Lorentz Shift

Normally, the charge carriers move straight to the electrodes under the influence of the electric field. If a magnetic field perpendicular to the electric field is present, as it is the case in the barrel part of the CMS Tracker, the charges are deflected from their track. A single charge $Q$ moving in electric and magnetic fields $E$ and $B$ with the velocity $v$ will experience the Lorentz^2.5 force $F$,

$$F=Q\,(E+v\times B)\quad.$$ (2.30)

Figure: Electrons and holes are deflected under the influence of an electric field. This Lorentz shift causes an offset between the particle track and the measured position.

This transverse force due to the magnetic field, also known as known as Hall^2.6 effect, results in inclined carrier movement relative to the electric field as shown in fig. . Electrons and holes are subjected to different shifts, since their drift velocities are different. The Hall effect has two consequences: The electrode target area of the charge widens proportional to the detector thickness and the target center is offset relative to the particle track. This ``Lorentz shift'' is usually expressed as an incline angle. With a CMS-like magnetic field of $4\,\rm T$, Lorentz angles of $31^{\circ}$ and $8^{\circ}$ have been measured for electrons and holes, respectively, in a silicon detector of $300\,\rm\mu m$ thickness [29].

The practical relevance of the Lorentz shift can be minimized by mechanically tilting the detectors such that the target areas on the electrodes of both electrons and holes coincide. With this choice, the equal Lorentz shifts of both carriers can be easily corrected by numerical offset subtraction. In the case of CMS, the corresponding tilt angle is $11.5^{\circ}$.