Phase of reflection of very low energy electrons at crystal surfaces

The amplitude coefficients of electron reflection at crystal surfaces are complex numbers, each of which may be characterized by a reflection intensity (the squared modulus of the coefficient) and a phase. The phase of reflection of very low energy (? 10 eV) electron reflection is described on the basis of existing theory, and experimental approaches to phase determination are reviewed. Theoretical properties of the phase are described on the basis of the two-beam dynamical theory of diffraction. The model considered is an idealized substrate crystal with an attached selvedge (surface region). The indirect effect of inelastic scattering (absorption) is included by going to complex values of the electron energy or of the surface-normal component K of the propagation vector. In the absence of a selvedge the phase is determined solely by the band structure of the substrate crystal. If a selvedge is present there are large additional effects on the phase associated with zeros of the amplitude coefficient of reflection on the complex K plane. The experimental approaches considered are: (1) measurement of the kinetic energy distributions of ions produced in the field ion microscope, and (2) measurement of the periodic deviations from the Schottky line in field-assisted thermionic emission and photo-emission. Recent results of phase determination for W (011) surface by method (1) are summarized and compared with theoretical expectations.