3 Reflection of an Unpolarized Beam from the Surface of a Bulk Magnetic Sample

An unpolarized incident electron beam is [10,11] an incoherent superposition of pure states representing all directions of the incident spin. The polarization of the reflected beam from any surface is, therefore, given by an average of the polarization over all polarization directions, weighted according to the intensity reflection coefficient for each polarization. This incoherent averaging process (section 7.2) gives this reflected polarization from a bulk surface (figure 1)

$$P = \frac{2y_1z_1}{3y_1^2+z_1^2}+O(\{y_1,z_1\})\protect$$

$$= -\frac{4e^2\hbar{}m_eV_1B_1}{12e^2m_e^2V_1^2+e^2\hbar^2B_1^2}+O(\{y_1,z_1\})\textrm{.}$$ (10)

Both the term given explicitly, and the next term in the binomial expansion, are in the direction of the magnetic flux density in the bulk material.

The most salient qualitative feature of this polarization formula is that, at high incident electron energies, the reflected polarization is dominated by a non-zero term, which is independent of the incident electron energy, and controlled by the balance between the electrostatic potential and the magnetic flux density, in the sample. This polarization can be as large as $\frac{1}{\sqrt{3}}$ in either direction.