Isometric lattices

Lattices with 3 or 6 Axes. The occurrence of six- and/or threefold axes in actual crystals occasions certain difficulties in classification. So long as we are concerned only with lattices, however, there need to be no ambiguity, if we proceed carefully and rigorously. For that reason we shall take a different (and more fundamental) approach in demonstrating that there are only two distinct lattice types consistent with the presence of three- or sixfold axes that are oriented in one direction only. The latter qualification is necessary to distinguish the present case from that of the isometric lattices where there... 

For all the isometric lattices each translation vector lies along a C4 axis, there are C2 axes bisecting each angle between them, and all the body di-... 

The geometry of all the various enclosing forms of the PA Sm core depends on only the same single lattice parameter u of the heptagonal isometric lattice... 

For all hexagonal clusters the form lattices are integral. Of 40 hexameric clusters, 27 form lattices are rationally equivalent with the isometric lattice l-A,, seven with the lattice and six with the lattice /2-A ,g. ... 

At T = 0 K, where any transformation of a pure substance tends to be isoen-tropic, phase stability can be related to the enthalpy and a phase transition occurs at those points in the phase diagram where two phases have equal enthalpy. Erom the computational point of view, it is possible to explore a range of crystalline volumes by isometric lattice deformations and obtain the corresponding values of pressure and, consequently, of enthalpy. It is intended that nuclei are allowed to relax to their equilibrium geometry after... 

Fractional Coordinates. In specifying the location of a point in a crystal lattice it is customary to employ coordinates, jt, y, z, that give the fraction of each principal vector distance (a, b, c), which define the unit cell. Thus, a point at the origin has the fractional coordinates 0,0,0 while the center of the cell has the coordinates 3,2,5. The face centers are 0, , , ,0, and , ,0 for the a, b, and c faces, respectively. It is to be emphasized that these fractional coordinates are not Cartesian except for isometric cells and are not even orthogonal for triclinic, monoclinic, or hexagonal lattices. 

DGS04] M. Deza, V. P. Grishukhin, and M. I. Shtogrin, Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices, World Scientific and Imperial College Press, 2004. 

Microscopic disorder We consider a lattice the sites of which have disordered resonance energies, with a distribution of width Ae, but have the same intersite interactions (same dipole orientation and oscillator strength) as the perfect lattice. This is the so-called substitutional disorder model.122 We assume the disorder width to be smaller than the excitonic bandwidth (4< Be) and examine the bottom of the excitonic band, where the emitting and the absorbing K 0 states lie. In a renormalization-group scheme, we split the lattice into isometric domains of n sites, on which the excitation is assumed to be localized, and write the substitutional-disorder hamiltonian in this basis we thereby obtain a new disorder width An Aen-1/2 and a... 

In all the cases considered so far, the lattices involved are integral. This property implies the existence of a metric tensor with integral entries, up to one real lattice parameter, as in the cubic case. One finds, in particular, isometric hexagonal lattices (with c = a). Similar and additional properties allow one to speak of strongly correlated biomacromolecular structures... 

Figure 11-3. Indexed isometric hexagonal form lattice of the hexameric R-phycoerythrin with hexa-...

With tq denoting the hexagonal radius of the central hole, the radius of the envelope is 4rg and equal to the height of the hexamer. It follows that the three-dimensional form lattice is hexagonal with lattice parameters a = c = and, therefore, isometric hexagonal because the axial ratio is y = cja = 1. This lattice, denoted... 

Figure 11-4. The indexed form lattice of the decameric cyclophilin A is isometric pentagonal, as...

The concept of a pentagonal lattice (and in general that of a polygonal lattice) has been introduced by Yamamoto in the context of axially symmetric aperiodic crystals [18], As this term might be unfamiliar to many people, some explanations are required before showing that the form lattice of cyclophilin is isometric pentagonal. 

Cyclophilin A is a simple example pentamer and decamer share the same isometric form lattice. Moreover, the mutual orientation of the pentamers in the decameric configuration is fixed through a linear scaling transformation leaving the form lattice invariant. The geometry of the entire biomacromolecular system and of its components is expressible in term of the single parameter tq, which is the shortest distance of the monomeric chain from the fivefold axis. 

Figure 11-5. The form lattice of the double heptameric ring of the pyrococcus abyssi Sm core in the free state conformation is isometric heptagonal. The lattice parameter is given by the distance u of each heptamer from a twofold axis (from [16], courtesy lUCr)...

In the particular case of the enclosing forms of the PVl hexamer shown in Figure 11-10, one finds = m = 4 and // = the corresponding form lattice becomes isometric hexagonal. The relation between this lattice and the external boundary is the same as for the R-phycoerythrin of Figure 11-3 and in two dimensions for the facet-like snow crystal of Figure 11-1. 

Figure 11-10. The cluster with symmetry 32 of a hexamer of the PVl coat protein in the human rhi-novirus has molecular forms with vertices at points of an isometric hexagonal lattice with parameters a = c = r/ 14 at which is the hexagonal radius of the capsid projected along the threefold axis. The boundary of the capsid projected along the y-axis is indicated with dashed lines...

Figure I. An isometric view of Hg2 se sFg showing the chains of mercury atoms (circles) running through the lattice composed of AsFg ions (octahedra)...

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