Crossover dimensionality

On the other hand, it is clear that in the classical regime, T> (T i is the crossover temperature for stepwise transfer), the transition should be step-wise and occur through one of the saddle points. Therefore, there should exist another characteristic temperature. r 2> above which there exist two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points atlT = T i. The existence of the second crossover temperature, 7, 2, for two-proton transfer has been noted by Dakhnovskii and Semenov [1989]. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Since the power 7 is easier to detect in two than in three dimensions, the first MC study [62] sampled a two-dimensional MWD in a range of temperatures (that is, of (L)), so that a change in the degree of interpenetration should trigger a crossover from dilute to semi-dilute regime at some density 0. Evidently, indeed, from Fig. 4, the MWD follows the form of Eq. (16). At 0 one observes a power 7eff 1.300 0.005 which comes closely to the expected one. Above 0 one finds 7eff —> 1, and the distribution (11) becomes relevant. 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

The crossover 2d 2d behavior can be described in a similar manner to the case of a tube confinement. For the chain, trapped between two parallel plates a distance D apart, one again has N/g blobs but they arrange to a two-dimensional random coil configuration ... 

I. Webman, J. L. Lebowitz, M. H. Kalos. Monte-Carlo studies of a polymer between planes, crossover between dimensionalities. J Physique 47 579-583, 1980. 

FIG. 13 Synthetic DN A motifs for the construction of DNA framework Three- [75] (13) and four-arm (14) DNA junction [8] DNA double-crossover (DX) molecules 15,16 were used for initial studies of enzymatic oligomerization [79]. The DX motif 17, containing four cohesive ends of individual nucleotide sequence, was used for the construction of two-dimensional DNA crystals [80]. 

Two-dimensional (or indeed n-dimensional) GA strings can be handled in the usual way, with an n-dimensional chunk of the strings being swapped by the crossover operator rather than a linear segment. However, care must then be taken to ensure that wraparound is applied in all n dimensions, not just one. (See also Problem 2 at the end of this chapter.)... 

Most interestingly, [Fe(btzp)3](Cl04)2 is the first one-dimensional Fe(II) spin crossover compound, which shows the LIESST effect, detected in this instance by 57Fe Mossbauer spectroscopy (Fig. 19). 

Increasing the length of the alkyl spacer in such a way as to yield 1,4-bis(tetrazol-l-yl)butane (abbreviated as btzb) (Fig. 16), changes the dimensionality of the Fe(II) spin crossover material [89]. In fact, [Fe(btzb)3] (C104)2 is the first highly thermochromic Fe(II) spin crossover material with a supramolecular catenane structure consisting of three interlocked 3-D networks [89]. Unfortunately, only a tentative model of the 3-D structure of [Fe(btzb)3](Cl04)2 could be determined based on the x-ray data collected at 150 K (Fig. 20). 

Solid [Fe(salacen)(l-methyl-imidazole)2]C104 displays a relatively complete, gradual spin crossover [180]. Measurements of both 57Fe Mossbauer spectra and magnetism indicate that the transition observed in the one-dimensional polymeric system [Fe(salacen)(l,l -tetramethylenediimidazole)] C104 is also gradual but incomplete at both 290 K (/ze/j=5.37 B.M.) and 4.2 K (Heff=337 B.M.) [180],... 

Deviations from the Forster decay (Eq. 9.29) arise from the geometrical restrictions. In the case of spheres, the restricted space results in a crossover from a three-dimensional Forster-type behavior to a time-independent limit. In an infinite cylinder, the cylindrical geometry leads to a crossover from a three-dimensional to a one-dimensional behavior. In both cases, the geometrical restriction induces a slower relaxation of the donor. 

Another important future area for diffusion layers is the use of three-dimensional catalyzed diffusion layers for liquid-based fuel cells. This allows the three-phase active zone to be extended into the diffusion layer to increase performance and utilization and reduce crossover [276,277]. Recent work by Lam, Wilkinson, and Zhang [278] has shown the scaleable use of this concept to create a membraneless direct methanol fuel cell. In other work by Fatih et al. [279], the... 

Figure 20. Two-dimensional arrays constructed from DAE and DAE+J motifs. Two arrays are shown, one containing two components, A and B, and a second containing four components, A, B, C, and D. Each of die starred components contains one or two hairpins perpendicular to the plane of the array. The double-crossover molecules are represented as two closed figures connected by two short lines. The helix axis of each domain is represented by a dotted line. The sticky ends are drawn schematically as complementary geometrical shapes, representing Watson-Crick complementarity. The horizontal repeat is two units in the top array and four in the bottom array the vertical repeat is a single unit in both arrays. The perpendicular hairpins are visible as stripes when this array is examined in the atomic force microscope.

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

Fig. 17 (a) Six-helix bundle consisting of six helices connected by double crossovers the strands shared by different helices are identified by colors, (b) AFM image of two-dimensional hexagonal arrangement of bundles (edge size is 324 nm). Adapted with permission from [74]... 

Fig. 18 (a) Tiles with colored edges are formed by DNA helices connected through double crossover motifs and interacting through sticky ends. The pairing rules determine a two-dimensional striped lattice, as observed in AFM imaging (b). Scale bar is 300 nm. Adapted with permission from [75]. 

Fig. 5. The low temperature crossover diagram of a one-dimensional CDW. t and K are proportional to the temperature and the strength of quantum fluctuations, respectively. The amount of disorder corresponds to a reduced temperature tu 0.1. In the classical and quantum disordered region, respectively, essentially the t = 0 behavior is seen. The straight dashed line separating them corresponds to At 1, i.e., K 1, where At is the de Broglie wave length. In the quantum critical region, the correlation length is given by At- Pinning (localization) occurs only for t = 0, K

Metal wires H20 (in the membrane), H2 and 02 crossover Capable of sub-mm resolutions, minimally affects cell performance Obtains one-dimensional data... 

This formula, aside from the prefactor, is simply a one-dimensional Gamov factor for tunneling in the barrier shown in Figure 2.10. The temperature dependence of k, being Arrhenius at high temperatures, levels off to kc near the crossover temperature, which, for AE = 0, is equal to kBTQ = hai/4. 

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2itIw. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] found that there are certain cases of strongly bent two-dimensional PES s in which the instanton period has a minimum at a finite amplitude. Therefore, the crossover temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At T>TC the trivial solution Q = Q (Q is the saddle point coordinate) replaces the instanton the action is S = pV (where V is the barrier height at the saddle point) and the Arrhenius dependence k exp(-/3V ) holds. 

---