A model for track parameters in collimated particle beams

The particles tracked through the beam telescope originate from a highly collimated particle beam. Our idea is to utilize the beam collimation for the alignment of the telescope from track data. More precisely, we outline the approach to align the z axis of the global coordinate system with the direction of the particle beam and to explicitly parametrize the first and second moments of the distribution of track parameters of beam particles. These moments can be used as an a priori statistics for the track fitter. In addition, the modeling of moments gives new constraints for the alignment of the beam telescope and allows to suppress the telescope distortions known as shearing and torsion.

3The telescope setup with all sensor distances is shown in Fig. 4.2

Constant 1236 ± 6.9

Figure 4.5.: Residuals in the localudirection for all Mimosa26 modules in the EUDET telescope. The modules are numbered along the beam line with No.1at the first position in the telescope.

A Gaussian is fitted to all residuals and the fitted mean and sigma values are shown. The residuals are centered around zero which shows that the alignment was successful. The width of the residuals depends on the position of the module in the telescope.

Z in mm

0 50 100 150 200

mµ in uσ

0 1 2 3 4 5 6 7 8 9 10

σu

M26 Resolution

σt

Tel. Resolution

σr

M26 Residuals

Figure 4.6.: The measured spatial resolution,σ_u, as a function of the telescopezcoordinate. The spatial resolution is computed by subtracting the telescope resolution from the measured residual width. As can be expected, the measured spatial resolution of all Mimosa26 modules is almost independent from thezposition in the telescope and lies around the expected value of3.5µm. The telescope resolution is5µm at the position of the DEPFET module between the two telescope arms.

Without loss of generality⁴, we consider the globalz = 0 plane as a further telescope layer placed before all other layers which follow at positions0< z1 <· · · < znalong thezaxis. At this telescope layer, the track parameters are given relative to thez= 0plane

λ0 =

The approach to align the global zaxis of the telescope with the direction of the particle beam can now be formulated more precisely as an equality constraint on the mean value of track parameters at the z= 0plane.

In other words, beam particles intersect thez = 0plane around the origin of the global coordinate system atx =y = 0and the average flight direction is parallel to thezaxis. To model the covariance matrix of track parameters, we assume that the track parameters decouple in thex−zandy−zplane.

In this case, the most general parameter covariance matrix can be written as

C₀ =

The covariance matrix in Eq. 4.10 assumes an elliptical beam spot having a spot size ofσ_xxandσ_yy in the horizontal direction (x) and the vertical direction (y). The angular spread of beam particles around thezaxis is described by the beam divergenceσ_tx,tx for thex−zplane andσ_ty,tyfor they−zplane.

The beam particle spread out along the beam direction due to multiple scattering and we can expect a positive correlation between the track intersection coordinates and the track incident angles. In the simplest case, the situation can be described by two linear beam correlation coefficients.

rx = σ_t²_x,x

σ_x,xσ_tx,x ≥0 (4.11)

r_y = σ_t²_y,y

σ_y,yσ_ty,y ≥0 (4.12)

In Fig. 4.7 and Fig. 4.8 we show the distribution of reconstructed track intersection coordinates, in-cident angles and their correlations for a3.75GeV electron run at DESY. The reconstruction of track parameters is performed in a fully aligned EUDET telescope operated with a single large scintillator in front. Within the telescope aperture, the measured beam spot size is10±1mm in the horizontal and

4In general, there is some degree of freedom regarding the choice of a global coordinate system. In particular, the reconstruc-tion results do not depend on global shift and rotareconstruc-tions of both the telescope sensors and the particle source.

x in mm

Figure 4.7.: Left: The distribution of track intersection coordinates at the z = 0plane for all recon-structed particle tracks with hits on all Mimosa26 sensors. The range of track intersections is limited to a rectangular field of10×20mm²defined by the active area of the Mimosa26 sensors. The central part of the beam spot is visible at the origin of the global coordinate system. Right: The distribution of track incident angles dx/dz anddy/dz at the z = 0 plane for reconstructed tracks. The beam axis is aligned to thezaxis during the telescope alignment.

6±1mm in the vertical direction. Similarly, the measured beam divergence is1.1×10⁻³rad in the x−zplane and0.9×10⁻³rad in they−zplane. As can be seen in Fig. 4.8, the track parameters show a linear correlation between track incident angles and intersection coordinates. The beam correlation coefficients are estimated to0.7±0.2.

This model can be employed in two ways. Firstly, the intersection of the beam axis with the sensor planes gives a reference flight path for the linearization of the track model. The strategy for the lineariza-tion of Kalman Filters for track fitting is described in [66]. Secondly, the collimalineariza-tion of beam particles is a property of the beam line and can be regarded as an independent further information about the beam particles. The moments< λ0 >andC0 are usable as a priori statistics for the track fitter in addition to the reconstructed hits. These properties of the particle beam will be used for the alignment of the telescope described in section 4.5.4.