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Now let us turn our attention to the r-z track fitting under
the influence of multiple scattering.
The treatment here provides a way to calculate the E0
of the last subsection including
multiple scattering effects in track fitting.
From now on, we take (y0,z0) = (0,0) and .The multiple scattering modifies the track parameter of the particle as
| |
(90) |
where
| |
(91) |
represents the change of the dip angle due to the multiple
scattering as a function of y.
This results in
| |
(92) |
where the parmeters with the suffix are those at the
initial point before multiple scattering.
Now, let us assume that we are given
(n+1) hit points, ,measured at fixed y positions, .Then the minimization of the :
| |
(93) |
The parameter vector that gives the minimum is
| |
(94) |
with
| |
(95) |
Through the above equation, the change in
due to the multiple scattering induces a change in :
| |
(96) |
From this we obtain the full error matrix including multiple scattering:
| |
(97) |
Again the problem reduces to the evaluation of the
correlation matrix in the coordinate measurements:
| |
(98) |
with
| |
(99) |
where the last expression is valid for .The full error matrix is then written in the form:
| |
(100) |
where the first term is from the coordinate measurement errors while the
second term from the multiple scattering:
| |
(101) |
In the case of equal space sampling with the same position resolution,
EM and EMS reduce to
| |
(102) |
with
| |
(103) |
in the large n limit.
Next: Examples of Applications
Up: Track Fitting in r-z
Previous: Pivot Transformation
Keisuke Fujii
12/4/1998