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Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group. These are the Bravais lattices in three dimensions: P primitive; I body centered (from the German Innenzentriert) F face centered (from the German Flächenzentriert) A centered on A faces only; B centered on ...
The face-centered cubic lattice (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of four lattice points per unit cell (1 ⁄ 8 × 8 from the corners plus 1 ⁄ 2 × 6 from the faces).
For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.
The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A-face centered or C-face centered lattices. There are seven rhombohedral space groups, with initial letter R.
For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.
The vertex arrangement of the 16-cell honeycomb is called the D 4 lattice or F 4 lattice. [2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; [3] its kissing number is 24, which is also the same as the kissing number in R 4, as proved by Oleg Musin in 2003.
The diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by 1 / 4 of the width of the unit cell in each dimension. Many compound semiconductors such as gallium arsenide , β- silicon carbide , and indium antimonide adopt the analogous zincblende structure , where each atom has nearest ...
The Wigner–Seitz cell of the face-centered cubic lattice is a rhombic dodecahedron. [9] In mathematics, it is known as the rhombic dodecahedral honeycomb . The Wigner–Seitz cell of the body-centered tetragonal lattice that has lattice constants with c / a > 2 {\displaystyle c/a>{\sqrt {2}}} is the elongated dodecahedron .
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