Advanced analysis techniques for X-ray reflectivities. Theory and application

The first part of this thesis adresses the phase problem in X-ray reflectivity. The analytical properties of the reflection coefficient imply that the phase is completely determined by the Hilbert transform of the logarithm of the modulus and the zeros in the upper half complex plane (UHP). To account in addition for interfacial roughness, a new formula for the Hilbert-phase is derived.In the following, the conditions for which the reflection coefficient has zeros in the UHP is discussed and the existing sufficient condition is extended to rough multi-layer systems. Procedures for locating these zeros are developed. The second part of this thesis introduces a new iterative inversion method for X-ray reflectivity. It expands the profile in a set of eigenfunctions, which are discrete approximations of the eigenfunction of the classical reconstruction problem of a compact supported function from its partially known Fourier-transform. In this work, piecewise constant functions, polygons and second-order B-splines are used to expand the density profile. The eigenvalue problems for the calculation of the above mentioned approximations are stated and solved. The formalism for the calculation of the reflection coefficient for these profiles is developed in dynamical and single-scattering theory. In the experimental part of this work iterative inverse schemes are applied to the analysis of X-ray reflectivity. Different sample systems are investigated: For two titanium-carbon samples tiny details at the Ti/C interface such as the formation of a thin TiC layer can be observed.The density profiles obtained from the reflectivities taken from nickel-carbon samples show the formation of SiC inside the Si sub strate. Finally, the new inversion scheme is applied to a series of reflectivities from a 700 AaSiGe film on a substrate.