Ultrasoft-X-Ray Reflection, Refraction, and Production of Photoelectrons (100-1000-eV Region)

The reflection, refraction, and associated production of photoelectrons by ultrasoft x rays (10-100 \AA{}) can be important bases for the determination of material constants such as the linear x-ray absorption coefficients and the electron mean free paths. These may then be used to establish directly the photoionization cross sections and the electron-collision cross sections which account for the dominant energy-absorbing processes within solids for this energy region. Because the effective sample depths for these interactions are typically less than 100 \AA{}, they constitute an important practical basis for surface characterization. By applying the exact theory for the reflection-refraction of a plane electromagnetic wave at an absorbing dielectric interface to the shorter-wavelength region (10 \AA{}), it can be shown that the conventional approximate theory of x-ray reflection is adequate. However, the more exact theory must be applied in the region of longer x-ray wavelengths (>50 \AA{}). Although the derivations of the exact theory are tedious, the results can be expressed in relatively simple form as functions of two material constants $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$, which are identifiable as the unit decrements to a complex dielectric constant, of the grazing-incidence angle, and of a parameter which is a function of this grazing angle and which becomes the angle of refraction for small angles of incidence. X-ray absorption coefficients and electron mean-free-path values have been determined from x-ray reflection and refraction and photoelectron excitation data. These values have been shown to agree reasonably well with such material constants as determined by transmission measurements through thin samples.