1 Alice TPC drift calibration using tracks

In the first aproximation there is a linear dependence of the z position on the drift time. In Alice TPC the expression on the A side and C side of the chambers have the same drift vlocity part v_{d} with opposite sign. The full drift length z_{0}A and z_{0}C are different. We suppose that the t_{0} offset givven by trigger arrival time is the same. In reality the t_{0} equalization is applied before, using the pad-by-pad calibratiom pulser measurement.

\begin{matrix}z_{A} = z_{0A} − v_{d}(t − t_{0}) \\ z_{C} = z_{0C} + v_{d}(t − t_{0}) \end{matrix}  (1)

Let’s the actual value of drift velocity v_{d} and the time offset t_{0} are shifted by some Δ value. Our starting drift velocity values and time offset are ṽ_{d} and t̃_{0}

\begin{matrix}v_{d} = ṽ_{d} + Δv_{d} \\ t_{0} = t̃_{0} + Δt_{0} \\ v_{c} = \frac{Δv_{d}}{ṽ_{d}} \\ Δz = Δt_{0}ṽ_{d} \end{matrix}  (2)

Than the actual z position is expressed using the starting z position mesurement z̃_{A} reps. z̃_{C}

\begin{matrix}z_{A} = z̃_{A} − \frac{Δv_{d}}{v_{d}} (z_{0A} − z̃_{A}) + Δt_{0}ṽ_{d} \\ = z̃_{A} − v_{c}(z_{0A} − z̃_{A}) + Δz \\ z_{C} = z̃_{C} − \frac{Δv_{d}}{v_{d}} (z̃_{C} − z_{0C}) − Δt_{0}ṽ_{d} \\ = z̃_{C} + v_{c}(z̃_{C} − z_{0C}) − Δz \end{matrix}  (3)

In previous expression we neglected second order correction

\begin{matrix}Δv_{d}Δt_{0} ≪ \frac{Δv_{d}}{v_{d}} (z_{0A} − z_{A}) \\ (t − t_{0}) ≈ \frac{(z_{0A} − z̃_{A})}{v_{d}} \end{matrix}  (4)

Combining the z masurement the track parameters can be fitted. Let’s assume linear track model:

\begin{matrix}z̃_{A} = ã_{A} + b̃_{A}x \\ z_{A} = a_{A} + b_{A}x \\ z̃_{C} = ã_{C} + b̃_{C}x \\ z_{C} = a_{C} + b_{C}x \end{matrix}  (5)

The relation between starting parameters ã, b̃ and corrected parameters a, b is linear.

\begin{matrix}a_{A} = ã_{A} − v_{c}(z_{0A} − ã_{A}) + Δz \\ b_{A} = b̃_{A}(1 + v_{c}) \\ a_{C} = ã_{C} + v_{c}(ã_{C} − z_{0C}) − Δz \\ b_{C} = b̃_{C}(1 + v_{c}) \end{matrix}  (6)

The inclination angle correction is the same on the A and C side.

Tracks crossing the central electrode, respectivally primary tracks can be used to monitor correction coeficients Δz and v_{c}. For tracks crossing the central electorde the a and b parameters at the crossing point fitted form A and C side are the same. In case of primary tracks, the z position at r-ϕ DCA are also the same:

\begin{matrix}a_{A} − a_{C} = 0 \\ Δã(1 − vc) + 2Δz − v_{c}(z_{A} − z_{C}) = 0 \\ Δã = \frac{v_{c}(z_{A} − z_{C}) − 2Δz}{1 − v_{c}} \end{matrix}  (7)

The measured difference

Combining infomation from A and C side the correction parameters, drift correction v_{c} and offset correction Δz can be fitted.

In case track crossed the central electrode the track parameters of the same track on A side and C side can be fitted. The actual track parameters a_{A} and a_{C} respectivally b_{A} and b_{C} are the same.

(8)