Statistics

The statistical error of the asymmetries are:

$$\Delta A_1^2 = \big({\frac{1}{P_tP_bf}}\big)^2 \big[{\big({\frac{1}{D(1+\eta\xi)... ...a A_{\parallel}}\big)^2 +\big({\frac{\eta}{d(1+\eta\xi)}A_{\perp}}\big)^2}\big]$$

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

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

where $N_\parallel$ ($N_\perp$) is the number of scattered electrons when the beam polarization is parallel or antiparallel (perpendicular) to the target polarization.

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

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

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