Using the information stored in the template with each hit:
- track length to the hit
- thickenss of the silicon traversed by the particle
it is possible to calculate analythically the deviations due
to the multiple scattering
if they are small and it is the only effect causing the
deflection of the particles from the ideal trajectory.
It can and was done with reasonable accuracy for the case
when the magnetic field can be neglected.
In all the following figures theorethical calculations are shown as full, black squares and are joined by a black line; results from the GEANT simulations are shown as open red circles and are joined by a red line
Example for a case of simulations without magnetic field:
- pi- at 1000 MeV/c, theta=40
(Postscript)
- pi- at 200 MeV/c, theta=40
(Postscript)
Usually we obtain also reasonable results with the magnetic field
present, but only for high momentum particles. For example
pi- at 1000 MeV/c and theta=42
- standard PHOBOS field
- 2T field in a cylinder
For lower energies however, the magnetic field significantly influences
the deviations calculated for lower momentum particles:
- pi- at 200 MeV/c theta=40,
standard PHOBOS field
- pi- at 200 MeV/c theta=40,
2T field in a cylinder
There is a signifficant dependence on the particle emission angle:
- pi- at 200 MeV/c theta=40
- pi- at 200 MeV/c theta=45
in one case the deviations from simulations are larger than theoretical at
about 40 cm, while in the second case they are much smaller there.
Of course there are large differences between deviations calculated for
particles with different charge:
- pi- at 200 MeV/c theta=40
- pi+ at 200 MeV/c theta=40
both the absolute values of deviations are different and the
dependence on the distance from the vertex.
The main feature is preserved: the deviations at the and of the track are smaller than calculated theoretically.
The first two factors are regular enough to be included in the theoretical calculations, the third one requires in general information not only from the points associated with the hits, but also arround the trajectory between the planes. It is thus not obvious if it will be possible to include this effect in the calculations correctly. However the magnitude of this effect is not yet known, so it may be negligible for our spectrometer.
Another problem in accurate determination of of deviations and covariance matrix, but only in the calculations using simulated tracks, is the complicated structure of the spectrometer. Because of this some of the tracks with largest deviation in a layer may become incompatible with the template. In this case the deviations will be signifficantly deformed.
