Periodic chain

Figure 3 Frequency-dispersion curves of the longitudinal polarizability per unit cell of infinite periodic chains of hydrogen molecules according to the method used (RPA (bottom) and UCHF (top)). AH the values are in a.u.. The position of the first excitation energies which corresponds to the poles is indicated by vertical bars.

The mean square displacements are calculated in the same spirit as for the simple Rouse model. If for simplicity we consider the periodic chain transform we get ... 

The Porath et al. experiment [14], reviewed in Sect. 2, reports nonlinear transport measurements on 10.4-nm-long poly(dG)-poly(dC) DNA, corresponding to 30 consecutive GC base pairs, suspended between platinum leads (GC-device). DFT calculations indicated that the poly(dG)-poly(dC) DNA molecule has typical electronic features of a periodic chain [58]. Thus, in both models (assuming dephasing or r-stack hybridization) the poly(dG)-poly(dC) DNA molecule is grained into a spinless linear TB chain. A generalization of the dephasing model to spin-transport has been proposed by Zwolak et al. [123]. 

Poly (benzyl ether) [G-2]-TEMPO, [G-3]-TEMPO, and [G-4]-TEMPO compounds have been synthesized and used as additives in the benzoyl peroxide initiated polymerization of styrene [127] (see Scheme 15c). After an induction period, chain growth is observed. However, the MWD is larger than in a dendrim-er-free TEMPO modulated system (Mw/Mn 2). The expectation that the den-drimer would isolate the growing chain end and prevent side reactions is not borne out. Polymerizations of methylmethacrylate, vinylacetate, and n-buty-lacrylate with the same initiator/TEMPO recipe are disappointing. 

As discussed above, a one-dimensional periodic chain (of atoms, molecules, or simply electrons localized at lattice site with a distance d between them) can be described by Bloch waves If there are two electrons (or atoms, or molecules, with two allowed spin states) per site, one calls this a "filled band" ... 

PDAs A Model of CPs Without Disorder In PDAs one would expect to have only two processes to consider transport along a really periodic chain, and all similar interchain hopping events. The case of PDAs illustrates the ambiguities of transport studies on CPs. Early time-of-flight experiments yielded mobilities ==5 cm2/V s along the chains, and 10 3 along the perpendicular directions [217]. A mobility of a few cm2/V s is typical of a molecular crystal, and the polymer character was not apparent. 

Figure 4.4. A one-dimensional periodic chain of p atomic orbitals. At the top is shown the sign combinations corresponding to the /r-point F (/t = 0), where A = oo. At the bottom are the sign combinations for the /r-point X k = Tr/d), where A = 2d.

The periodic chain offers the advantage that it is inherently uniform even in the perturbed state. As an example, the average distance between atoms separated by a fixed number k of skeletal bonds [k = j — h in Eqn. (2.1.36)] is the same whether the two atoms are placed towards a chain end or in the middle ... 

The open-chain transform should be used when dealing with the overall size of the chain and, in dynamics, when separate consideration of the first, most collective modes of motion is required. In the following, the more expedient periodic-chain transform will generally be adopted unless specified otherwise. 

Having found the independent linear combinations of bond vectors [see Eqs. (2.1.17), (2.1.27), and (2.1.28)], through the central-limit theorem it is easy to construct the Gaussian joint distribution and the associated quadratic potential. Adopting for simplicity the periodic-chain transform, we have... 

Expression (3.1.6) is exact for ring chains, whereas it is only approximate for open and periodic chains, as off-diagonal elements depending on pairs of different q values are neglected according to the Zimm-Hearst approach [20, 80]. For the ring and the periodic chain, where Q q,h) = Eqn. (3.1.6) reduces to... 

This equation implies perfect dynamical equivalence of all the chain atoms this is physically true only for the ring, whereas it is a model assumption in the periodic case. However, it should be noted that, apart from the first few collective modes, the periodic chain gives a good description of the open-chain dynamics and may be safely retained when investigating local chain motions, as suggested by Akcasu, Benmouna, and Han [81] and shown by us [82, 83]. The general solution of Eqn. (3.1.5) may be cast in the form... 

When the bead-and-spring chain is not in the ideal state, the intramolecular force is given in Eqn. (3.1.3). As it may be seen, in general, the force is not simply transmitted by first-neighboring atoms, but it has a long-range character. The relaxation times are given by Eqn. (3.1.11) after they are known, the dynamic viscosity i (cu) and the atomic correlation function B(k, t) are obtained from Eqs. (3.1.15) and (3.1.18) (for the periodic chain), and the complex modulus and dynamic structure factors are easily constructed. 

