Our software packages for x-ray reflectivity and dynamical x-ray diffraction

V. Holý

Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 5, 121 16 Praha, Czech Republic

holy@mag.mff.cuni.cz

Keywords: x-ray reflection, GISAXS method, dynamical x-ray diffraction, diffuse scattering

X-ray scattering methods are usually indirect so that a comparison of measured data with a suitably chosen structure model is necessary for the determination of the investigated structure. This comparison usually consists in the following steps:

(i) Formulation of a suitable structure model – this is the most difficult step that usually requires additional information on the investigated sample obtained by another methods, deduced from the sample growth procedure, etc.

(ii) Simulation of the process of x-ray scattering – in this step we have to make assumption concerning the scattering process (kinematical vs dynamical scattering theory, far-field limit). In the case of samples with a random structure we have to consider the process of ensemble averaging and possible ergodicity of the experimental data. We have also to include correctly the properties of the experimental set-up (coherence of the primary beam, resolution in real and reciprocal space, geometrical factors etc.).

(iii) Comparison of the simulated and measured data – this step is not trivial, but it is usually based on standard numerical algorithms (least-square fitting, genetic algorithms, neural networks) that are independent from the previous steps.

In the talk, I will deal only with the step (ii) and I will present several numerical programs for simulation of x-ray reflectivity (XRR), small-angle scattering (SAXS) and diffraction (XRD). The theoretical basis of all the programs can be found in Ref. [1].

XRR programs calculate the specular and diffuse intensities scattered from an arbitrary multilayer with randomly rough interfaces, taking into account statistical averaging over all microstates of the roughness profiles. The roughness is assumed fractal, i.e. each interface is described by its root-mean-square (rms) roughness, lateral correlation length and fractal dimension. The correlation of roughness profiles of different interfaces is described by a correlation matrix and it can depend on the space-frequency of the roughness. The scattering process is described within the distorted-wave Born approximation (DWBA).

The SAXS experiments are considered in grazing-incidence geometry (GISAXS method), in which the incidence angle of the primary beam is close to the critical angle of total external reflection. The programs presented in the talk describe GISAXS from two- and three-dimensional disordered arrays of scattering centers. The scattering factor of an individual center is calculated assuming a particular shape of the center (ellipsoidal or facetted). The correlation function of the center positions is calculated using various modifications of the well-known paracrystal model or ab-initio using the Monte-Carlo approach. A special attention is paid to the correlation of the center positions with their sizes; the programs include two correlation models denoted DA and LMA in the literature.

XRD programs are based on dynamical diffraction theory taking into account the two-beam approximation with exact dispersion surface of the 4^th order. Therefore, this approach does not suffer from tangential errors that occur usually in a standard formulation of the two-beam approximation with the simplified dispersion surface of the 2^nd order. The programs calculate the intensity diffracted from an arbitrary ideal pseudomorph single-crystalline superlattice, which does not contain misfit dislocations or other structure defects. The scattering geometry is general, i.e. the programs are not restricted to coplanar scattering and they can calculate the diffraction in any non-coplanar arrangement (including grazing-incidence geometry).

Compared to standard software available commercially or on internet, the programs include the following new features:

In XRR: correlation of roughness profiles of different interfaces giving rise to resonant-diffuse phenomena in reciprocal space. The correlation of the roughness can depend on the space frequency of the roughness profile; this property follows from the mechanism of the multilayer growth.

In GISAXS: the programs include various three-dimensional distributions of the scattering centers.

In XRD: Exact dispersion surface of the 4^th order makes it possible to calculate the diffracted intensity in any non-coplanar scattering geometry.

The programs use Matlab, especially its unique matrix manipulation capability, so that the calculation speed is comparable to Fortran or C++. The programs are not written in a user-friendly style but they are commented in the program head, so that their application is not complicated. The programs are available on request from the author of this talk.

[1] U. Pietsch, V. Holý, and T. Baumbach, High-Resolution X-Ray Scattering From Thin Films to Lateral Nanostructures, Springer-Verlag Berlin, Heidelberg, New York 2004.