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{{Short description|Type of crystal structure}}
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[[File:Diamond Cubic-F lattice animation.gif|thumb|250px|right|Unit cell of the diamond cubic [[Chemistry:Crystal structure|crystal structure]]]]
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[[File:Diamond cubic animation.gif|thumb|250px|Rotating model of the diamond cubic [[Chemistry:Crystal structure|crystal structure]]]]
[[File:diamond_lattice.stl|thumb|250px|3D ball-and-stick model of a diamond lattice]]
[[File:DiamondPoleFigure111.png|thumb|250px|[[Physics:Pole figure|Pole figure]] in [[Stereographic projection|stereographic projection]] of the diamond lattice showing the 3-fold symmetry along the [111] direction]]


Although often called the '''diamond lattice''', this structure is not a [[Lattice (group)|lattice]] in the technical sense of this word used in mathematics.

==Crystallographic structure==
[[File:visualisation diamond cubic.svg|thumb|upright|Visualisation of a diamond cubic unit cell: 1. Components of a unit cell, 2. One unit cell, 3. A lattice of {{nowrap|3 × 3 × 3}} unit cells]]
Diamond's cubic structure is in the Fd{{overline|3}}m [[Space group|space group]] (space group 227), which follows the [[Cubic crystal system|face-centered cubic]] [[Bravais lattice]]. The lattice describes the repeat pattern; for diamond cubic crystals this lattice is "decorated" with a ''motif'' of two [[Tetrahedron|tetrahedrally]] [[Chemistry:Chemical bond|bonded]] atoms in each [[Physics:Wigner–Seitz cell|primitive cell]], separated by {{sfrac|1|4}} of the width of the [[Chemistry:Crystal structure#Unit cell|unit cell]] in each dimension.<ref>{{citation |title=Diamond films: chemical vapor deposition for oriented and heteroepitaxial growth |first=Koji |last=Kobashi |publisher=Elsevier |year=2005 |isbn=978-0-08-044723-0 |page=9 |contribution=2.1 Structure of diamond}}.</ref> The diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by {{sfrac|1|4}} of the width of the unit cell in each dimension. Many compound semiconductors such as [[Chemistry:Gallium arsenide|gallium arsenide]], β-[[Chemistry:Silicon carbide|silicon carbide]], and [[Chemistry:Indium antimonide|indium antimonide]] adopt the analogous [[Cubic crystal system#Zincblende structure|zincblende structure]], where each atom has nearest neighbors of an unlike element. Zincblende's space group is F{{overline|4}}3m, but many of its structural properties are quite similar to the diamond structure.<ref>{{citation |title=Inorganic chemistry |first1=Egon |last1=Wiberg |first2=Nils |last2=Wiberg |first3=Arnold Frederick |last3=Holleman |publisher=Academic Press |year=2001 |isbn=978-0-12-352651-9 |page=1300}}.</ref>

The [[Physics:Atomic packing factor|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 <math>\tfrac{\pi\sqrt 3}{16} \approx 0.34,</math><ref>{{citation |title=The Science and Engineering of Materials |first1=Donald R. |last1=Askeland |first2=Pradeep Prabhakar |last2=Phulé |publisher=Cengage Learning |year=2006 |isbn=978-0-534-55396-8 |contribution=Example 3-15: Determining the Packing Factor for Diamond Cubic Silicon |page=82}}.</ref> significantly smaller (indicating a less dense structure) than the packing factors for the [[Cubic crystal system|face-centered and body-centered cubic lattices]].<ref>{{citation |title=Concise Dictionary of Materials Science: Structure and Characterization of Polycrystalline Materials |first=Vladimir |last=Novikov |publisher=CRC Press |year=2003 |isbn=978-0-8493-0970-0 |page=9}}.</ref> Zincblende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms.

The first-, second-, third-, fourth-, and fifth-nearest-neighbor distances in units of the cubic lattice constant are <math>\tfrac{\sqrt 3}{4}, \tfrac{\sqrt 2}{2}, \tfrac{\sqrt{11}}{4}, 1, \tfrac{\sqrt{19}}{4},</math> respectively.

==Mathematical structure==
Mathematically, the points of the diamond cubic structure can be given coordinates as a subset of a three-dimensional [[Integer lattice|integer lattice]] by using a cubic unit cell four units across. With these coordinates, the points of the structure have coordinates {{math|(''x'', ''y'', ''z'')}} satisfying the equations<ref name="ny09">{{citation |contribution=Neighborhood sequences in the diamond grid – algorithms with four neighbors |first1=Benedek |last1=Nagy |first2=Robin |last2=Strand |title=Combinatorial Image Analysis: 13th International Workshop, IWCIA 2009, Playa Del Carmen, Mexico, November 24–27, 2009, Proceedings |series=[[Lecture Notes in Computer Science]] |publisher=Springer-Verlag |year=2009 |volume=5852 |pages=109–121 |doi=10.1007/978-3-642-10210-3_9 |bibcode = 2009LNCS.5852..109N|isbn=978-3-642-10210-3}}.</ref>
<math display=block>\begin{align}
  x = y &= z\ (\text{mod }\ 2), \\ 
  x + y + z &= 0 \text{ or } 1\ (\text{mod }\ 4).
\end{align}</math>

