Spin-Dependence of the Scattering Cross-Section

By subjecting the general solution of the Dirac equation for a central potential to the condition that the electron wave-function is a superposition of a plane incident wave with a spherical (in wavefront geometry, not necessarily in the symmetry of the scattered amplitude) scattered wave, Kessler [31,32] finds it possible to express the effect of a spin-dependent potential correction on the scattering cross-section $\sigma$ (whether differential or integrated over a particular solid angle) as

$$\sigma =\sigma _0(1+SP)\textrm{,}$$ (3.2)

where $\sigma _0$ and $S$ are functions of the incident electron energy $E_b$, and of the scattering angle $\theta$, and $P$ is the component of the spin polarization of the incident beam along the normal to the scattering plane; that is to say, if the electrons were forced to collapse into states of well-defined spin component along this direction, the fractions with spin up $f_u$ and spin down $f_d$ would be such that

$$P=f_u-f_d\textrm{.}$$ (3.3)

$S$ is known as the asymmetry function [69,5,34], the Sherman function [31,32,5,34] or the analysing power [34,70]. The only assumption about the form of the potential that was required to obtain this result was its separability in spherical polar co-ordinates. All the finer details of the potential, be it that of a single atom (section 3.1.1) or that of a metal foil of the kind found in a real polarimeter (figure 3.1,) are contained in the functional forms of $\sigma _0$ and $S$. In particular, if there were no mechanism, of the kind described in equation 3.1, for spin-dependence in the scattering potential, then $S$ would vanish.

Equation 3.2 leads directly to the left-right asymmetry in scattering of a polarized electron beam. Two detectors are so placed as to collect scattered electron beams with scattering angles of the same magnitude, but in opposite senses (figure 3.1.) The scattering plane normals for the two detectors are therefore in opposite directions, i.e. the polarization component $P$ is of the same magnitude but opposite sign at the two detectors. For a scatterer of even quite low symmetry, $S$ will be the same for both detectors, with the result that the rates of electron arrival at the two detectors, $R_1$ and $R_2$, will be [5,34], from equation 3.2

$$R_1 = R_0(1+SP)\textrm{,}$$ (3.4)

and

$$R_2 = R_0(1-SP)\textrm{.}$$ (3.5)

In the case of the compact retarding-potential polarimeter (figure 3.1, chapter 3,) it is possible to obtain $S$ for some values of the energy of the incident electrons at the target and the energy window defined by the retarding potential from look-up tables [6]. Combining equations 3.4 and 3.5, the polarization is

$$P = \frac{R_1-R_2}{S(R_1+R_2)}\textrm{.}$$ (3.6)