Figure 8. Expansion ratio a (q) vs. z = zBjlnlq. First modes are separately shown (n, = qN/2n = 1, 2 for ring chain, = qN/n = 1 for open chain), whereas following ones are well approximated by continuous line also valid for periodic chain. (From ref. 68, by permission of the publishers, Butterworth Co. (Publishers) Ltd. ,)...

Figure 10. Real and imaginary part of complex modulus, G and G —cut/j, vs. loor for atactic polystyrene in dilute solution Oj- is shift factor. Experimental points are from refs. 102 and 103 best-fit continuous lines [from Eqn. (34) of ref. 12] superimposed on experimental points after rigid, parallel shift. [Model assumptions and parameters unperturbed periodic chain, N = 8000 (left) and N = 1300 (right), to/to = 47, R ff = 0.125 A.] (Reprinted with permission from ref. 12, Copyright 1981, American Chemical Society.)...

Figure 11. Dynamic correlation function B k, t) [from Eqn. (41) of ref 12] as function of t/to for three values of k as indicated on curves in presence of internal viscosity. Result with Tq = 0 (i.e., with no internal viscosity) are given by dashed lines. [Model assumptions and parameters periodic chain same as in Figure 10, N P 100.]...

Figure 13. Left half-peak time r,, 2 of coherent dynamic structure factor plotted vs. Q for poly(dimethylsiloxane) in C D. Continuous line, calculated results at T = 30 C dashed line, at T = 70 C. F.xperimental points are from ref. 115. Right characteristic time (j, vs. Q for molten sample at 7 =100°C. Experimental points are from ref. 115. [Model assumptions and parameters periodic chain, Tq = 0, R = 0.30 A, tB = 0.018 for solutions, R, = OA, xB = 0 for melt.] (Reprinted with permission from ref. 15, Copyright 1985, American Chemical Society.)...

This work focuses on the behaviour of an intrinsically fractal, binary chain, showing a rich structure in its vibrational spectrum, clearly distinct from that of conventional quasi-periodic chains. Furthermore, an approximate solution is given in the weak-coupling limit and the relationship with fractal chains, the discrete analogue of fractal strings, is emphasised. 

Motivated by the remarkable discovery of quasicrystalline ordering in solids in 1984 [1], wave propagation in deterministic non periodic media has been an area of intense research. Following the successful experimental realisation of a multitude of such structures through modem technologies, such as molecular beam epitaxy and laser ablation [2], their interest has increased ever since. The most widely known examples are quasi-periodic structures obtained by substitution rules, such as Fibonacci- or Thue-Morse-chains [3,4], Much less has been published on quasi-periodic chains constructed according to a Cantor-set algorithm, which are the subject of this note. 

Increasing the number of generations for the weakly coupled chain leads to its spectrum becoming similar to that of a more strongly coupled chain. For those, the appearance of the normalised spectrum remains basically the same independently of the generation number. In both cases, it is fundamentally different from that displayed by other quasi-periodic chains. 

DE Inosilicates with 2-periodic multiple chains 9.DG. Inosilicates with 3-periodic single and multiple chains 9.DH. Inosilicates with 4-periodic single chains 9.DJ. Inosilicates with 4-periodic double and triple chains 9.DK. Inosilicates with 5-periodic single chains 9.DL. Inosilicates with 5-periodic double chains 9.DM. Inosilicates with 6-periodic single chains 9.DN. Inosilicates with 6-periodic double chains 9.DO. Inosilicates with 7-, 8-, 10-, 12- and 14-periodic chains. 9.DP. Transitional ino-phyllosilicate structures 9.t)Q. Unclassified inosilicates 9.E Phyllosilicates... 

In describing the normal modes of a protein, it is instructive to compare them conceptually with those of a simple model of a polymer, such as a chain of atoms, both periodic and aperiodic. In a harmonic periodic chain, the normal modes carry energy without resistance from one end of the ID crystal to the other. On the other hand, the vast majority of normal modes of an aperiodic chain are spatially localized [138]. Protein molecules, which are of course not periodic, can be better characterized as an aperiodic chain of atoms, and most normal modes of proteins are likewise localized in space [111,112,126-128]. If a normal mode a is exponentially localized, then the vibrational amplitude of atoms in mode a decays from the center of excitation, Ro, as... 

The behavior of the vector si(n) indicates that if initially there was Si, the probability of remaining at this state after one step is si(l) = 0. After two steps (24 hours), si(2) = 1, i.e., there will be again a state of Day, as expected. According to section 2.1-5 Table 2-2, the above chain is defined as a periodic chain. 

Examples 2.43-2.45 treat a few models of periodic chains, also referred to as recurrent events. Generally speaking, a system undergoes some process as a result of which it occupies the states sequentially. The latter repeats its self ad infinitum or attains some steady state. 

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