Tracker Coordinate Measurement Improvement in the Tracker Analysis
The y coordinate provides better accuracy by design, in which the readout strips have much smaller pitch compared to the strip pitch in the x coordinate. We present the improvement in the accuracy of determination of y-coordinate, which is the most important for the determination of momentum (or rigidity).
When a charged particle crosses a layer of the silicon tracker, its coordinate is determined by taking the ratio between the signals induced on the two strips between which the particle passed (see Figure 1).
![Schematic of particle coordinate measurement in the silicon tracker.](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig1_.png)
The amplitude of the induced signals should be proportional to $Z^2$. For $Z>3$ the amplitude $A_1$ starts to become non-linear, which causes resolution degradation. This effect is illustrated in Figure 2, where the spatial resolution for helium is compared with the resolutions for carbon and oxygen.
![Coordinate resolution for helium, carbon and oxygen events](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig2_full.png)
We have developed an optimal way to correct for the non-linear effect. For this we need to identify a function $K(x$) which will restore the linearity of the amplitudes by taking into account amplitudes A1, A2 and the corrected position
$$ x=1/(1+(A_{1}+K(x))/A_{2}) \tag{1} $$
This is illustrated in Figure 3.
This work is published in [G. Ambrosi, V. Choutko, C. Delgado, A. Oliva, Q. Yan, and Y. Li, Nucl. Instrum. Methods Phys. Res., Sect. A 869, 29 (2017)].
![optimal way to correct for the non-linear effect.](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig3_.png)
To determine $K(x)$, we used the fact that cosmic rays are uniform and isotropic, so for an ideal tracker, without nonlinearity, we should see a uniform event position density (see Figure 4a), while in the non-linear case the event density is distorted (see Figure 4b).
![event position density](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig4_.png)
The function $K(x)$ was found to be different for different $|Z|$. Figure 5 shows the event density distribution for carbon nuclei before and after correction.
![event density before correction and after correction](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig5_.png)
This method improved the position resolution by a factor of 2 for carbon (see Figure 6) and more than a factor of 2 for heavier nuclei.
![comparison of the differences of the coordinates](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig6_.png)
Figure 7 shows the coordinate accuracy for nuclei from $Z=2$ to $Z=26$ using the new analysis technique. Note that, due to the design of the tracker readout amplifier, the maximum non-linearity occurs for $Z\sim9$. This leads to the coordinate accuracy as a function of $Z$ shown in Figure 7.
![AMS tracker residuals sigma as functions of nuclei charge Z](/sites/default/files/inline-images/Tracker%20coordinate%20measurement.Fig7_.png)