Multiple Scattering Correction

Unless we have an extra-tracking device in the new region, we do not know the scattering angle in the material. The best we can do is, therefore, to take into account the effect of the multiple scattering on the error matrix.

It can be shown (see Appendix A) that the error matrix for the track extended through the scatterer becomes

$$\begin{array} {lll} E_{\bf a'} = E_{\bf a} + E_{MS},\end{array}$$ (13)

where the second term on the right-hand side represents the correction to the error matrix due to the multiple scattering:

$$\begin{array} {lll} \left( E_{MS}\right)_{22} & = & \sigma_... ...} & = & \sigma_{MS}^2 \cdot ( 1 + \tan^2\lambda )^2 \end{array}$$ (14)

with all the other components being zero. The $\sigma_{MS}$ is given, as usual, by  

$$\begin{array} {lll} \sigma_{MS} = \frac{0.0141}{P({\rm GeV}... ...{X_L} \left( 1 + \frac{1}{9} \log_{10} X_L \right),\end{array}$$ (15)

where P, $\beta$, and X_L are the momentum, the velocity in units of the light velocity, and the thickness of the scatterer in units of its radiation length.

In summary, we can extrapolate a helical track through a material by first moving the pivot to the intersection of the track and the material and then making the aformentioned modifications to the track parameters and error matrix. If there are two or more scatterers, all we have to do is just repeat this process.

Once the error matrix for the extrapolated track is calculated, it is easy, for instance, to estimate the position error at a deflection angle $\phi$ from the error matrix given by

$$\begin{array} {lll} E_{\bf x}(\phi) = \left( \frac{\parti... ...rac{\partial {\bf x} }{\partial {\bf a}'} \right)^T.\end{array}$$ (16)