Local structuring

The three-dimensional synnnetry that is present in the bulk of a crystalline solid is abruptly lost at the surface. In order to minimize the surface energy, the themiodynamically stable surface atomic structures of many materials differ considerably from the structure of the bulk. These materials are still crystalline at the surface, in that one can define a two-dimensional surface unit cell parallel to the surface, but the atomic positions in the unit cell differ from those of the bulk structure. Such a change in the local structure at the surface is called a reconstruction. 

Figure A2.4.3. The localized structure of a hydrated metal cation in aqueous solution (the metal ion being assumed to have a primary hydration number of six). From [5].

For bulk structural detemiination (see chapter B 1.9). the main teclmique used has been x-ray diffraction (XRD). Several other teclmiques are also available for more specialized applications, including electron diffraction (ED) for thin film structures and gas-phase molecules neutron diffraction (ND) and nuclear magnetic resonance (NMR) for magnetic studies (see chapter B1.12 and chapter B1.13) x-ray absorption fine structure (XAFS) for local structures in small or unstable samples and other spectroscopies to examine local structures in molecules. Electron microscopy also plays an important role, primarily tlirough unaging (see chapter B1.17). 

A multitude of different variants of this model has been investigated using Monte Carlo simulations (see, for example [M])- The studies aim at correlating the phase behaviour with the molecular architecture and revealing the local structure of the aggregates. This type of model has also proven useful for studying rather complex structures (e.g., vesicles or pores in bilayers). 

M J1990. Calculation of Conformational Ensembles from Potentials of Mean Force. An Approach o the Knowledge-Based Prediction of Local Structures in Globular Proteins. Journal of Molecular Siology 213 859-883. 

MI Sippl. Calculation of conformational ensembles from potentials of mean force. An approach to the knowledge-based prediction of local structures m globular proteins. I Mol Biol 213 859-883, 1990. 

V De Filippis, C Sander, G Vriend. Pi edictmg local structural changes that result from point mutations. Protein Eng 7 1203-1208, 1994. 

C Bystroff, D Baker. Prediction of local structure m proteins using a library of sequence-structure motifs. J Mol Biol 281 565-577, 1998. 

In complexes with Cro, the overall bend and twist of the DNA are similar to those in the repressor complexes, but there is a significant difference in the local structure of two of the nucleotides in each half-site. Binding of 434 Cro or repressor fragment thus imposes a distinct local structure (Figure 8.13), as a result of differences in both the identity and conformations of various amino acid residues that interact with the DNA. The DNA conformational details are significant for the relative affinities of Cro and repressor for various sites, as we describe in a later section. 

EXAFS is a nondestructive, element-specific spectroscopic technique with application to all elements from lithium to uranium. It is employed as a direct probe of the atomic environment of an X-ray absorbing element and provides chemical bonding information. Although EXAFS is primarily used to determine the local structure of bulk solids (e.g., crystalline and amorphous materials), solid surfaces, and interfaces, its use is not limited to the solid state. As a structural tool, EXAFS complements the familiar X-ray diffraction technique, which is applicable only to crystalline solids. EXAFS provides an atomic-scale perspective about the X-ray absorbing element in terms of the numbers, types, and interatomic distances of neighboring atoms. 

One of the great advantages of Modulation Spectroscopy is its ability to fit the line shapes of sharp, localized structures, as illustrated in the lower part of Figure 1. These fits yield important relevant parameters, such as the value of the energy gap and the broadening parameter. 

M. Schoen. Rheology and local structure of thin films confined between thermally corrugated walls. Physica A 240 328-339, 1997. 

The order parameter field (r) characterizes the local structure of the phases we have investigated. The most interesting is the topology of the phases,... 

M. J. Vlot, S. Claassen, H. E. Huitema, J. P. v. d. Eerden. Monte Carlo simulation of racemic liquid mixtures thermodynamic properties and local structures. Mol Phys 97 19, 1997 M. J. Vlot, J. C. v. Miltenburg, H. A. Oonk, J. P. V. d. Eerden. Phase diagrams of scalemic mixtures. J Chem Phys 707 10102, 1997. 

In addition to unsaturated fatty acids, several other modified fatty acids are found in nature. Microorganisms, for example, often contain branched-chain fatty acids, such as tuberculostearic acid (Figure 8.2). When these fatty acids are incorporated in membranes, the methyl group constitutes a local structural perturbation in a manner similar to the double bonds in unsaturated fatty acids (see Chapter 9). Some bacteria also synthesize fatty acids containing cyclic structures such as cyclopropane, cyclopropene, and even cyclopentane rings. 

As well as charge-ordering in the system, out to two chloride shells, the specibc local structure shows strong interactions between the chloride and the ring hydrogens, as well as some interaction between the methyl groups of adjacent imidazoli-um cations. This is consistent with the crystal structure and implies that the molecular paclbng and interactions in the brst two or three coordination shells are similar in both the crystal and the liquid. 

Since the fine structure observed is only associated with the particular absorption edge being studied, and the energy of the absorption edge is dependent on the element and its oxidation state, EXAFS examines the local structure around one particular element, and in some cases, an element in a given oxidation state. A fuller picture can therefore be obtained by studying more than one absorbing element in the sample. 

The Bragg peaks indicated an ordered local structure within the sample film, and the interlayer spacings were reproduced compared with the bulk samples, with only... 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

Structurally Dyuamic CA the only generalizations mentioned so far were generalizations of either the rules or state space. Another intriguing possibility is to allow for the lattice C itself to become a full participant in the dynamical evolution of the system, much as the classically static physical space-time arena becomes a bona-fide dynamic element in general relativity. The idea is to study the behavior of systems evolving according to both value and local structure rules ... 

There are a number of additional local structures and properties that appear even in elementary CA systems. Grassberger [grass84b, for example, has observed that rule R22 actually harbors very complex long-range effects, similar to a critical phenomenon (see section 3.1.4). Since the majority of these findings require the use of more general and sophisticated correlation measures than we have defined thus far, we will pick up our discussion of them in chapter 4. 

An important hierarchy of parameterization schemes based on successive refinements to a mean-field theoretic description, called local structure theory, has been developed by Gutowitz, et. al. ([guto87a], [guto87b], and [guto88]), and is discussed in detail in chapter five. Below we summarize results of what is essentially a zeroth order local structure theory developed by Langton [lang90]. 

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