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Appendix

Note that there is a sign error in Eq. (12) of reference [3] - it should read

 
$\displaystyle {d \sigma \over d \Omega}$ = $\displaystyle {1 \over 4} ( \vert A_{\vert\vert}\vert^2 + \vert A_{\perp}\vert^...
...
(\vert A_{\vert\vert}\vert^2 - \vert A_{\perp}\vert^2 ) ( \xi_{1i} + \xi_{1f})$  
    $\displaystyle +
{1 \over 4}
( A_{\vert\vert} A^*_{\perp} + A^*_{\vert\vert} A_{...
...perp} - A^*_{\vert\vert} A_{\perp} )
( \xi_{3i} \xi_{2f} - \xi_{2i} \xi_{3f} ).$ (10)

This equation describes the differential cross section for elastic photon scattering from an unpolarized target in terms of the Stokes parameters and combinations of two complex amplitudes. It corresponds to Eq. (3) of this paper. The amplitudes A|| and $A_{\perp}$ are related to the M and N amplitudes through the relations

\begin{displaymath}A_{\vert\vert} = M \cos \theta - N \sin^2 \theta,
\;\;\; A_{\perp} = M.
\end{displaymath} (11)

A different notation for the Stokes parameters was used in [3]. In order to compare Eq. (10) with Eq. (3) the following substitutions should be made, relating the two sets of Stokes parameters:
$\displaystyle \xi_{1i} \rightarrow \xi_3^{(1)}, \;\;$ $\textstyle \xi_{2i} \rightarrow
\xi_1^{(1)}, \;\;$ $\displaystyle \xi_{3i} \rightarrow \xi_2^{(1)},$  
$\displaystyle \xi_{1f} \rightarrow \xi_3^{(2)}, \;\;$ $\textstyle \xi_{2f} \rightarrow
\xi_1^{(2)}, \;\;$ $\displaystyle \xi_{3f} \rightarrow \xi_2^{(2)}.$ (12)

We can then rewrite Eq. (A1) as

 \begin{displaymath}{d \sigma \over d \Omega} = d_1 + d_2 ( \xi_3^{(1)} +
\xi_3^{...
...} +
d_4 ( \xi_1^{(1)} \xi_2^{(2)} - \xi_2^{(1)} \xi_1^{(2)} ),
\end{displaymath} (13)

where
$\displaystyle d_1 = {1 \over 4} ( \vert A_{\vert\vert}\vert^2 + \vert A_{\perp}\vert^2 ),
\;\;$ $\textstyle \;\;$ $\displaystyle d_2 = {1 \over 4} ( \vert A_{\vert\vert}\vert^2 - \vert A_{\perp}\vert^2 ),$  
$\displaystyle d_3 = {1 \over 4} ( A_{\vert\vert}^* A_{\perp} + A_{\vert\vert} A^*_{\perp}),$ $\textstyle \;\;$ $\displaystyle d_4 = {i \over 4} ( A_{\vert\vert}^* A_{\perp} - A_{\vert\vert} A^*_{\perp}),$ (14)

and the identification with Eq. (3) of this paper is complete. A detailed discussion of other commonly used representations of the invariant amplitudes is given in [1].


next up previous
Next: Figures Up: Circular dichroism effects in Previous: Acknowledgments
Eoin Carney
1999-06-15