Derivation and Explanation of Equation 3.7

Why would one expect this formula to work? The underlying principle is the time-honoured one of making two measurements, in which the systematic error is of the same magnitude, but in opposite senses, then taking an average of the two measurements, to obtain a result free of systematic error. However, because the systematic errors under consideration here are multiplicative, rather than additive, ``average'' indicates geometric mean, rather than arithmetic mean. The application of this principle in the present case will now be examined in greater detail.

Let $\eta_i$ be the sensitivity (quantum efficiency) of detector $i$. The two sources of multiplicative systematic error mentioned above can be defined as conditions where $\eta_1 \neq{} \eta_2$, and where $\mathrm{d}s \neq{} 0$, where the misalignment $\mathrm{d}s$ is as defined in figure 3.3. The measured electron arrival rate at detector $i$, with incident polarization direction $j$, $R_{ij}$, is

$$R_{ij} = \eta_iQ_{ij}\textrm{,}$$ (3.8)

where $Q_{ij}$ is the actual rate at which electrons arrive at the detector.

Let $\Theta_i$ be the solid angle subtended by detector $i$ at the point where the electron beam strikes the thorium foil. The scattering probability per unit solid angle in the region of detector $1$ is (equation 3.4) $\sigma_0(1+SP_{\textrm{true}})$ for polarization direction $\uparrow{}$, and $\sigma_0(1-SP_{\textrm{true}})$ for polarization direction $\downarrow{}$, and vice versa (equation 3.5) in the region of detector $2$, i.e., if $Q_0$ is the rate at which electrons are incident on the thorium foil,

$$Q_{1\uparrow} = Q_0\sigma_0(1+SP_{\textrm{true}})\Theta_1\textrm{,}$$ (3.9)

$$Q_{1\downarrow} = Q_0\sigma_0(1-SP_{\textrm{true}})\Theta_1\textrm{,}$$ (3.10)

$$Q_{2\uparrow} = Q_0\sigma_0(1-SP_{\textrm{true}})\Theta_2\textrm{,}$$ (3.11)

and

$$Q_{2\downarrow} = Q_0\sigma_0(1+SP_{\textrm{true}})\Theta_2\textrm{.}$$ (3.12)

Therefore,

$$R_{1\uparrow} = Q_0\sigma_0(1+SP_{\textrm{true}})\Theta_1\eta_1\textrm{,}$$ (3.13)

$$R_{1\downarrow} = Q_0\sigma_0(1-SP_{\textrm{true}})\Theta_1\eta_1\textrm{,}$$ (3.14)

$$R_{2\uparrow} = Q_0\sigma_0(1-SP_{\textrm{true}})\Theta_2\eta_2\textrm{,}$$ (3.15)

and

$$R_{2\downarrow} = Q_0\sigma_0(1+SP_{\textrm{true}})\Theta_2\eta_2\textrm{.}$$ (3.16)

If $A$ is the cross-sectional area of a detector,

$$\Theta_1 = \frac{A}{r_1^2}\protect$$

$$= \frac{A}{(s+\mathrm{d}s)^2+h^2}\protect$$

$$= \frac{A}{s^2+h^2}\left(1+\frac{2s\mathrm{d}s+\mathrm{d}s^2}{s^2+h^2}\right)^{-1}\protect$$

$$\approx{} \frac{A}{s^2+h^2}\left(1-\frac{2s\mathrm{d}s}{s^2+h^2}\right)\textrm{,}$$ (3.17)

where $\vert\mathrm{d}s\vert \ll{} \frac{1}{2s(s^2+h^2)}$. Similarly,

$$\Theta_2 \approx{} \frac{A}{s^2+h^2}\left(1+\frac{2s\mathrm{d}s}{s^2+h^2}\right)\textrm{.}$$ (3.18)

Some cumbersome notation can be avoided by defining $\Theta_0 = \frac{A}{s^2+h^2}$ and $\mathrm{d}\Theta =\frac{2s\Theta_0\mathrm{d}s}{s^2+h^2}$. Then

$$\Theta_1 = \Theta_0-\mathrm{d}\Theta \textrm{,}$$ (3.19)

and

$$\Theta_2 = \Theta_0+\mathrm{d}\Theta \textrm{.}$$ (3.20)

The polarization obtained from equation 3.7, in the absence of random errors, is therefore,

$$P_{\textrm{formula}} = \frac{\sqrt{R_{1\uparrow}R_{2\downarrow}}-\sqrt{R_{1\downarrow}R_... ...qrt{R_{1\uparrow}R_{2\downarrow}}+\sqrt{R_{1\downarrow}R_{2\uparrow}})}\protect$$

$$= \left(\sqrt{Q_0^2\sigma_0^2(1+SP_{\textrm{true}})^2\eta_1\eta_2(\Theta_0+\mathrm{d}\Theta )(\Theta_0-\mathrm{d}\Theta )}\right.\protect$$

$$\left.-\sqrt{Q_0^2\sigma_0^2(1-SP_{\textrm{true}})^2\eta_1\eta_2(\Theta_0+\mathrm{d}\Theta )(\Theta_0-\mathrm{d}\Theta )}\right)\protect$$

$$/\left(\sqrt{Q_0^2\sigma_0^2(1+SP_{\textrm{true}})^2\eta_1\eta_2(\Theta_0+\mathrm{d}\Theta )(\Theta_0-\mathrm{d}\Theta )}\right.\protect$$

$$\left.+\sqrt{Q_0^2\sigma_0^2(1-SP_{\textrm{true}})^2\eta_1\eta_2(\Theta_0+\mathrm{d}\Theta )(\Theta_0-\mathrm{d}\Theta )}\right)\protect$$

$$= P_{true}\textrm{.}$$ (3.21)