Infinite Arrays

J. G. Powles, M. J. D. Mallett, G. Rickayzen, W. A. B. Evans. Exact analytical solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc Royal Soc London 457 391, 1992. 

This box is embedded in an infinite array of boxes, all with the same geometrical arrangement of particles. 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

When reacting Ba dissolved in liquid Na with nitrogen at temperatures not too far from the melting point of the solvent, there was evidence for the formation of Ba-rich nitride clusters prior to the deposition of solid Ba2N [8]. Using both Na as well as a K-Na alloy as solvents, an extensive class of new subnitrides can be prepared. They contain [M6N] octahedra, either discrete or condensed into finite clusters and infinite arrays, respectively. 

The symmetry operations, G, of the space group acting on an atom placed at an arbitrary point in space will generate a set of mo equivalent atoms in the unit cell. Operation of the lattice translations, R, acting on this set generates an infinite array of such atoms, with the finite set of ma atoms being repeated at each point on the lattice. This is illustrated in Fig. 10.1 in which nia = 4 and each of the rectangles defined by the horizontal and vertical lines represents a unit cell that is identical with the one outlined with heavy lines. 

Like its sodium and potassium analogues, RbN(SiMe3)2 when reacted with ferrocene forms an infinite array of formula [( RbN(SiMe3)2 2 Fe(ri5-C5H5)2 )oo]. Its structure resembles those of its sodium and potassium counterparts in that the RbN(SiMe3)2 2... 

Figure 18.1 Random walk of an object through an infinite array of discrete boxes numbered by m = 0, 1, + 2,.... At time t = 0 the object is located in box m = 0 (probability 1) and then moves with equal probability to the two adjacent boxes m = 1 (probabilities 1/2). The time steps are numbered by n. The resulting occupation probabilities, p(n,m), of being in box m after time step n are the Bernoulli coefficients (Eq. 18-1). Curve A shows a typical individual path. Curve B represents the unlikely case in which the object jumps six times in the same direction.

All of the other chapters in this book deal with the symmetries of finite (discrete) objects. We now turn to the symmetry properties of infinite arrays. The end use for the concepts to be developed here is in understanding the rules governing the structures of crystalline solids. While an individual crystal is obviously not infinite, the atoms, ions, or molecules within it arrange themselves as though they were part of an infinite array. Only at, or very close to, the surface is this not the case this surface effect does not, in practice, diminish the utility of the theory to be developed. 

The. vectors a and b can be considered as two edges of a parallogram called a unit cell. The lattice can also be thought of as an infinite array of unit cells, all neatly fitted together so as to leave no gaps. 

To summarize, a lattice is an infinite array of identical points (i.e., each one has exactly the same environment of other points) and the points are obtained one from another by translations only (no rotations or reflections being required). 

Before examining these electronic effects, wc should delve a little more deeply into the theory and terminology of solid slate chemistry lhan was done on pages 269-272. Specifically, if there is an infinite array of identical orbitals (say H l.v) represented by < . related by translational symmetry and spaced at... 

These solutions are appropriate to an infinite array of hexagonal passages. For certain finite arrangements solutions can be found by the principle of reflection. 

In any work setting there is the opportunity for unforeseen circumstances to occur. Weather, mechanical failures, emergency situations, etc. can all produce conditions that can create physiological disruption. This might come from a call to report unexpectedly or early to work, extending hours on the job, a road traffic accident that extends the 1-hr commute to 2 hr, or any one of an almost infinite array of possibilities. 

A similar mixed ox a- thia macrocycle incorporating a rigid horseshoeshaped aromatic moiety, l,ll,21-trioxa-8,14-dithia[2,9,2]paracyclophane, L40, has been synthesized and reacted with copper iodide in MeCN solution (Scheme 19).160 Its polymeric copper iodide complex [Cu4I4(L40)2] 40 crystallizes, in a 2 1 metal to ligand ratio, as an ID infinite array of cubane-like units consisting of four copper atoms, four i3-iodine atoms, and four sulfur atoms, stemming from four different macrocycles. The Cu—Cu distances are about 2.731 A. Unfortunately, the photophysics of this compound have not been studied. 

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