There are eight points ([[Modular arithmetic|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 {{mvar|x, y, z}} coordinates of these eight points. Adjacent points in this structure are at distance {{tmath|\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 {{mvar|a}} of units across by multiplying all coordinates by&nbsp;{{math|{{sfrac|''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<ref name="ny09"/><ref name="e09">{{Citation |contribution=Isometric Diamond Subgraphs |last=Eppstein |first=David |arxiv=0807.2218 |title-link=International Symposium on Graph Drawing |title=Graph Drawing: 16th International Symposium, GD 2008, Heraklion, Crete, Greece, September 21–24, 2008, Revised Papers |series=Lecture Notes in Computer Science |volume=5417 |year=2009 |pages=384–389 |editor-last=Tollis |editor-first=Ioannis G. |editor2-last=Patrignani |editor2-first=Maurizio |publisher=Springer-Verlag |doi=10.1007/978-3-642-00219-9_37 |isbn=978-3-642-00219-9 |s2cid=14066610}}.</ref>
<math display=block>(a,b,c,d) \to (a+b-c-d,\ a-b+c-d,\ -a+b+c-d).</math>
Because the diamond structure forms a [[Isometry|distance-preserving]] subset of the four-dimensional integer lattice, it is a [[Partial cube|partial cube]].<ref name="e09" />

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.<ref name="pk01">{{citation |title=A unified formulation of honeycomb and diamond networks |last1=Parhami |first1=B. |first2=Ding-Ming |last2=Kwai |journal=IEEE Transactions on Parallel and Distributed Systems |volume=12 |issue=1 |pages=74–80 |year=2001 |doi=10.1109/71.899940}}.</ref>

The diamond cubic is sometimes called the "diamond lattice" but it is not, mathematically, a [[Lattice (group)|lattice]]: there is no [[Physics:Translational symmetry|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 (geometry)|congruence]] of [[Euclidean space]]. Moreover, the diamond crystal as a network in space has a strong isotropic property.<ref>{{citation|title=Topological Crystallography -With a View Towards Discrete Geometric Analysis-|first=Toshikazu|last=Sunada|publisher=Springer|year=2012|isbn=978-4-431-54176-9}}</ref> Namely, for any two vertices {{mvar|x, y}} of the crystal net, and for any ordering of the edges adjacent to {{mvar|x}} and any ordering of the edges adjacent to {{mvar|y}}, there is a net-preserving congruence taking {{mvar|x}} to {{mvar|y}} and each {{mvar|x}}-edge to the similarly ordered {{mvar|y}}-edge. Another (hypothetical) crystal with this property is the [[Laves graph]] (also called the K<sub>4</sub> crystal, (10,3)-a, or the diamond twin).<ref>{{citation|title=Crystals that nature might miss creating|first=Toshikazu|last=Sunada|journal=Notices of the AMS|volume=55|year=2008|pages=208–215}}</ref>

==Mechanical properties==
The compressive strength and hardness of [[Chemistry:Diamond|diamond]] and various other materials, such as [[Chemistry:Boron nitride|boron nitride]],<ref>Blank, V.; Popov, M.; Pivovarov, G.; Lvova, N. et al. (1998). "Ultrahard and superhard phases of fullerite C60: comparison with diamond on hardness and wear". Diamond and Related Materials 7 (2–5): 427. [https://www.sciencedirect.com/science/article/abs/pii/S092596359700232X]</ref> (which has the closely related [[Cubic crystal system#Zincblende structure|zincblende structure]]) is attributed to the diamond cubic structure.

[[File:Lorimerlite framework.JPG|thumb|150px|Example of a diamond cubic truss system for resisting compression]]
Similarly, [[Physics:Truss|truss]] systems that follow the diamond cubic geometry have a high capacity to withstand compression, by minimizing the unbraced length of individual [[Engineering:Strut|strut]]s.<ref>Lorimer, A. "The Diamond Cubic Truss", Interior World: Design & Detail, vol.121, 2013, pp. 80–81</ref> The diamond cubic geometry has also been considered for the purpose of providing [[Physics:Structural rigidity|structural rigidity]]<ref>R. Kraft. Construction Arrangement, USA, United States Patents, US3139959, 1964 [https://www.google.com/patents/US3139959]</ref><ref>Gilman, J. Tetrahedral Truss, USA, United States Patents, US4446666, 1981 [https://www.google.com/patents/US4446666]</ref> though structures composed of skeletal [[Triangle|triangle]]s, such as the [[Engineering:Space frame|octet truss]], have been found to be more effective for this purpose.

==See also==
*{{annotated link|Chemistry:Allotropes of carbon}}
*{{annotated link|Physics:Crystallography}}
*{{annotated link|Laves graph}}
*{{annotated link|Triakis truncated tetrahedral honeycomb}}

==References==
{{Reflist}}

==External links==
* [http://www.math.rutgers.edu/~vidit/software.html Software] to construct self-avoiding random walks on the diamond cubic lattice

[[Category:Crystal structure types]]
[[Category:Crystallography]]
[[Category:Cubes]]
[[Category:Infinite graphs]]
[[Category:Lattice points]]
[[Category:Minerals in space group 227]]
[[Category:Regular graphs]]

{{Sourceattribution|Diamond cubic}}