To reconstruct an ionising particle track, such as in High Energy Physics Experiments, the positions of these particles within space must be detected. The technique involves using many sensor planes to identify particle positions and recreate their paths through interpolation. Sensors in these situations rely heavily on spatial resolution to reconstruct particle crossing points.
The spatial resolution of a particle sensor is evaluated by irradiating the sensor under test with a high-energy particle beam and measuring the discrepancies between the measured and real hit spots. It is obvious that the true impact places of incoming particles must be known; one solution to this difficulty is to employ a well-established tracking system (commonly referred to as a telescope) to measure these positions. However, a solution based solely on the sensors under test can be applied. We can monitor the particle using the same sensors that we use to measure resolution.
Figure 1 graphically depicts this mechanism. For simplicity, consider a one-dimensional system. We can use at least three sensors to detect the coordinates x1, x2, and x3 of the particle trajectory at three different positions, d1, d2, and d3. By interpolating 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 denote the two sensors used to estimate the position of the third, that is c, such an anticipated position will be:
and the difference with the measured one (in the following called residue also) mathematically is
Assuming the uncertainty in the location measurement has a Gaussian probability density function with a standard deviation σ, e also has a Gaussian probability density function 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 above example, however, is an ideal instance; in real life, there are other causes of uncertainty. Multiple scattering occurs when incoming particles are deflected by the sensor’s thickness. Multiple scattering causes an additional aleatory contribution to the measured position’s coordinates. Multiple scattering causes an increase in the distribution of residues, which can be approximated using the term σs.
where σr is the real resolution.
Other causes of uncertainty arise from the imperfect alignment of the various parts. Each sensor, as a solid body, has six degrees of freedom, three for translation and three for rotation. The two translations perpendicular to the particle’s trajectory cause an offset in the coordinates of the impact spot. The sole effect on the residue distribution is a shift in the Gaussian peak. Translation along the particle direction adds uncertainty to coordinate d. However, if the distance between sensors is greater than the position uncertainty, this component can be omitted.
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 small (a few degrees), the adjustment can be overlooked. The θd tilt is more complex as each sensor’s x and y coordinates are connected to the coordinates of another sensor (refer to Figure 4(b)).
The tilt around the axis parallel to the particles’ beam direction can be described mathematically as follows:
The effect of all the rotation on the residue is to widen its distribution, but from the analysis of the coordinates x2 and y2 detected on a sensor as a function of the coordinates x1 and y2 detected on another sensor, the tilt can be estimated and corrected.
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 is the technique used to define the particles’ crossing points with each individual detector. Using charge sharing between adjacent pixels can increase position reconstruction resolution. The barycenter algorithm, often known as COG (Centre of Gravity), is commonly employed. Due to the limited nature of the detector and the size of the cluster, the COG, although allowing for a lower resolution limit (i.e. better resolution) than the pixel dimension, also introduces a systematic inaccuracy.
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
