Lattices 254 INDEX

The identity of several rows and columns in the matrix in Eq. (4.3) shows that the same information is contained in a more compact matrix. This matrix can be written with the four distinguishable states on the 2nnd lattice indexing the rows and columns, in the order in which they are listed in Table 4.1 [153] ... 

In crystallographic notation it is customary to place a bar over lattice index numbers to indicate negative direction relative to the origin of the axes.)... 

By interchanging the summation over the reciprocal lattice (index j) and the summation over the angular momentum in Eqs. (8) and (10), one can write ... 

Indices of diffusion anisotropy - fractional anisotropy (EA) and lattice index (LI), calculate the degree of differences in diffusion in different directions... 

In crystals, it is necessary to denote the plane directions and is done conventionally either by Miller s indices or by lattice planes. Directions are given as the lowest vector in referring to the coordinate system, x a), y b) and z c). A vector parallel to the chain axis is denoted [001]. The first plane intersects the origin of the coordinate system. The next plane intersects the three axes at x = ajh, y = b k and z = cll. The task is to find an integral combination of h, k and / that is finally presented in parentheses (hkl). All planes containing the chain axis, i.e. those parallel to the chain, have the general formula (/z, k, 0). The lattice index system indicates not only the orientation of the planes but also the shortest distance between planes. The set of planes denoted (010) is a subset of (020). The orientation of the two sets of planes is the same but the interplane distances d k) are different /qio = b and J020 = bjl. 

Here, the subscript h indexes different helices in the unit cell. In this convention, the fractional coordinates for each unique helix in the unit cell are specified with the helix axis directed along the z-coordinate and passing through the origin. Each helix may have a distinct setting angle, o)/, and h and k may take fractional values to indicate helices at nonprimitive lattice positions. The lattice index / is replaced by the helix repeat index, n. 

The overlap integrals, in turn, fall off like exponentials of the distance from the center of the participating atomic functions, the actual rate of decay being fixed by the exponent and nature (s,p,d,...) of the functions. The superscripts j,h,l on these integrals measure the interdistance (in integer numbers of the ID lattice parameter a) between the atomic functions which are reasonably localized in the direct space. Thus, one can anticipate a fast decay of these integrals and a small interaction range for cells. This is indeed the case except when the same lattice index.occurs on.both terms of a product of atomic functions, e.g. 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Figure Bl.21.2. Atomic hard-ball models of stepped and kinked high-Miller-index bulk-temiinated surfaces of simple metals with fee lattices, compared with anfcc(l 11) surface fcc(755) is stepped, while fee...

Figure C2.5.10. The figure gives tire foldability index ct of 27-mer lattice chains witli sets containing different number of amino acids. The sets are generated according to scheme described in [27], The set of 20 amino acids is taken as a standard sample. Each sequence witli 20 amino acids is optimized to fulfil tire stability gap [5]. The residues in tire standard samples are substituted witli four different sets containing a smaller number of amino acids [27]. The foldability of tliese substitutions is indicated by tire full circles. The open diamonds correspond to tire sequences witli same composition. However, tire amino acids are chosen from tire reduced representation and tire resultant sequence is optimized using tire stability gap [5].

We assign an index number to each of the polymer molecules and pick up the analysis of the problem after i polymer molecules have already been placed on an otherwise empty lattice. Our first question, then, concerns the number of ways the (i + l)th polymer molecule can be placed in the lattice. The polymer is to be positioned one repeat unit at a time, so it is an easy matter to count the number of available positions for the first segment of the (i + l)th molecule. Since the total lattice consists of N sites and ni of these are already occupied, the first segment of the (i + l)th molecule can be placed on any one of the N - ni remaining sites. 

Indexings and Lattice Parameter Determination. From a powder pattern of a single component it is possible to determine the indices of many reflections. From this information and the 20-values for the reflections, it is possible to determine the unit cell parameters. As with single crystals this information can then be used to identify the material by searching the NIST Crystal Data File (see "SmaU Molecule Single Stmcture Determination" above). 

Another desirable property for a ceramic color is a high refractive index. For example, valuable pigments are based on spinels [1302-67-6] ( 2jj = 1.8) and on zircon ( 2j = 1.9), but no valuable pigments are based on apatite ( 2j = 1.6), even though the lattice of apatite is as versatile for making ionic substitutions as that of spinel. 

The crystal group or Bravais lattice of an unknown crystalline material can also be obtained using SAD. This is achieved easily with polycrystalline specimens, employing the same powder pattern indexing procedures as are used in X-ray diffraction. ... 

JCPDS-ICDD Elemental and Lattice Spacing Index ilDDO). This index is available from JCPDS-International Centre for Diffraction Data, 1601 Park Lane Swarthmore, PA 19081. 

For an analytical treatment of Eq. (18) we make a mean-field approximation in layers, where the index i is now decomposed into the layer index k and lattice position j within the layer Si s /.. The mean-field approximation in the layer leads to the layer order parameter = (T. Its evolution is obtained from (18) as... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

Consider a size N one-dimensional lattice with site variables Si t) = 1, i = On even (or odd) time steps, either even-indexed (or odd-indexed) sites evolve according to fixed probabilistic peripheral rules that is, according to rules that depend only on the values of a given site s neighboring sites. Such rules are completely specified by a set of four conditional probabilities, 0 < Ui < 1, i = T...4 ... 

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