Structural order metrics

The first crystal-independent structural order metric that we will explore is the translational parameter l w an integral measure of the amplitude of the material s total correlation function h(r),... 

Figure 4 The path traversed in structural order-metric space as liquid water (SPC/E) is compressed isothermally at two different temperatures. Filled diamonds represent T = 260 K, and open triangles represent T = 400 K. The arrows indicate the direction of increasing density. A and C are states of maximum tetrahedral order at the respective temperatures, whereas B is a state of minimum translational order. Reprinted with permission from Ref. 29. 

If one has a priori knowledge of the types of structural order relevant to a system of interest, one can then generally construct metrics that are capable of detecting and quantifying that order. Such metrics are often designed to report the deviation of a structure from a reference arrangement of particles. This information can be especially useful for studying the behavior of supercooled liquids and related systems that exhibit transient structural precursors of the stable crystalline phase.34 The structural order... 

Continuing with the mini-theme of computational materials chemistry is Chapter 3 by Professor Thomas M. Truskett and coworkers. As in the previous chapters, the authors quickly frame the problem in terms of mapping atomic (chemical) to macroscopic (physical) properties. The authors then focus our attention on condensed media phenomena, specifically those in glasses and liquids. In this chapter, three properties receive attention—structural order, free volume, and entropy. Order, whether it is in a man-made material or found in nature, may be considered by many as something that is easy to spot, but difficult to quantify yet quantifying order is indeed what Professor Truskett and his coauthors describe. Different types of order are presented, as are various metrics used for their quantification, all the while maintaining theoretical rigor but not at the expense of readability. The authors follow this section of their... 

To describe quantitatively the disorder present in a material, it is often convenient to introduce a structural order parameter. This term refers to a metric that can detect the development of order in a many-body system, perhaps by employing the tools of pattern recognition (Brostow et al., 1998). In many cases, such a measure is constructed to serve as a reaction coordinate for a thermodynamic phase transition (van Duijneveldt and Frenkel, 1992). However, since the form of the order parameter clearly depends on the phenomenon of interest, the development of such measures can be a difficult and subtle matter. 

In contrast to the bond-orientational order parameters mentioned above, scalar measures for translational order [that is, of the tendency of particles (atoms, molecules) to adopt preferential pair distances in space] have not been well studied. However, a number of simple metrics have been introduced recently (Truskett et al., 2000 Torquato et al., 2000, Errington and Debenedetti, 2001) to capture the degree of spatial ordering in a many-body system. In particular, the structural order parameter t. 

One aim of studying olfaction has been to gain an understanding of how the physical features of odor molecules correlate with their perceptual qualities. Unfortunately, the perceptual qualities do not often follow an easily ordered metric that can be simply related to molecular structures. We might, therefore, speak of a dual problem in attempting to relate odor perception to the features of the corresponding molecules no obvious metric is available to describe either the space of odor perceptions or the space of odor chemistry. 

Since the starting structure and the initial atom pair was casually selected, distance matrix generation and random metrization should be performed several times in order to get an ensemble of metric matrices. 

Another noteworthy example is x-ray absorption fine structure (EXAFS). EXAFS data contain information on such parameters as coordination number, bond distances, and mean-square displacements for atoms that comprise the first few coordination spheres surrounding an absorbing element of interest. This information is extracted from the EXAFS oscillations, previously isolated from the background and atomic portion of the absorption, using nonlinear least-square fit procedures. It is important in such analyses to compare metrical parameters obtained from experiments on model or reference compounds to those for samples of unknown structure, in order to avoid ambiguity in the interpretation of results and to establish error limits. 

It is noteworthy that the value of g is different from zero and relatively large. This result suggests that the electric and magnetic contributions to the nonlinearity are essentially of the same order of magnitude. This large value of the magnetic contributions is probably due to the near centrosym-metric arrangement of the monomer units in the helical polymer structure... 

Third, the metric tensor is determined by the variables 4>, //, A. On the other hand and v never appear in Eqs.(6)-(9) (reflecting the fact that x° and x5 constant dilatations are always possible without harming the commutator relations for the Killing motions), so these equations are of first order on 4>, / and A. However, the equations can be rearranged resulting in the following symbolic structure ... 

One relevant concern has been to prioritize the order of screening, or to decide which compound libraries to purchase for screening. One approach that has been used relies on the complementary concepts of diversity and similarity. Given two compounds, how do you quantitate how divergent the two structures are. One major problem is the choice of a relevant metric, what parameters are considered, how are the parameters scaled, and so on. Similarity, like beauty, is clearly in the eye of the beholder. There is no generally relevant set of parameters to explain all observations and one should expect that a given subset of parameters will be more relevant to one problem than to another. 

A QMS should be comprised of all the processes supporting that business and include an effective management review of those process metrics. Management needs to be aware of and understand process performance through structured metrics review programs in order to take appropriate action, providing resources and capital to improve the QMS. This hierarchy is illustrated in Figure 2. 

Since the G-action on // 1(C) is free, the slice theorem implies that the quotient space gTl((,)/G has a structure of a C°°-manifold such that the tangent space TaAtt-HO/G) at the orbit G x is isomorphic to the orthogonal complement of Vx in Txg 1( ). Hence the tangent space is the orthogonal complement of Vx IVX JVX KVX in TxX, which is invariant under I, J and K. Thus we have the induced almost hyper-complex structure. The restriction of the Riemannian metric g induces a Riemannian metric on the quotient g 1(()/G. In order to show that these define a hyper-Kahler structure, it is enough to check that the associated Kahler forms u>[, u) 2 and co z are closed by Lemma 3.32. 

Although the linearity of the chain-rule differential expressions (10.5) confers primitive affine-type spatial structure on thermodynamic variables, it does not yet provide a sense of distance or metric on the space (other than what might be displayed in an arbitrarily chosen axis system). In order to bring intrinsic geometrical structure to the thermodynamic space, we need to define the scalar product (R RJ) [(9.29)] that dictates the spatial metric on Ms- The metric on Ms should reflect intrinsic physical properties of the thermodynamic responses, not merely generic chain rule-type mathematical properties of their differential representation. At the same time, we must exhibit how the space Ms is explicitly connected to the physical measurements of thermodynamic responses. Because such measurements assign scalar values to physical properties, it is natural to associate each scalar product of Ms with the scalar value of an experimental measurement. How can this be done ... 

Distinguishing Space Groups by Systematic Absences. From the symmetry and metric properties of an X-ray diffraction pattern we can determine which of the 6 crystal systems and, further, which of the 11 Laue symmetries we are dealing with. Since we need to know the specific space group in order to solve and refine a crystal structure, we would still be in a highly unsatisfactory situation were it not for the fact that the X-ray data can tell us still more. 

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