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 [5,34]. As Kessler [31,32] 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 3.1.) The mechanism for this can be understood classically, as the additional, position-dependent 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 [5,34]. Gay & Dunning [5] and Dunning [34] have developed a quantitative version of this model for a single scattering nucleus, which results in the expression for the spin-dependent potential contribution

$$V_{so}'=\frac{Ze\mu _B\mathbf{L}.\mathbf{S}}{8\pi\epsilon _0\hbar m_ec^2\vert\mathbf{r}\vert^3}\textrm{,}$$ (3.1)

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 [45]. Kessler [31,32] also makes some helpful comments on this model.