138:
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52:
43:
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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
635:
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
626:
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
31:
548:
393:
426:
793:
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
114:, which have a similar structure, with one kind of atom (such as silicon in cristobalite) at the positions of carbon atoms in diamond but with another kind of atom (such as oxygen) halfway between those (see
1223:
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.
768:
295:
431:
599:
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
250:
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
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systems that follow the diamond cubic geometry have a high capacity to withstand compression, by minimizing the unbraced length of individual
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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.
1049:
Combinatorial Image
Analysis: 13th International Workshop, IWCIA 2009, Playa Del Carmen, Mexico, November 24–27, 2009, Proceedings
1298:
253:
994:
Askeland, Donald R.; Phulé, Pradeep
Prabhakar (2006), "Example 3-15: Determining the Packing Factor for Diamond Cubic Silicon",
543:{\displaystyle {\begin{aligned}x=y&=z\ ({\text{mod }}\ 2),\\x+y+z&=0{\text{ or }}1\ ({\text{mod }}\ 4).\end{aligned}}}
301:. Zincblende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms.
80:
is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was
403:
Mathematically, the points of the diamond cubic structure can be given coordinates as a subset of a three-dimensional
304:
The first-, second-, third-, fourth-, and fifth-nearest-neighbor distances in units of the cubic lattice constant are
1101:
Graph
Drawing: 16th International Symposium, GD 2008, Heraklion, Crete, Greece, September 21–24, 2008, Revised Papers
1047:
Nagy, Benedek; Strand, Robin (2009), "Neighborhood sequences in the diamond grid – algorithms with four neighbors",
802:. Moreover, the diamond crystal as a network in space has a strong isotropic property. Namely, for any two vertices
388:{\displaystyle {\tfrac {\sqrt {3}}{4}},{\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {11}}{4}},1,{\tfrac {\sqrt {19}}{4}},}
407:
by using a cubic unit cell four units across. With these coordinates, the points of the structure have coordinates
1313:
1303:
59:
1328:
1318:
141:
Visualisation of a diamond cubic unit cell: 1. Components of a unit cell, 2. One unit cell, 3. A lattice of
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790:
220:
164:. The lattice describes the repeat pattern; for diamond cubic crystals this lattice is "decorated" with a
176:
104:
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158:
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Lorimer, A. "The
Diamond Cubic Truss", Interior World: Design & Detail, vol.121, 2013, pp. 80–81
1021:
Concise
Dictionary of Materials Science: Structure and Characterization of Polycrystalline Materials
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200:
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Parhami, B.; Kwai, Ding-Ming (2001), "A unified formulation of honeycomb and diamond networks",
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The diamond cubic is sometimes called the "diamond lattice" but it is not, mathematically, a
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1115:
1064:
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All of the other points in the structure may be obtained by adding multiples of four to the
224:
1098:(2009), "Isometric Diamond Subgraphs", in Tollis, Ioannis G.; Patrignani, Maurizio (eds.),
297:
significantly smaller (indicating a less dense structure) than the packing factors for the
239:, where each atom has nearest neighbors of an unlike element. Zincblende's space group is F
911:
799:
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404:
228:
161:
126:
70:
1104:, Lecture Notes in Computer Science, vol. 5417, Springer-Verlag, pp. 384–389,
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3m, but many of its structural properties are quite similar to the diamond structure.
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851:
632:
571:
coordinates of these eight points. Adjacent points in this structure are at distance
172:
93:
1137:
51:
1282:
884:. The diamond cubic geometry has also been considered for the purpose of providing
775:
111:
85:
63:
1119:
636:
coordinates may be transformed into three-dimensional coordinates by the formula
110:
in any proportion. There are also crystals, such as the high-temperature form of
1244:
R. Kraft. Construction
Arrangement, USA, United States Patents, US3139959, 1964
1068:
946:
Diamond films: chemical vapor deposition for oriented and heteroepitaxial growth
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893:
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169:
154:
55:
199:
in each dimension. The diamond lattice can be viewed as a pair of intersecting
17:
1256:
1245:
42:
1183:
Topological
Crystallography -With a View Towards Discrete Geometric Analysis-
100:
1271:
1255:
Gilman, J. Tetrahedral Truss, USA, United States
Patents, US4446666, 1981
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771:
847:
96:
81:
1205:
Sunada, Toshikazu (2008), "Crystals that nature might miss creating",
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to construct self-avoiding random walks on the diamond cubic lattice
627:
in coordinate values between any two points (their four-dimensional
1110:
881:
877:
866:
806:
of the crystal net, and for any ordering of the edges adjacent to
136:
107:
830:-edge. Another (hypothetical) crystal with this property is the
969:
Wiberg, Egon; Wiberg, Nils; Holleman, Arnold
Frederick (2001),
89:
62:
of the diamond lattice showing the 3-fold symmetry along the
763:{\displaystyle (a,b,c,d)\to (a+b-c-d,\ a-b+c-d,\ -a+b+c-d).}
290:{\displaystyle {\tfrac {\pi {\sqrt {3}}}{16}}\approx 0.34,}
896:, have been found to be more effective for this purpose.
