DYNAMICAL THEORY OF X-RAY-DIFFRACTION FOR THE STUDY OF CRYSTAL-SURFACES

The Darwin dynamical theory is extended to the asymmetric diffractions of the Bragg and Laue cases in a form suited for treating the crystal with a different structure on its surface by using the characteristic matrix method. For ideal crystals, the exact analytical solutions valid for any incidence condition are obtained as a function of the general expression of the deviation parameter within the two-beam approximation. It is shown that the exact expressions are reduced to the expressions given by the conventional theories using the appropriate approximations in the cases of CTR (crystal truncation rod) scattering, specular reflectivity and Bragg reflection including the condition of a Bragg angle of 90 degrees. Next the Darwin theory is extended to the coplanar three-beam case to study the effect of other beams on CTR scattering. It is shown that the reflectivity of CTR scattering obtained by the two-beam approximation holds good except when the other beam satisfies the Bragg condition, and the modulation in the reflectivity on such conditions is sensitive enough to get information of the surface structure.