Evaluation of the roughness of a crystal surface by X-ray scattering. I. Theoretical considerations (original) (raw)
Related papers
MRS Proceedings, 1994
ABSTRACTGrazing incidence X-ray reflectivity may be used to measure surface topography including roughness and correlation lengths to sub-nanometre precision. A study is made of a technically important surface, a carefully-polished specimen of Zerodur glass-ceramic, which has been measured by diffuse scatter of CuKα X-radiation and atomic-force profilometry methods. The data have been analysed in terms of a fractal representation of the surface correlation function. Results from the two methods agreed within their estimated errors, with the X-ray data showing roughnesses of 1.3 nm, correlation length of 1 μm and fractal parameter (bandwidth) of 0.35. The X-ray methods have a lower cut-off length, are much more rapid for averaged information and are both non-contacting and non-destructive. They also show potential for the study of interface roughness in thin films.
Crystal truncation rod X-ray scattering: exact dynamical calculation
Journal of Applied Crystallography, 2008
A new method is presented for a calculation of the reciprocal-space distribution of X-ray diffracted intensity along a crystal truncation rod. In contrast to usual kinematical or dynamical approaches, the method is correct both in the reciprocal-lattice points and between them. In the method, the crystal is divided into a sequence of very thin slabs parallel to the surface; in contrast to the well known Darwin dynamical theory, the electron density in the slabs is constant along the surface normal. The diffracted intensity is calculated by a matrix formalism based on the Fresnel reflection and transmission coefficients. The method is applicable for any polarization of the primary beam and also in a noncoplanar scattering geometry.
Acta Crystallographica Section A Foundations of Crystallography, 2011
An attempt is made to go beyond the distorted-wave Born approximation addressed to the grazing-incidence small-angle X-ray (GISAX) scattering from a random rough surface. The integral wave equation adjusted with the Green function formalism is applied. To find out an asymptotic solution of the nonaveraged integral wave equation in terms of the Green function formalism, the theoretical approach based on a self-consistent approximation for the X-ray wavefunction is elaborated. Such an asymptotic solution allows one to describe the reflected X-ray wavefield everywhere in the scattering (, ') angular range, in particular below the critical angle cr for total external reflection (is the grazing scattering angle with the surface, ' is the azimuth scattering angle; 0 is the grazing incidence angle). Analytical expressions for the reflected GISAX specular and diffuse scattering waves are obtained using the statistical model of a random Gaussian surface in terms of the r.m.s. roughness and two-point cumulant correlation function. For specular scattering the conventional Fresnel expression multiplied by the Debye-Waller factor is obtained. For the reflected GISAX diffuse scattering the intensity of the R dif (, ') scan is written in terms of the statistical scattering factor ; 0 ð Þ and Fourier transform of the two-point cumulant correlation function. To be specific for isotropic solid surfaces, the statistical scattering factor ; 0 ð Þ and Fourier transform of the two-point cumulant correlation function parametrically depend on the root-meansquare roughness [ ; 0 ð Þ = 0 for = 0] and cumulant correlation length ', respectively. The reflected R dif (, ') scans are numerically simulated for the typical-valued { 0 , , '} parameters array.
Physical Review B, 1999
The morphology of free surfaces of polyetherimide ͑PEI͒ samples has been characterized using atomic force microscopy in noncontact mode ͑NC-AFM͒, x-ray reflectivity ͑XRR͒, and x-ray diffuse scattering ͑XDS͒. PEI slabs have been prepared by molding pellets onto float glass. In order to characterize the roughness and morphology mathematically, we considered the surfaces as being self-affine with a rms roughness and Hurst coefficient H up to a finite correlation length. NC-AFM topographs with ϭ61Ϯ6.3 Å showed the presence of blobs on the surface having a slight tendency to form clusters. These features are reflected in the heightheight correlation function which shows the presence of ''strong'' short-range (1 ϭ1064 Å, H 1 ϭ0.46) and ''weak'' long-range (2 ϭ14537 Å, H 2 ϭ0.99) correlations. The detection of the specularly reflected contribution in XRR scans only gives access to , the lateral fluctuations being averaged out over the coherence area. Scans in XDS include the detection of the diffuse ͑off-specular͒ scattering arising from lateral correlations in the surface fluctuations profile and are hence sensitive to , , and H. The XDS data have been analyzed by simultaneous fits using the distorted-wave Born approximation up to the first order. Calculations were performed using previously published approximations. Best fits to the experimental data provided strongly different correlations length values (7570 Åрр1245 Å) indicating that the estimation of the latter by XDS is somewhat ambiguous. This is in agreement with the strong discrepancy between the correlation lengths determined with x rays and NC-AFM, which can be accounted for by the limits of the perturbation theory for interfaces with high rms roughnesses. ͓S0163-1829͑99͒04032-1͔
Self-consistent approach to x-ray reflection from rough surfaces
Physical Review B, 2007
A self-consistent analytical approach for specular x-ray reflection from interfaces with transition layers ͓I. D. Feranchuk et al., Phys. Rev. B 67, 235417 ͑2003͔͒ based on the distorted-wave Born approximation ͑DWBA͒ is used for the description of coherent and incoherent x-ray scattering from rough surfaces and interfaces. This approach takes into account the transformation of the modeling transition layer profile at the interface, which is caused by roughness correlations. The reflection coefficients for each DWBA order are directly calculated without phenomenological assumptions on their exponential decay at large scattering angles. Various regions of scattering angles are discussed, which show qualitatively different dependence of the reflection coefficient on the scattering angle. The experimental data are analyzed using the method developed.
