Delaunay cells

Figure 2. A Delaunay cell within the Finney pack is a tetrahedron that groups together four nearest neighbor spheres. Each apex of the tetrahedron (labeled S, T, U, V) is the center of a sphere. Only the segments of the spheres contained within the cell are shown the rest of each sphere is contained in adjacent cells. The void area in each face of the cell is a local minimum in cross-section and thus controls access of the meniscus between nonwetting and wetting phases into the cell volume. The center of the throat in face TUV is labeled W, and the center of the pore body is labeled X.

Conventional Unstructured Grid Methods In general any grid that is not structured is an unstructured grid. Of particular importance are Voronoi tessellations and their dual the Delaunay tessellation. In three dimensions Voronoi cells are convex polyhedra and Delaunay cells are tetrahedra. In two dimensions Voronoi cells are convex polygons and Delaunay cells are triangles. 

If no external evidence is available, it is still possible to determine the unit cell dimensions of crystals of low symmetry from powder diffraction patterns, provided that sharp patterns with high resolution are avail able. Hesse (1948) and Lipson (1949) have used numerical methods successfully for orthorhombic crystals. (Sec also Henry, Lipson, and Wooster, 1951 Bunn 1955.) Ito (1950) has devised a method which in principle will lead to a possible unit cell for a crystal of any symmetry. It may not be the true unit cell appropriate to the crystal symmetry, but when a possible cell satisfying all the diffraction peaks on a powder pattern lias been obtained by Ito s method, the true unit cell can be obtained by a reduction process first devised by Delaunay (1933). Ito applies the reduction process to the reciprocal lattice (see p. 185), but International Tables (1952) recommend that the procedure should be applied to the direct space lattice. 

There is also a numerical reduction process first introduced by Delaunay (1933), which leads automatically from the possible unit cell to the smallest unit cell. Moreover, when the reduction process is complete, the characteristics of the smallest cell, revealed in diagrams which are used as part of the reduction process, show the symmetry and indicate how the true unit cell is related to the smallest unit. Details of the process are given in International Tables (1952). (See also Patterson and Love (1957).)... 

Hereditary spherocytosis (HS) comprises a group of inherited hemolytic anemias characterized by chronic hemolysis with a broad spectrum of severity (Hassoun et al, 1997). The principal cellular defect is the loss of erythrocyte surface area relative to the intracellular volume, although increased osmotic frailty is also a factor. A distinctive spherical red blood cell (RBC) morphology is observed in sufferers of HS and splenic destruction of these abnormal erythrocytes is the primary cause of the hemolysis experienced (Delaunay, 1995 Palek and Jarolim, 1993). 

Delaunay, J. (1995). Genetic disorders of the red cell membranes. FEES Lett. 369, 34-37. 

V= 534.9 A Z = 2,d = 3.98 g cm- andF(000) = 559.86. Single-crystal precession and Weissenberg photographs indicated that the space group is triclinic. A Delaunay reduction of the cell chosen failed to show additional symmetry. The structure was successfully refined in the space group FI. 

There are two broadly accepted methods of unit cell reduction. One of them was introduced by Delaunay and then applied to a transformation of a... 

Figure 5.13. The schematic of Delaunay-Ito transformation. Left - the four unit cell vectors (vj, V2, V3 and V4 = -V1-V2-V3) are associated with the comers of the tetrahedron, while the six scalar products between the corresponding pairs of the vectors (s through S34) are linked to the edges of the tetrahedron. Assuming that > 0, the transformation is carried out as shown on the right the sign of 12 is changed its value is subtracted from that on the opposite edge and added to those on all adjacent edges the direction of vi (or V2) is reversed and the new vectors v 3 and v 4 are determined as V3+V1 and V4+V1 (or V3+V2 and V4+V2, respectively).

From the four possible triplets of resulting vectors (vi, v, V3 to V2, V3, V4), the one that has shortest vectors is selected because the angles among these vectors are closest to 90°. The Delaunay-Ito reduced primitive cells can be classified into 24 types according to the relationships between unit cell vectors and their scalar products. They are easily converted into one of the 14 Bravais lattices. For example, if V = V2 and s u = s 23 = 0 (an = a23 = 90°), the standard unit cell is orthorhombic C-centered, with lattice vectors, v° calculated as follows ... 

The unit cell reduction using Delaunay-Ito method can be easily automated as is done in the ITO indexing computer code, which is discussed in section 5.11. The Delaunay-Ito reduced unit cell, however, may not be the one with the shortest possible vectors, although the latter is conventionally defined as a standard reduced unit cell. 

Regardless of which indexing method was employed, the resulting unit cell (especially when it is triclinic) shall be reduced using either Delaunay-Ito or Niggli method in order to enable the comparison of different solutions and to facilitate database and literature searches. Furthermore, the relationships between reduced unit cell parameters must be used to properly determine the Bravais lattice. The Niggli-reduced cell is considered standard and therefore, is preferable. 

This algorithm realizes a zone search indexing method combined with the Delaunay-Ito technique (see section 5.10.1) for the reduction of the most probable unit cell. The most commonly used versions of computer codes are ITO 13 and IT015. The program arrives at a solution by using the following algorithm ... 

Reduces the resultant unit cell by using the Delaunay-Ito method and then transforms it into one of the 14 Bravais lattices if the lattice is not primitive. 

Delaunay A, Pfheger D, Barrault MB, Vinh J, Toledano MB. A thiol peroxidase is an h(2)o(2) receptor and redox-transducer in gene activahon. Cell 2002 111 471-481. 

The unit cells given for DHDK 0.4 CHCI3 and DHDK 0.5 C2H5OH are both the Dirichlet and Delaunay reduced triclinic cells. 

The unit cells given for DHDK CH3COOH and DHDK m-dinitrobenzene are both Dirichlet (but not Delaunay) reduced cells. 

Pablos, J. L., Amara, A., Bouloc, A., Santiago, B., Caruz, A., Galindo, M., Delaunay, T., Virelizier, J. L., and Arenzana-Seisdedos, F. (1999). Stromal-cell derived factor is expressed by dendritic cells and endothelium in human skin. Am. J. Pathol. 155, 1577-1586. 

While this two-scale model allows for a natural connection between the two scales (i.e., the inclusion-boundary behavior and the atomistic cell behavior are connected via the common strain transformation), it also is limited by the required uniformity of strain in the atomistic inclusion (through the periodic continuation conditions on the atomistic cell) and in each of the tetrahedra of the Delaunay tessellation. 

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