Ernst Mally - Revision history

en>1982vdven at 22:52, 6 October 2014

2014-10-06T22:52:13Z

← Older revision		Revision as of 00:52, 7 October 2014
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	~~[[File:Diamond Cubic-F lattice animation.gif\|thumb\|250px\|right\|Unit cell of the diamond cubic [[crystal structure]]]]~~		Name: Marlon Alanson<br>Age: 36 years old<br>Country: Italy<br>City: Reggio Calabria <br>Post code: 89125<br>Address: Via Giulio Camuzzoni 12<br><br>Also visit my blog - [http://Www.ddlion.com/?p=569 fifa 15 Coin Generator]
	~~[[File~~:~~DiamondPoleFigure111.png\|thumb\|250px\|[[Pole figure]] in [[stereographic projection]] of the diamond lattice showing the 3-fold symmetry along the [[Miller index\|[111] direction]].]]~~

	The '''diamond cubic''' [[crystal structure]] is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was [[diamond]], other elements in [[Carbon group\|group 14]] also adopt this structure, including [[tin\|α-tin]], the [[semiconductor]]s [[silicon]] and [[germanium]], and silicon/germanium [[alloy]]s in any proportion.

	~~==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 3 x 3 x 3 unit cells]]~~
	Diamond cubic is in the Fd{{overline\|3}}m [[space group]], which follows the face-centered [[Cubic crystal system\|cubic]] [[bravais lattice]]. The lattice describes the repeat pattern; for diamond cubic crystals this lattice is "decorated" with a ''motif'' of two [[tetrahedron\|tetrahedrally]] [[Chemical bond\|bonded]] atoms in each [[Wigner–Seitz cell\|primitive cell]], separated by 1/4 of the width of the [[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> Many [[compound semiconductor]]s such as [[gallium arsenide]], β-[[silicon carbide]] and [[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 [[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>\frac{\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~~> ~~Zinc blende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms.~~

	The first, second, third and fourth nearest-neighbor distances in units of the cubic lattice constant are <math>\frac{\sqrt{3}}{4}</math>, <math>\frac{\sqrt{2}}{2}</math>, <math>\frac{\sqrt{11}}{4}</math>, and <math>1</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]] by using a cubical unit cell four units across. With these coordinates, the points of the structure have coordinates (''x'', ''y'', ''z'') satisfying the equations
	~~:''x'' = ''y'' = ''z'' (mod 2), and~~
	:~~''x'' + ''y'' + ''z'' = 0 or 1 (mod 4).~~<~~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 }}.</ref>
	~~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'', and ''z'' coordinates of these eight points. 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. 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 such that sum to 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
	:(''a'', ''b'', ''c'', ''d'') → (''a'' + ''b'' − ''c'' − ''d'', ''a'' − ''b'' + ''c'' − ''d'', −''a'' + ''b'' + ''c'' − ''d'').<ref name="ny09"/><ref name="e09">{{citation\|contribution=Isometric diamond subgraphs\|first=David\|last=Eppstein\|authorlink=David Eppstein\|arxiv=0807.2218\|title=[[International Symposium on Graph Drawing\|Proc. 16th International Symposium on Graph Drawing, Heraklion, Crete, 2008]]\|series=Lecture Notes in Computer Science\|volume=5417\|year=2009\|pages=384–389\|publisher=Springer-Verlag\|doi=10.1007/978-3-642-00219-9_37}}.</ref>
	~~Because the diamond structure forms a [[Isometry\|distance-preserving]] subset of the four-dimensional integer lattice, it is a [[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 [[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]].

	~~==Manufacturing considerations==~~
	~~{{unreferenced section\|date=March 2011}}~~
	~~[[File:Diamond structure.png\|framed\|left\|A diamond cubic crystal viewed from a [[Crystallography#Notation\|<110>]] direction.]]~~

	Since materials with diamond or zinc blende structures are important for formation of [[Wafer (electronics)\|wafers]] used in the fabrication of modern [[electronics]], it is important to know that they present open, hexagonal [[ion channel]]s when [[ion implantation]] is carried out from any of the <~~110~~> ~~directions (that is, 45° from one of the cube edges). Their open structure also results in a volume reduction upon melting or [[amorphous solid\|amorphization]], as is also seen in [[ice]].~~

	They display [[octahedron\|octahedral]] [[cleavage (crystal)\|cleavage]], which means that they have four planes—directions following the faces of the octahedron where there are fewer [[Chemical bond\|bonds]] and therefore points of structural weakness—along which single crystals can easily split, leaving smooth surfaces. Similarly, this lack of bonds can guide chemical [[industrial etching\|etching]] of the right chemistry (i.e., [[potassium hydroxide]] solutions for Si) to produce pyramidal structures such as mesas, points, or etch pits, a useful technique for [[microelectromechanical systems]].
	~~{{Clear}}~~

	~~==See also==~~
	*[[Crystallography]]
	*[[Triakis truncated tetrahedral honeycomb]]

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	*{{Commons-~~inline\|Diamond cubic}}~~
	* [http://~~octettruss~~.~~kilu~~.de/~~diamond.html diamond 3D animation]~~
	* [http://www.math.rutgers.edu/~vidit/software.html Software] to construct self avoiding random walks on the diamond cubic lattice

