2.1 The Spin-Dependence of the Scattering Potential

The differential cross-section as a function of scattering angle in Mott scattering, i.e. the scattering of electrons by nuclei at small impact parameters, exhibits a dependence on the electron spin direction [10,14]. As Kessler [2,5] points out, in addition to simplifying the potential calculation, the use of small impact parameters, i.e. high energies and large scattering angles, increases the analysing power of the polarimeter as a result of the $\frac{1}{\vert\mathbf{r}\vert^3}$ dependence of the spin-orbit correction (equation 2.1.) The mechanism for this can be understood classically as the additional term in the scattering potential resulting from the torque exerted on the electron's spin magnetic moment by the magnetic field due to the presence of a moving, charged nucleus in the electron's rest frame [10,14]. Gay & Dunning [10] and Dunning [14] have developed a quantitative version of this model for a single scattering nucleus, which results in the expression for the spin-dependent potential contribution

$$V_{so}'=\frac{Ze\mu _B\mathbf{L}.\mathbf{S}}{8\pi\epsilon _0\hbar m_ec^2\vert\mathbf{r}\vert^3}$$ (2.1)

Where $Z$ is the proton number of the nucleus, $\mu _B$ is the Bohr magneton, $\mathbf{L}$ is the orbital and $\mathbf{S}$ the spin angular momentum of the electron, in the rest frame of the nucleus, and $\mathbf{r}$ is the displacement of the electron from the nucleus. This potential form is equivalent to the fine splitting by the spin-orbit interaction familiar from the analysis of bound electronic states in atomic physics [16]. Kessler [2,5] also makes some helpful comments on this model.