Classical-Field Theory of Electron Waves as a Polarized Radiation Probe   of Magnetic Surfaces
  Recently, there has been a revival of interest in mechanisms for changing the spin polarization of an electron beam on transmission through, or reflection from, a magnetic surface. An understanding of these mechanisms would allow the use of an electron beam as a polarized radiation probe for magnetic characterization, like light in MOKE and neutrons in PNR. Here, a mechanism is described which, unlike simultaneously occurring processes proposed elsewhere, polarizes an unpolarized incident beam without recourse to inelastic processes.   A magnetic field leads to a Zeeman term in an electron's Hamiltonian, which depends on the angle theta between the electron's spin vector and the magnetic flux. As a result, when an electron wave is incident on the surface of a bulk magnetic material (figure 1,) the wave-number of the transmitted wave depends on theta. When the conditions of continuity of the wave-function, and of its first spatial derivative, at the surface, and conservation of particles, are applied, an electron reflection coefficient is obtained which also depends on theta. Therefore, some polarizations are preferentially reflected, while others are preferentially transmitted. The amplitude reflection and transmission coefficients can readily be converted to intensity coefficients, and averaged over an incoherent superposition of electron waves of different theta, e.g. an unpolarized incident beam. The reflected polarization is P = -(2 e mu_B V B)/(3 e^2 V^2 + mu_B^2 B^2), which can take values -1/sqrt(3) <= P <= 1/sqrt(3), depending on the balance between V and B.   The analysis can be extended to multi-layers using the theory of Fabry-Perot etalons. 
D. C. Hatton and J. A. C. Bland