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Statistical errors and optimization

The statistical error of the asymmetries are:

$\displaystyle \Delta A_1^2$ $\textstyle =$ $\displaystyle \big({\frac{1}{P_tP_bf}}\big)^2
...a A_{\parallel}}\big)^2

The statistical error of $A_\parallel$ and $A_\perp$ can be obtained from:

$\displaystyle \Delta A_\parallel = \sqrt{\frac{1-A^2_\parallel}{N_\parallel}}
\sim\frac{1}{\sqrt{N_\parallel}}$   $\displaystyle \hspace{1cm}
\Delta A_\perp = \sqrt{\frac{1-A^2_\perp}{N_\perp}}

where $N_\parallel$ ($N_\perp$) is the number of scattered electrons when the beam polarization is parallel or antiparallel (perpendicular) to the target polarization. They can be estimated from the cross section of inclusive deep inelastic electron scattering, as shown in section 3.

The error of $A_1$ for each kinematics can be written as:

$\displaystyle \Delta A_1^2$ $\textstyle =$ $\displaystyle \alpha^2\Delta A_\parallel^2+\beta^2\Delta A_\perp^2$  

To minimize $\Delta A_1$ we need $t_\parallel=\frac{\alpha}{\beta}t_\perp$.

Xiaochao Zheng 2001-06-09