Diamond cubic

Mathematical structure

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. With these coordinates, the points of the structure have coordinates (x, y, z) satisfying the equations $${\displaystyle \begin{array}{cc}x=y& =z(\text{mod }2),\\ x+y+z& =0\text{ or }1(\text{mod }4).\end{array}}$$

There are eight points (modulo 4) that satisfy these conditions:

(0,0,0), (0,2,2), (2,0,2), (2,2,0),

(3,3,3), (3,1,1), (1,3,1), (1,1,3)

All of the other points in the structure may be obtained by adding multiples of four to the x, y, z coordinates of these eight points. Adjacent points in this structure are at distance ⁠${\displaystyle \sqrt{3}}$⁠ apart in the integer lattice; the edges of the diamond structure lie along the body diagonals of the integer grid cubes. 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.

Alternatively, each point of the diamond cubic structure may be given by four-dimensional integer coordinates whose sum is either zero or one. Two points are adjacent in the diamond structure if and only if their four-dimensional coordinates differ by one in a single coordinate. The total difference in coordinate values between any two points (their four-dimensional Manhattan distance) gives the number of edges in the shortest path between them in the diamond structure. The four nearest neighbors of each point may be obtained, in this coordinate system, by adding one to each of the four coordinates, or by subtracting one from each of the four coordinates, accordingly as the coordinate sum is zero or one. These four-dimensional coordinates may be transformed into three-dimensional coordinates by the formula $${\displaystyle (a,b,c,d)\to (a+b-c-d,a-b+c-d,-a+b+c-d).}$$ Because the diamond structure forms a distance-preserving subset of the four-dimensional integer lattice, it is a partial cube.

Yet another coordinatization of the diamond cubic involves the removal of some of the edges from a three-dimensional grid graph. In this coordinatization, which has a distorted geometry from the standard diamond cubic structure but has the same topological structure, the vertices of the diamond cubic are represented by all possible 3d grid points and the edges of the diamond cubic are represented by a subset of the 3d grid edges.

The diamond cubic is sometimes called the "diamond lattice" but it is not, mathematically, a lattice: there is no translational symmetry that takes the point (0,0,0) into the point (3,3,3), for instance. However, it is still a highly symmetric structure: any incident pair of a vertex and edge can be transformed into any other incident pair by a congruence of Euclidean space. Moreover, the diamond crystal as a network in space has a strong isotropic property. Namely, for any two vertices x, y of the crystal net, and for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the similarly ordered y-edge. Another (hypothetical) crystal with this property is the Laves graph (also called the K₄ crystal, (10,3)-a, or the diamond twin).

Mechanical properties

The compressive strength and hardness of diamond and various other materials, such as boron nitride, (which has the closely related zincblende structure) is attributed to the diamond cubic structure.

Similarly, truss systems that follow the diamond cubic geometry have a high capacity to withstand compression, by minimizing the unbraced length of individual struts. The diamond cubic geometry has also been considered for the purpose of providing structural rigidity though structures composed of skeletal triangles, such as the octet truss, have been found to be more effective for this purpose.