Patterson radial

Figure 5-5V. Turbine with three, four, or six radial blades. Handles wide range of applications. Courtesy of International Process Equipment Co., Div. of Patterson Foundry and Machine Co.

Figure 9.6 Radial Patterson function. (a jLinear triatomic molecule. (b) Patterson function constructed from (a). For construction procedure, see Chapter 6, Section IILC and Figure 6.10. (c) Patterson function (b) averaged by rotation, the expected result of calculating a Patterson function from diffraction intensities measured from an amorphous sample.

Another comparison of experimental and theoretical data is shown in Fig. 19.1 IB and C. Here, we compute the radial Patterson inversions, U(K), of the experimental data (Fig. 19.1 IB) for comparison with the simulations (Fig. 19.11C). 

The radial Patterson inversion of the anomalous difference signal is calculated using... 

Figure 19.11 Comparison of anomalous signals obtained for Rb-RNA and Rb-DNA. Panel (A) shows the ASAXS signals for both DNA and RNA displayed on a log scale. Panels (B) and (C) illustrate an alternative method for comparing measured with predicted ion distributions. A radial Patterson inversion can be applied both to the data and to the curve generated from simulation. Although it is challenging to interpret this curve directly (see text), it is instructive to compare experiment with prediction in this form. When an ion probe radius of 3 A is used, the differences in ion distribution between RNA and DNA are clearly mirrored by simulation.

The Patterson function is a map that indicates all the possible relationships (vectors) between atoms in a crystal structure. It was introduced by A. Lindo Patterson " in 1934, inspired by earlier work on radial distribution functions in liquids and powders. In crystals the directionality as well as the lengths of vectors between atoms (atomic distances) can be deduced. By contrast, in liquids and powders the geometric information that can be obtained is limited to interatomic distances, because in these the molecules are randomly oriented. While the use of the Patterson function revolutionized the determination of crystal structures of small molecules in the 1930s to 1950s, direct methods are now the most widely used methods for obtaining structures of small organic molecules. The Patterson function, however, continues to play an essential part in the determination of crystal structures of inorganic compounds and macromolecules. It is also very useful when the structure of a small molecule proves difficult to solve by direct methods. 

In the case of well-ordered crystals, It Is possible to deduce their atomic structures by appropriate manipulation of diffraction Intensities. In the case of x-ray scattering by liquids, direct use of measured intensities yields, at best, very limited structural Information (radial distribution functions). For ordered liquids, however, it is possible to posit structural models and to calculate what their scattering Intensities would be so that it is more productive to conduct the comparisons in diffraction space. To this end, it is possible to devise a point model to represent the spatial repetition of the constituent units in the ordered array and to compare its scattering maxima to the observed ones (6,9). More sophisticated analyses (10-12) make use of the complete electron densities (or projections onto the chain axis z), usually by calculating their Patterson functions P(z) since the scattering intensity function is its Fourier transform. 

The autocorrelation function is sometimes referred to simply as the correlation function. Among those working in crystal structure analysis, the autocorrelation function is known as the Patterson function. Many of the distribution functions obtained from scattering intensity data are in the nature of the correlation function, with possible differences in the normalization constant or a constant term. Functions in this vein include the pair correlation function or the radial distribution function (and its uniaxial variant cylindrical distribution function), discussed in Chapter 4. 

Figure 15. Diffuse reflectance distributions used to measure tissue absorption and scattering properties non-invasively. (a,b) principle of the technique, showing light entering a point on the tissue surface and the measured radial distribution of the diffusely reflected (backscattered) light that depends on the tissue absorption and scattering properties, (c) external surface probe (courtesy Dr M. Patterson, Hamilton, Canada), (d) endoscopic probe (courtesy Dr R. Bays and colleagues, Lausanne, Switzerland) in this case the distribution is measured along the probe from light input at the end, with the probe placed flat on the tissue (e.g. esophagus) surface.

A more common impeller classification is by flow leaving the impeller zone. Impellers can be classified into radial or axial flow impellers. Some examples of radial flow impellers are the Narcissus impeller (NS), concave blade disc turbine (Figure 6.11), (Chemineer) BT-6, and the multibladed disc turbine. A six-bladed disc turbine, shown in Figure 6.12, is often referred to as a Rush ton-type turbine (RT) (Ulbrecht and Patterson, 1985). The standard blade is DJ long and D,/5... 

Axial flow impellers used in low bottom clearance tanks can also create radial flow if the direction of the flow is downward. In this case, the flow can leave the impeller zone only by flowing in the radial direction (Tatterson, 1991). This phenomenon is usually not observed since the standard bottom clearance for low viscosity impellers in gas-liquid dispersions is between one impeller diameter and one half the tank diameter (Ulbrecht and Patterson, 1985). The hydrofoil impellea-... 

Comparison of the radial distribution function calculated tor the 5.1-residue helical configuration, with inclusion of a jS carbon atom per residue, and the experimental radial distribution function for carbonmonoxyhemo-globin, as calculated from the three-dimensional Patterson function given by Perutz. 

P(x) is the radial distribution function (RDF) as introduced in Section I.I.2.I.4. It is the product of electronic densities attached to two points of the real space separated by vector x, with x being the distance between a pair of atoms in a set of atoms arranged disorderly. P(x) is thus a probability it is a generalization of the Patterson function of a crystal [10]. It corresponds to the Fourier transform of I(s). Reciprocally to Equation (1.8),... 

Two impeller types were simulated the standard six-blade disk turbine and a 45° pitched blade turbine. The outflow from the disk turbine was simulated by fixing the tangential velocities at the blade tip locus (the FIX option in Fluent). The radial velocities and k/e ratios generated were close to the values that have been measured by Wu and Patterson (1989). The outflow velocities and turbulence energy from the pitched blade turbine were fixed at the bottom locus of the impeller blades using the data of Fort et al. (1999). The resulting flow patterns were close to the data measured. 

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