	~~{{DEFAULTSORT:Diamond Cubic}}~~
	~~[[Category:Articles containing video clips]]~~
	~~[[Category:Crystallography]]~~
	~~[[Category:Lattice points]]~~
	~~[[Category:Cubes]~~]

en>Ohconfucius: Script-assisted fixes: per MOS:NUM, MOS:CAPS, MOS:LINK

2013-10-20T13:10:43Z

Script-assisted fixes: per MOS:NUM, MOS:CAPS, MOS:LINK

← Older revision		Revision as of 15:10, 20 October 2013
Line 1:		Line 1:
	~~I am Tamie from Annemasse~~. ~~I am learning~~ to ~~play~~ the ~~Post horn~~. ~~Other hobbies~~ are ~~People watching~~.<br><br>~~Check out my~~ page: [~~http~~://~~plmodel~~.vn/vi/36-~~thi~~-~~sinh~~/~~detail~~/1/13.html ~~how~~ to ~~get free fifa 15 coins~~]		[[File:Diamond Cubic-F lattice animation.gif\|thumb\|250px\|right\|Unit cell of the diamond cubic [[crystal structure]]]]
			[[File:DiamondPoleFigure111.png\|thumb\|250px\|[[Pole figure]] in [[stereographic projection]] of the diamond lattice showing the 3-fold symmetry along the [[Miller index\|[111] direction]].]]

			The '''diamond cubic''' [[crystal structure]] is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was [[diamond]], other elements in [[Carbon group\|group 14]] also adopt this structure, including [[tin\|α-tin]], the [[semiconductor]]s [[silicon]] and [[germanium]], and silicon/germanium [[alloy]]s in any proportion.

			==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 3 x 3 x 3 unit cells]]
			Diamond cubic is in the Fd{{overline\|3}}m [[space group]], which follows the face-centered [[Cubic crystal system\|cubic]] [[bravais lattice]]. The lattice describes the repeat pattern; for diamond cubic crystals this lattice is "decorated" with a ''motif'' of two [[tetrahedron\|tetrahedrally]] [[Chemical bond\|bonded]] atoms in each [[Wigner–Seitz cell\|primitive cell]], separated by 1/4 of the width of the [[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> Many [[compound semiconductor]]s such as [[gallium arsenide]], β-[[silicon carbide]] and [[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 [[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>\frac{\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> Zinc blende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms.

			The first, second, third and fourth nearest-neighbor distances in units of the cubic lattice constant are <math>\frac{\sqrt{3}}{4}</math>, <math>\frac{\sqrt{2}}{2}</math>, <math>\frac{\sqrt{11}}{4}</math>, and <math>1</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]] by using a cubical unit cell four units across. With these coordinates, the points of the structure have coordinates (''x'', ''y'', ''z'') satisfying the equations
			:''x'' = ''y'' = ''z'' (mod 2), and
			:''x'' + ''y'' + ''z'' = 0 or 1 (mod 4).<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 }}.</ref>
			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'', and ''z'' coordinates of these eight points. 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. 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 such that sum to 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
			:(''a'', ''b'', ''c'', ''d'') → (''a'' + ''b'' − ''c'' − ''d'', ''a'' − ''b'' + ''c'' − ''d'', −''a'' + ''b'' + ''c'' − ''d'').<ref name="ny09"/><ref name="e09">{{citation\|contribution=Isometric diamond subgraphs\|first=David\|last=Eppstein\|authorlink=David Eppstein\|arxiv=0807.2218\|title=[[International Symposium on Graph Drawing\|Proc. 16th International Symposium on Graph Drawing, Heraklion, Crete, 2008]]\|series=Lecture Notes in Computer Science\|volume=5417\|year=2009\|pages=384–389\|publisher=Springer-Verlag\|doi=10.1007/978-3-642-00219-9_37}}.</ref>
			Because the diamond structure forms a [[Isometry\|distance-preserving]] subset of the four-dimensional integer lattice, it is a [[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 [[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]].

			==Manufacturing considerations==
			{{unreferenced section\|date=March 2011}}
			[[File:Diamond structure.png\|framed\|left\|A diamond cubic crystal viewed from a [[Crystallography#Notation\|<110>]] direction.]]

			Since materials with diamond or zinc blende structures are important for formation of [[Wafer (electronics)\|wafers]] used in the fabrication of modern [[electronics]], it is important to know that they present open, hexagonal [[ion channel]]s when [[ion implantation]] is carried out from any of the <110> directions (that is, 45° from one of the cube edges). Their open structure also results in a volume reduction upon melting or [[amorphous solid\|amorphization]], as is also seen in [[ice]].

			They display [[octahedron\|octahedral]] [[cleavage (crystal)\|cleavage]], which means that they have four planes—directions following the faces of the octahedron where there are fewer [[Chemical bond\|bonds]] and therefore points of structural weakness—along which single crystals can easily split, leaving smooth surfaces. Similarly, this lack of bonds can guide chemical [[industrial etching\|etching]] of the right chemistry (i.e., [[potassium hydroxide]] solutions for Si) to produce pyramidal structures such as mesas, points, or etch pits, a useful technique for [[microelectromechanical systems]].
			{{Clear}}

			==See also==
			*[[Crystallography]]
			*[[Triakis truncated tetrahedral honeycomb]]

			==References==
			{{reflist}}

			==External links==
			*{{Commons-inline\|Diamond cubic}}
			* [http://octettruss.kilu.de/diamond.html diamond 3D animation]
			* [http://www.math.rutgers.edu/~vidit/software.html Software] to construct self avoiding random walks on the diamond cubic lattice

			{{DEFAULTSORT:Diamond Cubic}}
			[[Category:Articles containing video clips]]
			[[Category:Crystallography]]
			[[Category:Lattice points]]
			[[Category:Cubes]]

en>RjwilmsiBot: /* External links */Adding Persondata using AWB (7393)

2010-11-17T23:10:37Z

External links: Adding Persondata using AWB (7393)

New page

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