129:
in the technical sense of this word used in mathematics.
774:
subset of the four-dimensional integer lattice, it is a
603:
of units across by multiplying all coordinates by
369:
346:
329:
312:
258:
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of the width of the unit cell in each dimension. Many
1153:
IEEE Transactions on Parallel and Distributed Systems
1055:, vol. 5852, Springer-Verlag, pp. 109–121,
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429:
310:
256:
944:Kobashi, Koji (2005), "2.1 Structure of diamond",
762:
589:
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858:) is attributed to the diamond cubic structure.
914: – Scientific study of crystal structures
814:, there is a net-preserving congruence taking
299:face-centered and body-centered cubic lattices
8:
47:3D ball-and-stick model of a diamond lattice
810:and any ordering of the edges adjacent to
1109:
908: – Materials made only out of carbon
846:The compressive strength and hardness of
838:crystal, (10,3)-a, or the diamond twin).
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924:Triakis truncated tetrahedral honeycomb
888:though structures composed of skeletal
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770:Because the diamond structure forms a
149:Diamond's cubic structure is in the Fd
1090:
1088:
850:and various other materials, such as
157:(space group 227), which follows the
88:also adopt this structure, including
7:
116:Category:Minerals in space group 227
34:Rotating model of the diamond cubic
631:) gives the number of edges in the
560:(0,0,0), (0,2,2), (2,0,2), (2,2,0),
926: – Space-filling tessellation
563:(3,3,3), (3,1,1), (1,3,1), (1,1,3)
556:4) that satisfy these conditions:
25:
1053:Lecture Notes in Computer Science
854:, (which has the closely related
203:lattices, with each separated by
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998:, Cengage Learning, p. 82,
973:, Academic Press, p. 1300,
826:-edge to the similarly ordered
920: – Periodic spatial graph
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1120:10.1007/978-3-642-00219-9_37
1324:Minerals in space group 227
1069:10.1007/978-3-642-10210-3_9
865:Example of a diamond cubic
590:{\displaystyle {\sqrt {3}}}
1345:
1019:Novikov, Vladimir (2003),
133:Crystallographic structure
125:, this structure is not a
121:Although often called the
423:satisfying the equations
27:Type of crystal structure
1023:, CRC Press, p. 9,
552:There are eight points (
60:stereographic projection
1299:Crystal structure types
948:, Elsevier, p. 9,
221:compound semiconductors
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237:zincblende structure
235:adopt the analogous
195:of the width of the
84:, other elements in
1061:2009LNCS.5852..109N
971:Inorganic chemistry
886:structural rigidity
772:distance-preserving
201:face-centered cubic
159:face-centered cubic
1207:Notices of the AMS
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834:(also called the K
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1192:978-4-431-54176-9
1179:Sunada, Toshikazu
1165:10.1109/71.899940
1129:978-3-642-00219-9
1078:978-3-642-10210-3
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122:
120:
112:cristobalite
74:
68:
918:Laves graph
894:octet truss
876:Similarly,
871:compression
832:Laves graph
155:space group
56:Pole figure
1293:Categories
931:References
796:congruence
145:unit cells
1213:: 208–215
1111:0807.2218
890:triangles
822:and each
749:−
731:−
719:−
707:−
692:−
686:−
671:→
521:mod
461:mod
279:≈
262:π
197:unit cell
143:3 Ă— 3 Ă— 3
101:germanium
64:direction
1283:Software
1181:(2012),
1138:14066610
900:See also
223:such as
86:group 14
1057:Bibcode
848:diamond
787:lattice
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569:x, y, z
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168:of two
127:lattice
97:silicon
82:diamond
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882:struts
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554:modulo
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231:, and
173:bonded
108:alloys
103:, and
92:, the
73:, the
1309:Cubes
1134:S2CID
1106:arXiv
878:truss
867:truss
166:motif
90:α-tin
1187:ISBN
1124:ISBN
1073:ISBN
1025:ISBN
1000:ISBN
975:ISBN
950:ISBN
804:x, y
282:0.34
246:The
227:, β-
99:and
1161:doi
1116:doi
1065:doi
818:to
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58:in
1295::
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1087:^
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471:)
468:2
457:(
451:z
448:=
441:y
438:=
435:x
421:)
419:z
415:y
411:x
409:(
383:,
377:4
366:,
363:1
360:,
354:4
343:,
337:2
333:2
326:,
320:4
316:3
285:,
267:3
241:4
214:4
211:/
208:1
190:4
187:/
184:1
151:3
20:)
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