Simple lattice

An unexpected conclusion from this fonuulation, shown in various degrees of generality in 1970-71, is that for systems that lack tlie synunetry of simple lattice models the slope of the diameter of the coexistence curve... 

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

DA Elmds, M Levitt. Exploring conformational space with a simple lattice model for protein structure. J Mol Biol 243 668-682, 1994. 

To present briefly the different possible scenarios for the growth of multilayer films on a homogeneous surface, it is very convenient to use a simple lattice gas model language [168]. Assuming that the surface is a two-dimensional square lattice of sites and that also the entire space above the surface is divided into small elements, forming a cubic lattice such that each of the cells can be occupied by one adsorbate particle at the most, the Hamiltonian of the system can be written as [168,169]... 

A particularly simple lattice model has been utilized by Harris and Rice [129] and subsequently by Stettin et al. [130] to simulate Langmuir mono-layers at the air/water interface chains on a cubic lattice which are confined to a plane at one end. Haas et al. have used the bond-fluctuation model, a more sophisticated chain model which is common in polymer simulations, to study the same system [131]. Amphiphiles are modeled as short chains of monomers which occupy a cube of eight sites on a cubic lattice and are connected by bonds of variable length [132], At high surface coverage, Haas et al. report various lattice artefacts. They conclude that the study... 

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

The explicit form of the Gc topology is analytically accessible for only a very few specific systems, most notably those defined by additive rules and relatively simple lattices. The method of calculation of these topologies will be presented in some detail iii chapter 5. 

Draw a heterogeneous lattice, using circles and squares to indicate atom positions in a simple cubic lattice. Indicate both Schottky and Frenkel defects, plus the simple lattice defects. Hint- use both cation and anion sub-lattices. 

Potzel and Kalvius were the first to investigate zinc compounds with Zn Mossbauer spectroscopy. The first measurements were carried out with Znp2 powder at 4.2 K [80]. The observed quadrupole splitting could not be explained by a simple lattice sum calculation. More detailed measurements were carried out at... 

From these observations, we have noticed the similarity of the simple lattice inclusions to the more sophisticated assemblies of molecules (e.g. cyclodextrins 76 and proteins 78 where the formation of H-bonded loops was first detected and described. Conclusively the motive for the formation of simple inclusion crystals and of more complex associates between high and low molecular weight compounds, such as enzyme-substrate complexes, can be traced back to the same source. 

Because simple lattice models take no account of local directional preferences, they fail to model these important local restraints on protein structure. Instead, they rely almost entirely on long-range interactions to encode the most stable conformation(s) (Dill et al., 1995). Thus the ability of lattice models to reproduce protein-like behavior must be called into question. And though their simplicity makes them intellectually attractive, their use in teaching and modeling protein-like behavior must be qualified with a caveat that local directional preferences have been ignored. 

Abstract In this review, we consider a variety of aspects of polymer crystallization using a very simple lattice model. This model has three ingredients that give it the necessary flexibility to account for many features of polymer crystallization that have been observed experimentally. These ingredients are (1) a difference in attraction between neighboring (nonbonded) components, (2) attraction between parallel bonds, and (3) temperature-dependent flexibility due to the energy cost associated with kinks in the... 

As another example illustrating an explicit switch to normal coordinates, we consider a three-dimensional monoatomic simple lattice. In such a system, masses of all particles are the same and the positions of their stable equilibria are at the lattice sites which are given by radius vectors n (called lattice vectors). Instead of an unsystematic particle numbering (i = 1,2,..., N), it is now convenient to distinguish them by the lattice sites they belong to and to designate them by the index n. The... 

Kieflfer S. W. (1979a). Thermodynamics and lattice vibrations of minerals, 1 Mineral heat capacities and their relationships to simple lattice vibrational models. Rev. Geophys. Space Phys., 17 1-19. 

Fig. 6. (A) The simple lattice myosin filament arrangement that occurs in all the...

Luther et al. (1996) conducted a systematic study of the occurrence of the simple lattice and superlattice across the vertebrate kingdom. Superlattices are present in the muscles of all the higher vertebrates, namely, in mammals (including humans), in amphibians, in birds, in reptiles, and in some muscles of cartilaginous fish. Simple lattices occur in all the teleost (bony fish) muscles so far studied, in some muscles of cartilaginous fish, and also in some primitive fish such as sturgeons and bowfin. 

The A-band lattices in different kinds of striated muscles have distinct arrangements. As shown in Fig. 3 and reproduced in simpler form in Fig. 10A and B, vertebrate striated muscle A-bands have actin filaments at the trigonal points of the hexagonal myosin filament array. As discussed prevously, this array also occurs in two types, the simple lattice and superlattice. The ratio of actin filaments to myosin filaments in each unit cell is 2 1. In both cases the center-to-center distance between adjacent myosin filaments is 70 A, but this varies as a function of overlap, becoming smaller as the sarcomere lengthens, giving an almost constant volume to the sarcomere (April et al, 1971). 

Figure 24A shows three cross-sectional slices through the fish simple lattice M-band reconstruction of Luther and Crowther (1984). Here it... 

Fig. 24. M-band structure from electron microscopy of both simple lattice and superlattice muscles. (A) 3D Reconstruction of fish muscle M-band. Three distinct layers were observed in the reconstruction, at each of the M-bridge levels M4, Ml, and M4 (M and B label the myosin filaments and M-bridges, respectively.) The observed 32-point group symmetry has been imposed on the 3D map. (B) Part of the M-band as modeled by Luther and Squire (1978). Ml and M4 bridges are seen connecting adjacent myosin filaments. Halfway along the M-bridges and running parallel to the myosin filaments are the M-filaments. M3 marks a further level of secondary Y-shaped bridges. (C) A slice...

Fig. 15. Intensity profiles along the equator of the bony fish muscle low angle X-ray diffraction pattern from muscles at rest (A), fully active (B), and in rigor (C). The indexing in (A) is based on the hexagonal A-band lattice, and the arrows indicate peaks that come from the Z-band. (C) to (F) are computed electron density maps based on the amplitudes of the A-band peaks in (A) to (A), respectively. The simple lattice unit cell is outlined in (D). (From Harford and Squire, 1997.)...

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