Reciprocal space techniques

Macroscopic experiments, both reciprocal-space techniques and direct-space techniques, conclude that the (110) surfaces of Pt, Ir and Au reconstruct to a (1 x 2) structure.30 Several atomic models have been proposed,31 of which the simple missing row model agrees better with various experimental data. In this model every other [110] atomic row is missing from the reconstructed (1 X 1) surface. To transform a (1 x 1)... 

In infinite periodic systems, an attractive alternative to the use of a cut-off distance is the Ewald sum technique, first described for chemical systems by York, Darden and Pedersen (1993). By using a reciprocal-space technique to evaluate long-range contributions, the total electrostatic interaction can be calculated to a pre-selected level of accuracy (i.e., the Ewald sum limit is exact) witli a scaling that, in tlie most favorable case (called Particle-mesh Ewald , or PME), is AlogA. Prior to the introduction of Ewald sums, the modeling of polyelectrolytes (e.g., DNA) was rarely successful because of the instabilities introduced... 

In infinite periodic systems, an attractive alternative to the use of a cut-off distance is the Ewald sum technique, first described for chemical systems by York, Darden and Pedersen (1993). By using a reciprocal-space technique to evaluate long-range contributions, the total... 

Once the content of the unit cell has been established, a model of the crystal structure should be created using either direct or reciprocal space techniques, or a combination of both. Direct space approaches do not mandate immediate use of the observed integrated intensities, while reciprocal space methods are based on them. 

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... 

Based on the method of analysis employed, one can distinguish between real or reciprocal-space techniques. The first class includes optical microscopy, transmission electron microscopy (TEM), scanning electron microscopy (SEM) and atomic force microscopy (AFM). Techniques such as light scattering and neutron and X-ray scattering are part of the latter category. All these experimental methods are discussed in the following sections. 

We surreptitiously introduced reciprocal space in Figure 4.1, in an attempt to show its usefulness before giving the formal definitions. It is so very helpful in the interpretation of many diffraction experiments that we need to understand it more fully. We shall use it extensively in the discnssion of triple-axis experiments, in which reciprocal space mapping is an essential technique. 

In this chapter we show how restricting the angular acceptance of the detector adds another dimension to the information available from high resolution diffraction techniques, enabling strains and tilts in a sample to be identified separately. We explain the reciprocal space representation of triple-axis diffraction maps, and give several examples of its application to materials characterisation. 

Unfortunately, the small-angle scattering techniques used in the investigations of Nafion morphology generally probe but a small region of reciprocal space and Fourier inversion methods of analyzing the... 

Since Ap is the Fourier transform of AF, Eq. (5.12) implies that minimization of J (Fobs - Pcaic )2 dr and of J (Fobs - Fcalc)2 dS are equivalent. Thus, the structure factor least-squares method also minimizes the features in the residual density. Since the least-squares method minimizes the sum of the squares of the discrepancies in reciprocal space, it also minimizes the features in the difference density. The flatness of residual maps, which in the past was erroneously interpreted as the insensitivity of X-ray scattering to bonding effects, is an intrinsic result of the least-squares technique. If an inadequate model is used, the resulting parameters will be biased such as to produce a flat Ap(r). 

EM provides local structural information about the samples in both real and reciprocal space, for example local structural information about the surface and the bulk of the sample at the atomic level, together with chemical, electronic and three-dimensional structural information are now routinely available. Some of these methods are described in this chapter. Electron-sample interactions and scattering are fundamental to EM. EM is a diffraction technique in which crystals diffract electrons in accordance with Bragg s law, nX = Idhki sin 9, where X is the... 

For each crystal, reflections from two intensity-equivalent regions of reciprocal space (hki, hSkS t, and khi, k h S ) were examined using the 0-26 scan technique. (For Ba]Naio-A, the two regions examined were hkl and hik.) Each reflection was... 

Film diffraction data can be obtained by several techniques flat plate camera, cylindrical camera or precession camera. We chose the latter, because it records a relatively undistorted "slice" of diffraction transform (reciprocal space) (19,20). We felt that this characteristic of the precession technique outweighed the disadvantages of longer exposure times (this was lessened by use of a Helium atmosphere in the camera enclosure). For potassium bromide amylose, the fiber studied, this required exposure of a 79y x 79y x 300y fiber specimen for 45 hours (35kv,... 

Since crystal and film are fixed, only a small part of reciprocal space can be observed at a time, but a reciprocal point of the diffracting volume scatters during the whole exposure time. By contrast, scanning techniques such as Weissenberg or precession cameras see a larger part of reciprocal space, but each reciprocal point stays only a short time on the Ewald sphere. 

Crystal structures are determined by X-ray diffraction—you know, Bragg s Law and all that reciprocal space stuff. If you ve yet to study this technique, then all you really need to know for now is that a monochromatic (i.e., one wavelength or frequency) X-ray beam, when passed through a perfect single crystal, will give a pattern of spots which (in the old days) were recorded on a photographic plate (Figure 8-40). From this pattern of spots one can, with some effort, reconstruct the crystal structure. 

Diffraction patterns from amorphous materials consist of a few vague halos. Accurate measurement of the intensity distributions of these halos in reciprocal space provides information about the distribution of atoms in real space. Reviews of the experimental methods and data handling techniques for X-ray-, electron-, and neutron-scattering is given in the literature The mathematical tool which is used to relate the... 

---