Angular dependence of the absorption-induced nodal plane shifts of X-ray stationary waves

The classical equations of dynamical theory in the Bragg case are rederived in a way which is valid even in the presence of absorption where every quantity used is complex. The properties of the phase and the amplitude of the diffracted wave are discussed in detail. It is shown that for non-centrosymmetric crystals the position of the nodes at the centre of the reflection domain is strongly absorption dependent. The limit of the phase of the ratio of the diffracted to the incident amplitude far from the reflection domain is calculated to be equal to φ_h + π and φ_h, below and above the reflection domain, respectively, where φ_h is the phase of the structure factor calculated taking into account anomalous dispersion and the imaginary part of the form factor. It is therefore absorption dependent and so is the position of the nodes of stationary waves which is never invariant with the angular position of the crystal, even outside the total reflection domain. The example of a non-centrosymmetric crystal, GaAs, opposite reflections, 111 and , and of wavelengths close to the absorption edges of gallium and arsenic is used to illustrate the results. It shows that the absorption-induced shifts of the nodal planes are in general directed towards the surface of the crystal, independently of the sense of the diffraction vector. The variation of the penetration depth within the total reflection is interpreted in the non-absorbing case by means of a new surface which completes the dispersion surface within the Bragg gap.