Rhombic pattern

Fig. 38 Transmission electron micrographs of gold nanoparticles organized in 2D arrays through binding of two types of three-branched DNA motifs, (a) Array where only one tile contains 5 nm particles. The resulting arrangement has two different characteristic length scales, (b) Array where both tiles contain 5-nm particles, yielding an equally spaced pattern, (c) Array where one tile contains a 5-nm particle and the other tile contains a 10-nm particle. The pattern of alternating smaller and bigger spheres mimics the rhombic pattern of the tile array. Adapted with permission from [164]...

In this section, we have used the rotational invariance to predict the form of the phase equations. Recently, a lot of effort has been devoted to the development of amplitude equations - or order parameter equations - in a rotationally invariant form [15, 85, 86]. According to Gunaratne [86], supplemental terms introduced in the equations to meet this constraint could account for the stabilization of otherwise unstable states, in particular rhombic patterns which have been observed both in experiments [87] and in numerical simulations [29]. This could also explain the stabilization of the mixed states of Equation (16), also observed in experiments [51] and simulations [30]. Nevertheless, alternative interpretations can be proposed in terms of finite size effects in the numerical simulations, and in terms of three-dimensional effects for the experiments. Mixed states observed in our laboratory more likely correspond to projections of two exactly superposed layers, the first one with striped symmetry, and the second one with hexagonal symmetry. More generally, one expects that the strong transverse gradients present in the experiments can produce an hybridization between otherwise flawless structures. 

Fig. 7. Time evolution of an unstable pattern (a) an imposed perfect rhombic pattern with a characteristic angle of 50° (b) and (c) patterns formed 0.5 h and 20 h, respectively, after the perturbation was removed. The time asymptotic pattern achieved by 20 h has domains with different characteristic angles. The region shown is 6 mm x 6 mm. The control parameters were the same as in Figure 3b except that [CH2(COOH)2]o = 32 mM. (From [13])...

Fig. 8. Predicted and observed bifurcation diagrams for the transition from a uniform state to rhombic patterns, (a) The band of allowed values ofA6 = 0- 60° for linearly stable rhombs, as a function of the bifurcation parameter n. The bifurcations from the uniform state to rhombic patterns (or regular hexagons, 0 — 60°) and from rhombic patterns (or hexagons) to stripes are both slightly subcritical. (b) The root-mean-square width of the distribution, A rms, of angles measured for the observed Turing patterns, as a function of malonic acid concentration. Other control parameters were the same as in Figure 3b. (From [13])...

The general theory of rhombic patterns recently developed by Gunaratne [13,47] has resolved some mysteries around our previous reports of a mixed state , which is now realized to be a type of rhombic patterns [46]. Our experimental results on rhombic pattern formation and selection are in good accord with the theory [13, 47]. Rhombic patterns should occur widely in other nonequilibrium systems, including surface tension driven convection, ferrofluids, Rayleigh-B nard convection, and driven capillary waves, but we know of no other laboratory studies of rhombic patterns. We hope that our work will stimulate a search for rhombic patterns in other systems. 

Calculations of a similar kind, carried out by Stoeckli and Stoeckli-Evans for argon on rhombic sulphur likewise reveal a marked difference in the patterns of energy distribution of sites, as between the (Oil) and the (111) faces of the sulphur. 

Preliminary X-ray work on CrSBr crystals, probably formed via a chemical-transport reaction, shows rhombic symmetry (181). Powder patterns of the iodides can be indexed with respect to a hexagonal lattice, and are similar to those of Cris (17, 181). 

FIGURE 5.4 Anisotropy in absorption and derivative powder-type ERP spectra. (A) axial intensity pattern (B) axial EPR absorption (C) axial EPR derivative (D) rhombic EPR absorption (E) rhombic EPR derivative (F) the spectrum of horse heart cytochrome c, a rhombic EPR derivative with anisotropic broadening (Hagen 2006). (Reproduced by permission of The Royal Society of Chemistry.)... 

In order to illustrate the general applicability of the methodology we have extended our approach to other large zeolite crystals, such as SAPO-34, SAPO-5 and ZSM-5. Our study on the rhombic SAPO-34 crystals reveals a four-pointed star fluorescence pattern at 445 K, which is transformed into a square-shaped feature at 550 K. This is illustrated in Figure 4a. Confocal fluorescence slices, summarized in Figures 4b-d, recorded at different temperatures show the cubical pattern, which proceed from the exterior of the crystal inwards. Both observations are consistent with a model which involves six components of equal tetragonal pyramids as illustrated in Figure 3b. 

K) which exhibited rhombic EPR spectra (gi =2.0015, g2 = 2.0046, and g3 = 2.0105). Only the smallest g value showed a hyperfine coupling pattern, a triplet to the unique N-atom = 2.84mT) this further split into 1 2 3 2 1... 

---