Other One-Dimensional Systems

Our discussion so far has focused on polyacetylene, the hydrogen chain and related examples. The broad results, however, are transferable to many other systems. For a band describing a chain of orbitals the e(k) versus k diagram will look a little different. The phase factor at k = 0 requires (equation 13.6) all the atomic coefficients equal to +l (13.34). This is the point where maximum 

At k = 7tla the phase difference between adjacent orbitals is —I and maximum bonding results. Apart from this rather simple difference the band structure is identical to our earlier example. The bands run down from left to right in the e(k) versus k diagram. Now, of course, p represents p — pa rather than p — p interactions. Consequently, the band spread in an absolute sense is larger since Po- overlap is larger than p overlap, that is, PA P. - 

Plots of e(k) versus k, DOS and S-S COOP for a one-dimensional, linear chain of sulfur atoms. An expansion of the DOS around the s region is given in the box at the top. 

Many other one-dimensional examples exist [17], with perhaps very different chemical compositions, but which are understandable in an exactly analogous v/ay [18]. One particularly important series are the organic metals made by stacking planar molecules on top of one another [19]. Tetrathiofulvalene (TTF) shown in 13.39a is one example. The tetramethylated derivative (TMTTF) and its selenium 

A little more complicated compound but isoelectronic to SN, is BaMgo.iLio.9Si2, prepared by Wengert and Nesper [20]. Its structure is shown in 13.40. Here, the black spheres are Si and the grey ones correspond to Ba (large) and Li along with Mg (small). 

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For chemists, it is probably not a big surprise that a one-dimensional chain of hydrogen atoms does not exist and that it will immediately decompose into isolated H2 molecules. The Peierls distortion has important consequences for one-dimensional systems, such as polyacetylene with C-C bond-length alterations (instead of equal C-C distances) [74], infinite molecules with platinum-platinum bonding such as Krogmann s salts K2[Pt(CN)4]Xo,3 3H2O with X = Cl or Br [75], or other one-dimensional systems [76], and it also affects three-dimensional systems, in particular elemental structures (see Section 3.4). From a group-theoretical point of view, Peierls distortions are characterized by a loss of translational symmetry in the above example, the nonequidistant chain of H atoms is less symmetrical (in terms of translational symmetry) than the equidistant one. 

Caution must be exercised in interpretation of the physical data for the tetracyanoplatinate complexes (as well as all other one-dimensional systems) because purity and morphology are extremely critical for one-dimensional systems. For example, a 1.00 x 0.01 x 0.01 mm perfect needle crystal of K2Pt(CN)4Xo.3 would contain — lx 10 parallel strands each of 3.5 x 10 collinear platinum atoms. Thus, purity (foreign impurities, end groups, and/or crystalline defects) levels of one part per million indicate that each strand averages more than three defects, which may drastically alter some (and in particular transport) measurements. Besides the intrinsic purity problem of one-dimensional systems, the physical properties of K2Pt(CN)4-Xo.3(H20)a are a strong function of hydration. Dehydration alters the crystal structure and thus properties of the complexes (78). Care must be maintained to ensure that dehydration is not caused by the measurement technique. For... 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Although not very commonly used in the separation of nentral hpids, two-dimensional systems have been nsed to separate hydrocarbons, steryl esters, methyl esters, and mixed glycerides that move close to each other in one-dimensional systems. Complex neutral lipids of Biomphalaria glabrata have been first developed in hexane diethyl ether (80 20), dried, and the plates have been turned 90°, followed by the second development in hexane diethyl ether methanol (70 20 10) for complete separation of sterol and wax esters, triglycerides, free fatty acids, sterols, and monoglycerides [54]. 

One dimensional conjugated carbon polymers can occur in many configurations as depicted in Figure 2 where also we included some chains with nitrogen and sulfur for later reference. Also included there are inorganic one dimensional semiconductors, like SbSI and SbSBr for later comparison. Besides the depicted one-dimensional system others like TCNQ- and KCP-salts could be included here as well but rough measurements of their nonlinear coefficients gave deceptively small values which combined with their ill-characterisation make them poor candidates for nonlinear optical devices. 

The primary structure of DNA is a one-dimensional system similar to four-letter text and can be subjected to the simplest combinatory rules. The particular motifs can be combined with one or several other motifs in away similar to using building blocks. For instance, G-rich motif can be added to one or both ODN flanks. A certain sequence, e.g., a sequence containing unmethylated deoxyribodi-nucleotide CpG motifs that mimic prokaryotic DNA (I), can be placed between similar or different motifs, like GC-rich palindrome and/or G-rich motifs (Fig. I). Various motif combinations will yield a number of putative DNA sequence variants that can be used for further tests and selection of perspective ODN compounds (see Notes 1-4). 

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

In the other case - with weak screening - Vc(k) pc Vr°(fc) shows the dispersion given in (12) and in general, the details of the fc-dependence are not only up to the transverse extension ( of the quasi one-dimensional system under consideration but also to the screening length [3, 11, 46, 34]. 

This review will be concerned with recent progress made towards an understanding of conduction phenomena in typical homomolecular crystals, e.g. anthracene and the phthalocyanines, with certain charge-transfer complexes, selected biological systems, certain novel one-dimensional systems and other materials which serve to illustrate a particular theoretical approach or the value of an experimental technique. Little attention will be given to experimental procedures other than when these are not in common use and have not been adequately described in the earlier reviews. 

Fig. 7.1. Schematic illustration of indirect photodissociation for a one-dimensional system. The two dashed potential curves represent so-called diabatic potentials which are allowed to cross. The solid line represents the lower member of a pair of adiabatic potential curves which on the contrary are prohibited to cross. The other adiabatic potential, which would be purely binding, is not shown here. More will be said about the diabatic and the adiabatic representations of electronic states in Chapter 15. The right-hand side shows the corresponding absorption spectrum with the shaded bars indicating the resonance states embedded in the continuum. The lighter the shading the broader the resonance and the shorter its lifetime.

Chapter 5 described the important details of dynamic (nonsteady-state) zone formation in one-dimensional systems. Omitted were other important aspects of zone formation and structure, including steady-state zones, two-dimensional zones, and the problem of statistical zone overlap. These topics will be examined in this chapter. 

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