Instrumental aberrations

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

There is no reason for an experimental Bragg peak to be exactly described by a simple analytical function. Bragg peaks are generally very complicated objects. In the early usual simplified formalism, still in use, the experimental function h x) describing a broadened Bragg peak profile observed on a powder diffraction pattern is due to the convolution of the instrumental aberration function g(x) with the sample function fix) ... 

X-Ray powder diffractograms contain in encrypted form information about the structure of the sample material. The positions and intensities of diffraction peaks reveal the information about an ideal crystal structure. The form of the peaks reflects the information about defects in the strncture. Instrumental aberrations affect the apparent peak positions (especially at low and high scattering angles) and intensities of the diffraction peaks as well as the form of the peaks.Hence, properly taking into account the instrumental contributions is essential both for studies aimed at obtaining information about the ideal crystal structure of the material and information about deviations from this ideal structure. 

The line profile in X-ray powder diffraction for a monochromatic beam is determined by sample broadening and instrumental aberration. Figure 6.1 shows schematically contributions to the observed profile h (p) from instrumental aberration g (p) and physical profile ftp) for monochromatic X-rays. Measurements of a sample of the material without physical broadening ( ideal sample ) with the diffractometer without instrumental aberration would give the profile as a Dirac d-function. 

The application of hi resolution electron microscopy to the determination of structures in systems with actual or potential use in selective oxidation catalysis is discussed. Problems of image interpretation, arising from instrumental aberrations and multiple scattering, are outlined, and exan les of its use are given in the systems Bi-Mo-W-0, Bi-Mo-Nb-0 and Bi-W-Nb-0. In all three, intermediate phases are revealed which show either potential or actual structural adaptability to varying stoichiometry, particularly with regard to oxygen content. 

As we have already mentioned, the development of high resolution X-ray diffraction devices made it possible to show that Gaussians did not fit the experimental profiles well. Generally speaking, when instrumental aberrations lead to symmetrical increases in peak width, the peaks can usually be approximated as Gaussian or Loientzian functions. The latter are expressed as follows ... 

The dispersion of the distribution, or diffraction line broadening, is measured by the full width at half the maximum intensity (FWHM) or by the integral breadth ifi) defined as the integrated intensity (J) of the diffraction profile divided by the peak height (/3 = ///q). Line broadening arises from the convolution of the spectral distribution with the functions of instrumental aberrations and sample-dependent effects (crystallite size and structure imperfections). 

The specimen is immersed in the next lens encountered along the column, the objective lens. The objective lens is a magnetic lens, the design of which is the most crucial of all lenses on the instrument. Instrumental resolution is limited primarily by the spherical aberration of the objective lens. 

Run-of-the-mill instruments can achieve a resolution of 5-10 nm, while the best reach 1 nm. The remarkable depth of focus derives from the fact that a very small numerical aperture is used, and yet this feature does not spoil the resolution, which is not limited by dilfraction as it is in an optical microscope but rather by various forms of aberration. Scanning electron microscopes can undertake compositional analysis (but with much less accuracy than the instruments treated in the next section) and there is also a way of arranging image formation that allows atomic-number contrast, so that elements of different atomic number show up in various degrees of brightness on the image of a polished surface. 

The focusing of radiation within the instrument was formerly done by means of lenses, but these suffer from chromatic aberration and particularly in respect of the relationship between the visible and ultraviolet parts of the spectrum. Focusing is now usually carried out by means of suitably curved mirrors having a reflecting surface coated with aluminium which is protected by a silica film. 

It can be readily anticipated that the new instrumentation, having extended the point resolution of the microscope up to its information limit [117], will provide even better high resolution images of nanoclusters, and also that it will not supersede, but emphasize, the role of EH, as the relevant structural information encoded in the phase (which is still completely lost in the recording process) can be retrieved corrected by all coherent aberrations. 

We first define the geometry and instrumental parameters common to high resolntion diffractometry. As a reference, we then develop the dnMond diagram for visualisation of X-ray optics and use it to discuss practical beam conditioners. Next we treat the principal aberrations of high resolution diffractometry tilt, curvature and dispersion. We discuss the requirements on X-ray detectors, and finally show how to set up a high resolution measurement in practice. 

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