Two points are adjacent in the diamond structure if and only if their four-dimensional coordinates differ by one in a single coordinate. This structure may be scaled to a cubical unit cell that is some number a of units across by multiplying all coordinates by a / 4.Īlternatively, each point of the diamond cubic structure may be given by four-dimensional integer coordinates whose sum is either zero or one. Adjacent points in this structure are at distance √ 3 apart in the integer lattice the edges of the diamond structure lie along the body diagonals of the integer grid cubes. There are eight points (modulo 4) that satisfy these conditions: X = y = z (mod 2), and x + y + z = 0 or 1 (mod 4). With these coordinates, the points of the structure have coordinates ( x, y, z) satisfying the equations Mathematically, the points of the diamond cubic structure can be given coordinates as a subset of a three-dimensional integer lattice by using a cubic unit cell four units across. The first-, second-, third-, fourth-, and fifth-nearest-neighbor distances in units of the cubic lattice constant are √ 3 / 4, √ 2 / 2, √ 11 / 4, 1 and √ 19 / 4, respectively. Zincblende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms. The atomic packing factor of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the vertices of the structure and are as large as possible without overlapping) is π √ 3 / 16 ≈ 0.34, significantly smaller (indicating a less dense structure) than the packing factors for the face-centered and body-centered cubic lattices. Zincblende's space group is F 43m, but many of its structural properties are quite similar to the diamond structure. Many compound semiconductors such as gallium arsenide, β- silicon carbide, and indium antimonide adopt the analogous zincblende structure, where each atom has nearest neighbors of an unlike element. 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. The lattice describes the repeat pattern for diamond cubic crystals this lattice is "decorated" with a motif of two tetrahedrally bonded atoms in each primitive cell, separated by 1 / 4 of the width of the unit cell in each dimension. A lattice of 3 × 3 × 3 unit cellsĭiamond's cubic structure is in the Fd 3m space group (space group 227), which follows the face-centered cubic Bravais lattice.
Visualisation of a diamond cubic unit cell: 1.
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