High-resolution x-ray scattering measurements: I. Surfaces
Reports on Progress in Physics, 2000
X-ray scattering investigations of surfaces and interfaces in soft-condensed matter are reviewed. Both high-resolution structural determinations in the direct space and investigations of fluctuations with long-range correlations requiring a high resolution in the Fourier space are discussed. All the scattering cross-sections for diffraction or diffuse scattering are derived within a unified frame, and the experimental aspects related to their measurement are discussed in detail. The general principles are illustrated by various examples of studies of the liquidvapour interface, Langmuir and Langmuir-Blodgett films, wetting films, polymer films, liquid crystals and liquid-liquid interfaces.
Applied Optics, 1989
XUV and x-ray scattering by a LiF crystal is measured. The angular distribution of the scattered radiation (ADSR) reveals characteristic features, side peaks or asymmetry. The surface of the sample is statistically characterized by a microdensitometer analysis of electron micrographs resolving the short spatial wavelengths of the surface roughness. This analysis shows that the surface has a large microroughness with an autocovariance function which is Gaussian in its initial portion. The first-order perturbation vector theory of the roughness-induced scattering leads to an interpretation of the ADSR features in terms of the modulation of the surface power spectral density function associated with the microroughness by an optical factor. The possibility of obtaining short scale roughness characterization from XUV or x-ray measurements is discussed. 1. Introduction A rough surface can be intuitively regarded as a random collection of sinusoidal gratings. Each grating can diffract radiation of wavelength X0 according to the condition: PX 0 = l(sinO 2-sinO 1), (1) where 02 and 0 stand for the angle of the diffracted beam and the incident angle, respectively, both taken with respect to the normal of the mean surface; 1 is the period of the grating and p is the order of diffraction. The incident radiation is mainly diffracted to the zeroth and first orders corresponding, respectively, to the specular and scattered radiation. This qualitative approach is at the origin of mathematical treatments of the roughness-induced scattering using a statistical description of the roughness profile in conjunction with scalar or vector diffraction theory of the electromagnetic field.
Optics for EUV, X-Ray, and Gamma-Ray Astronomy V, 2011
Several hard X-ray imaging telescopes of the next future will be characterized by a high angular resolution. To this end, it is necessary to produce optics with a very low surface microroughness, as this is responsible of X-ray scattering, which results in image quality degradation especially at the higher energies. To the smooth surface approximation, it is possible to compute the X-Ray Scattering (XRS) from the Power Spectral Density (PSD) of the surface roughness. Indeed, multilayers coatings will be used to reflect X-rays beyond 10 keV; in this case the scattering pattern is more complicated but it can still be computed from the PSDs of each interface and the cross-correlation functions of the rough profiles. A growth model able to describe the roughness evolution of the surfaces enables us to compute the XRS of the multilayer, which can be directly compared to the experimental data. With this approach we aim to validate the roughening model assumed and to accurately predict the scattering pattern we expect on the focal plane. In this work we show the application of this formalism: direct measurements of the PSDs for the substrate and the outermost layer of a multilayer sample are used as input for the code to model the PSD growth. XRS measurements of that sample, performed at the energy of 8.05 keV, are presented to validate the modeling achieved.
Applied Optics, 1998
Grazing-incidence specular reflectance and near-specular scattering were measured at Al-K ␣ ͑1. 486-keV, 8.34-Å͒ radiation on uncoated dielectric substrates whose surface topography had been measured with a scanning probe microscope and a mechanical profiler. Grazing-incidence specular reflectance was also measured on selected substrates at the Cu-K ␣ ͑8.047-keV, 1.54-Å͒ wavelength. Substrates included superpolished and conventionally polished fused silica; SiO 2 wafers; superpolished and precision-ground Zerodur; conventionally polished, float-polished, and precision-ground BK-7 glass; and superpolished and precision-ground silicon carbide. Roughnesses derived from x-ray specular reflectance and scattering measurements were in good agreement with topographic roughness values measured with a scanning probe microscope ͑atomic force microscope͒ and a mechanical profiler that included similar ranges of surface spatial wavelengths. The specular reflectance was also found to be sensitive to the density of polished surface layers and subsurface damage down to the penetration depth of the x rays. Density gradients and subsurface damage were found in the superpolished fused-silica and precision-ground Zerodur samples. These results suggest that one can nondestructively evaluate subsurface damage in transparent materials using grazing-incidence x-ray specular reflectance in the 1.5-8-keV range.