Extraction of the Topological Information

For the scattering of an isotropic material we already know the result of the separation and a method to obtain it the result is the scattering of the ideal multiphase system as sketched on p. 123 in Fig. 8.9. A way to obtain the result is the classical Porod-law analysis (Sect. 8.3.2). 

The fundamental problem of the classical method is the fact that there is no viable procedure to extend it to the scattering of anisotropic materials. Moreover, the required manual processing is cumbersome, slow and may yield biased results. 

Th Interference Function. The function sketched in Fig. 8.9 can be understood as s Iid (i), the intensity of the ideal multiphase system multiplied by a power 

Equation (8.59) defines the ID interference function of a layer stack material. Gi (s) is one-dimensional, because p has been chosen in such a way that it extinguishes the decay of the Porod law. Its application is restricted to a layer system, because misorientation has been extinguished by Lorentz correction. If the intensity were isotropic but the scattering entities were no layer stacks, one would first project the isotropic intensity on a line and then proceed with a Porod analysis based on p = 2. For the computation of multidimensional anisotropic interference functions one would choose p = 2 in any case, and misorientation would be kept in the state as it is found. If one did not intend to keep the state of misorientation, one would first desmear the anisotropic scattering data from the orientation distribution of the scattering entities (Sect. 9.7). 

The addressed types of interference functions are the starting point for the evaluations described in Sects. 8.53-8.5.5. 

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The quantitative analysis of spirals yields, on the one hand, a detailed picture of the concentration distribution inside and outside the spiral core. This information can be used to clarify details in the mechanistic steps of the reaction coupled with diffusion. On the other hand, a large number of topological scenarios in the core region have been explored by now that are based on algorithms for the extraction of the essential dynamic features. These are commonly represented by the trajectory of the spiral tip. Unfortunately, in the experimenal evaluation the precise location of the tip is still a rather ill defined quantity and each experimentalist has his own procedure to trace the tip trajectories. In numerical simulation a clearer definition can be provided in terms of the concentration levels of the two variables. 

However, the key to our method consists of computing the colored or reactive versions of and for the distance matrix D, as required by the above computational scheme, see Table 3.8. These operators, also-called topo-electronegativity and topo-chemical hardness matrices, properly carry the chemical information leading to the reactive forms of the topological index for electronegativity T(x) and chemical hardness T(tj). The extraction of both the T(x) and T(tj) reactive indices from the newly defined operators and f may follow several mathematical routes. One is described in what follows (Putz et al., 2013b). 

Automated Extraction of Interference Functions. For the classical synthetic polymer materials it is, in general, possible to strip the interference function from the scattering data by an algorithm that does not require user intervention. Quantitative information on the non-topological parameters is lost (Stribeck [26,153]). The method is particularly useful if extensive data sets from time-resolved experiments of nanostructure evolution must be processed. Background ideas and references are presented in the sequel. 

Linear representations are by far the most frequently used descriptor type. Apart from the already mentioned structural keys and hashed fingerprints, other types of information are stored. For example, the topological distance between pharmacophoric points can be stored [179, 180], auto- and cross-correlation vectors over 2-D or 3-D information can be created [185, 186], or so-called BCUT [187] values can be extracted from an eigenvalue analysis of the molecular adjacency matrix. 

So far the emphasis has been on solving the crystal structure from the knowledge of the unit cell and ionic content. The motivation for this work is to provide an automated procedure to help determine or solve the crystal structure of new compounds that are synthesised in a powder form. Of course the methods developed can generate other structural topologies and perhaps a new, yet to be synthesised, crystal structure. However, the task has been to solve a particular structure and so one might extract more information from the experimental data to aid the prediction process (e.g. use of symmetry elements). Thus, the number of unwanted possible (meta)stable structures, or polymorphs, that could be generated may be reduced. In this section, the emphasis is on finding all the important polymorphs for a particular chemical formula. 

Topological information about an arbitrary spin system can be extracted based on a Taylor series expansion of experimental coherence-transfer functions (Chung et al., 1995 Kontaxis and Keeler, 1995) [see Eq. (190)]. Undamped magnetization-transfer functions between two spins i and j are an even-order power series in t, . The first nonvanishing term is of order 2rt if the spins i and j are separated by n intervening couplings (Chung et al., 1995). 

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