Real structures, direct-space techniques

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

The averaging procedure inherent in optical diffractometry, as well as the direct, real-space crystallographic power of high resolution electron microscopy combine to yield a uniquely powerful set of techniques for structural characterisation. 

At present the only technique available for specific (sometimes called real-space) structure determination is high resolution electron microscopy (HREM). At first sight this appears to be an ideal method, as the direct imaging of the structure avoids the phase problem normally associated with diffraction methods, and can be applied to all materials, whatever their state of long range order. Compared with diffraction methods, however, the accuracy is relatively poor, as the available resolution is limited to not much better than 2S, well above the theoretical limit. Furthermore, severe problems of image interpretation occur, but within certain limitations, these can be overcome and the technique applied successfully. The object of this paper is to illustrate the use of these direct imaging methods in systems with possible catalytic application. 

ESDIAD is obviously not a diffraction technique such as LEED, but it gives direct information about surface structure in real space. The sensitivity is to local bonding geometry, and long-range order is not necessary as in LEED. ESDIAD is especially sensitive to the orientation of hydrogen atoms in surface complexes, which is difficult to observe by any other technique. 

The AFM techniques briefly introduced above provide direct real-space access on structures and phases on all relevant length scales of hierarchical ordering of polymers. The subsequent sections shall not encompass all possible structures and examples of polymer morphology, but rather focus on selected examples that illustrate the rich information extracted from AFM. These examples include classic morphologies, as well as more recently reported polymeric nano- and microstructures. 

The above theoretical framework was developed in real space. However, computing the correlation functions in real space can be carried out only for simple geometries, such as the lamellar phase [31]. In order to apply the theory to more complex structures, efficient methods other than the direct real-space computation have to be developed. One particularly useful method is the reciprocal-space technique, which utilizes the symmetries of the ordered phases. The key observation is that the mean-field solution w (r) = is a periodic... 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

The ability of the STM to achieve atom-resolved real-space images of localized regions of a surface and to directly resolve the local atomic-scale structure has provided essential insight into the active sites on catalysts and emphasized the importance of edges, kinks, atom vacancies, and other defects, which often are difficult to detect with other techniques (46-49). It is evident, however, that STM cannot be used to image real catalysts supported on high-surface-area, porous oxide carriers. 

Calculation of the multiple scattering by means of a real-space cluster approach is considerably more flexible than band-structure methods. Since this technique does not rely on crystal periodicity, it can readily be applied to interpret data for materials of arbitrary atomic arrangements. The sensitivity to higher order correlations has been shown. Fujikawa et al. (94,96) favor short-range-order multiple-scattering XANES theory, in which atoms are not divided into shells but the scattered waves are classified into a direct term and a fully multiple-scattering term. 

The application of the scanning tunneling microscope (STM) and the atomic force microscope (AFM) to electrochemical problems over the past seven years has greatly enlivened electrochemical surface science. The salient feature of these techniques is their ability to obtain real-space images of electrode surfaces with atomic resolution. The facility of obtaining this kind of direct structural information has both confirmed... 

The major role of TOF-SARS and SARIS is as surface structure analysis techniques which are capable of probing the positions of all elements with an accuracy of <0.1 A. They are sensitive to short-range order, i.e. individual interatomic spacings that are <10 A. They provide a direct measure of the interatomic distances in the first and subsurface layers and a measure of surface periodicity in real space. One of its most important applications is the direct determination of hydrogen adsorption sites by recoiling spectrometry [12, H]. Most other surface structure techniques do not detect hydrogen, with the possible exception of He atom scattering and vibrational spectroscopy. 

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