10.1016/j.nuclphysbps.2011.04.016

D. Biagetti et al. Beam
test results for the RAPS03 nonepitaxial CMOS active pixel sensor,
Nucl. Instr and Meth A 628 (2011) 230–233
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