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 one-dimensional 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 *σ*which
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 the*x* 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) 228-231, 10.1016/j.nuclphysbps.2011.04.016 - D. Biagetti et
al.
*Beam test results for the RAPS03 non-epitaxial CMOS active pixel sensor*, Nucl. Instr and Meth A 628 (2011) 230–233

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