Trigonal lattice

Frustration and symmetry breaking are two different but related concepts. Frustration implies symmetry breaking because it occurs when three isochiral threefold chains having different azimuthal orientations, hence not related by any element of symmetry, are packed in a trigonal lattice. The symmetry breaking is probably a more general concept because it has been found even in the case of orthorhombic and tetragonal unit cells.33,146,149... 

Figure 47 The crystal structure of Sr3NiPtOe along the chain direction showing the trigonal lattice of chains. Note also the relatively close interchain separation. (Reprinted with permission from Ref 101, 1999 American Chemical Society)...

Independently, a two-chain helical structure was found at 3 GPa and 673 K by Chrichton et al. [135]. A hexagonal unit cell with o=709.08(5) and c=430.28(6) pm has been obtained which was refined to a trigonal lattice... 

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

Around the same time, Murthy et al. [94] also carried out a study of oriented, Li-doped polyacetylene, prepared by Lugli et al. [42]. These diflfiaction data can be indexed on a trigonal lattice, space group R3c, <7= 12.4 A, c = 7.5 A (Figure 1.10). Importantly, the... 

As each cryptand core inherits threefold symmetry, there arises two possibilities by which these trigonal molecules can crystallize (Fig. 34) a planar centrosymmetric hexagonal lattice formed by the interactions between identical groups resulting in SHG inactive molecules or noncentrosymmetric trigonal lattice formed by the interaction between different groups leading to the SHG... 

Such nonprimitive or centered cells can occur in all crystal systems higher than triclinic and introduce an extra seven lattice types to those already defined by the crystal system. The conventional types of nonprimitive lattice are C-centered (C), with an extra lattice point in the center of the (001) face at xmit cell position (V2,V2,0) body-centered (I),with an extra lattice point at the unit cell center at ( /2,V2,V2,) face-centered (F), with extra lattice points in the center of the unit cell faces at positions (V2V2O), (V2OV2), (O a A) and the rhombohedrai R setting of the trigonal lattice. There are also nonstandard settings of these, such as A-centered monoclinic cells, but these do not represent distinct lattice types. 

The stracture of overlayers is certainly a function of the preparation conditions these have to be included for sensible comparison of the literature. Ag/Cu(lll) is an example where several metastable structures are created as a function of increasing deposition temperature. Figure 11 shows that this system forms a moire overlayer (Tdep = 225 K), while deposition at 300 K leads to a trigonal lattice of dislocation loops in the underlying substrate, substitution takes place at 775 K deposition temperature [97Bes]. 

A number of selected crystal structures of polymers are shown in this section in projection along the helix axes. The poly(ethylsilylethylene) of Fig. 5.23 has its silicon atoms marked by solid circles. Neighboring helices are related by a center of symmetry, so that they must be enantiomorphous and also anticlined, i.e., the helix pairs have different handedness (d-RH and f-LH helices) and opposite inclinations of side-groups (up- and down-helices) as discussed in Sect. 5.1.8. The coordination number for the helices is three instead of the expected four because the 31 and 32 screw axes of the 2 3/1 helices match the trigonal lattice symmetry and permit a closer overall packing with CN = 3 rather than 4 (see Fig. 5.21). 

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