Effects of Thin Layer Multiple Scattering

  Consider a particle passing through a thin layer, whose track parameter vector changes from ${\bf a}$ to ${\bf a'}$ due to multiple scattering there. We assume here that the track parameter vector of the particle is measured after the multiple scattering to be ${\bf a}_M$with its error matrix $E_{{\bf a}_M}$. In the Gaussian approximation valid for small angle multiple scattering, the probability of getting ${\bf a}_M$ when the true track parameter vector before the multiple scattering is ${\bf a}$ is given by  

$$\begin{array} {lll} P\left({\bf a}_M;{\bf a}\right) & = & ... ...left( {\bf a} - {\bf a}'' \right) \right) \right],\end{array}$$ (1)

where E'_MS is the diagonalized error matrix corresponding to multiple scattering N is a normalization factor depending only on $E_{{\bf a}_M}$ and E'_MS, and

$$\begin{array} {lll} \nonumber \left( {\bf a} - {\bf a}' \ri... ...& = & ( 1 + B ) \left( {\bf a} - {\bf a}'' \right)\end{array}$$

with B being a matrix whose only nonzero component is $(B)_{35} = \kappa\tan\lambda/(1+\tan^2\lambda)$.Making use of the fact that B is nilpotent, i.e. B² = 0, we can easily eliminate ${\bf a}''$ from the exponent of Eq.A.1:

$$\begin{array} {lll} & ~& -\frac{1}{2} \left( \left( {\bf a... ... \cdot \left( {\bf a} - {\bf a}_M \right) \right) ,\end{array}$$ (3)

where we have defined

$$\begin{array} {lll} {\bf a}'' \equiv {\bf a}' - \left( E_{... ...\cdot {\bf a}_M + E_{MS}^{-1} \cdot {\bf a} \right)\end{array}$$ (4)

and

$$\begin{array} {lll} E_{MS}^{-1} & \equiv & (1-B)^T {E'}_{MS... ... & = & \left[ (1+B) {E'}_{MS} (1+B)^T \right]^{-1}.\end{array}$$ (5)

Changing the integration variable from ${\bf a'}$ to ${\bf a}''$, Eq.A.1 now becomes

$$\begin{array} {lll} P\left({\bf a}_M;{\bf a}\right) & = & ... ...1} + E_{MS}^{-1} \right) \cdot {\bf a}'' \right] .\end{array}$$ (6)

Notice that the last line is a constant independent of ${\bf a}$ and ${\bf a}_M$,which proves that the error matrix including the multiple scattering has to be

$$\begin{array} {lll} E_{\bf a} & = & E_{{\bf a}_M} + E_{MS} \cr & = & E_{{\bf a}_M} + (1+B) {E'}_{MS} (1+B)^T\end{array}$$ (7)

The derivation of the explicit form of E'_MS is straightforward. The multiple scattering changes the helix parameter vector from ${\bf a}$ to ${\bf a'}$and, when the pivot is chosen to be the point of the multiple scattering, the tangential vector thereat changes as

$$\begin{array} {lll} \left. \frac{d}{d\phi} \right\vert _{\p... ...ial y} + \tan\lambda' \frac{\partial}{\partial z} .\end{array}$$ (8)

Then the direction change of the track in space $\Delta \theta$ is given by

$$\begin{array} {lll} ( \Delta \theta )^2 \simeq \sin^2 \Del... ... + \frac{ (\Delta\phi_0)^2 }{ (1+\tan^2\lambda) } , \end{array}$$ (9)

where $\Delta \phi_0$ and $\Delta \tan\lambda$ are defined as components of $\Delta {\bf a} \equiv {\bf a'} - {\bf a}$.The above equation determines E'_MS through

$$\begin{array} {lll} \chi^2_{MS} & = & \left(\frac{\Delta\th... ...{\bf a}^T \cdot {E'}_{MS}^{-1} \cdot \Delta {\bf a}.\end{array}$$ (10)