Lattice:Hexagonal diamond - Revision history

KevinYager: /* Particle Positions */

2018-01-09T15:08:15Z

‎Particle Positions

KevinYager: /* Particle Positions */

2018-01-09T14:43:50Z

‎Particle Positions

KevinYager: /* Alternating Hexagonal Diamond */

2014-10-14T15:48:26Z

‎Alternating Hexagonal Diamond

KevinYager: /* See Also */

2014-10-14T15:47:52Z

‎See Also

KevinYager at 15:44, 14 October 2014

2014-10-14T15:44:39Z

KevinYager at 15:44, 14 October 2014

2014-10-14T15:44:02Z

KevinYager: Created page with "The '''hexagonal diamond''' unit cell is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional Lattice:diamond diamond (a.k.a..."

2014-10-14T15:43:00Z

Created page with "The '''hexagonal diamond''' unit cell is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional Lattice:diamond diamond (a.k.a..."

New page

The '''hexagonal diamond''' unit cell is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional [[Lattice:diamond diamond]] (a.k.a. cubic diamond) exists within a cubic unit cell, hexagonal diamond exists within a hexagonal unit cell. In both cases, elements are bonded tetrahdrally. However, in cubic diamond, the six-membered rings are all in the [http://en.wikipedia.org/wiki/Chair_conformation chair conformation], whereas in hexagonal diamond, some six-membered rings are in the [http://en.wikipedia.org/wiki/Boat_conformation boat conformation].

[[Image:Lonsdaleite unit cell.png|thumb|500px|center|The four distinct positions in the unit cell are each given a different color. From: "Structural and vibrational properties of the 6H diamond: First-principles study" [http://dx.doi.org/10.1016/j.diamond.2006.03.013 doi: 10.1016/j.diamond.2006.03.013] ]]

==Canonical Hexagonal Diamond==
A canonical hexagonal diamond lattice (single atom/particle type arranged as shown above) has symmetry ''Fd3m''.
===Symmetry===
* Crystal Family: Hexagonal
* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: Hexagonal
* [http://en.wikipedia.org/wiki/Bravais_lattice#Bravais_lattices_in_3_dimensions Bravais Lattice]: hexagonal
* Crystal class: Hexoctahedral
* Space Group: P6<sub>3</sub>/mmc
* Particles per unit cell: <math>n=4</math>
* Volume of unit cell: <math>V_d=a^2 c \sin(60^{\circ}) = a^2 c \frac{\sqrt{3}}{2}</math>
* Dimensionality: <math>d=3</math>

===Structure===
TBD
===Particle Positions===
There are 14 positions. In total there are 4 particles in the unit cell.
====Fractional====
Positions are given in terms of fractional coordinates relative to the unit-cell edge-vectors:
::: <math>\mathbf{a}=(a,0,0)</math>
::: <math>\mathbf{b}=(b/2,\sqrt{3}b/2,0)</math>
::: <math>\mathbf{c}=(0,0,c)</math>
* <math> 4 \, \mathrm{bottom\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{1}{2}</math>
** <math>\left(0,0,0\right),(1,0,0),(0,1,0),(1,1,0)</math>
* <math> 4 \, \mathrm{mid\,\, layer}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 1</math>
** <math>\left(0,0,\frac{5}{8}\right),\left(1,0,\frac{5}{8}\right),\left(0,1,\frac{5}{8}\right),\left(1,1,\frac{5}{8}\right)</math>
* <math> 2 \, \mathrm{internal\,\, strut}: \, 1 + 1 = 2</math>
** <math>\left(\frac{1}{3},\frac{1}{3},\frac{1}{8}\right),\left(\frac{1}{3},\frac{1}{3},\frac{4}{8}\right)</math>
* <math> 4 \, \mathrm{top\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{1}{2}</math>
** <math>\left(0,0,1\right),(1,0,1),(0,1,1),(1,1,1)</math>

====Absolute====
* <math> 4 \, \mathrm{bottom\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{1}{2}</math>
** <math>\left(0,0,0\right),(a,0,0),\left(\frac{b}{2},\frac{\sqrt{3}b}{2},0\right),\left(a+\frac{b}{2},\frac{\sqrt{3}b}{2},0\right)</math>
* <math> 4 \, \mathrm{mid\,\, layer}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 1</math>
** <math>\left(0,0,\frac{5c}{8}\right),\left(a,0,\frac{5c}{8}\right),\left(\frac{b}{2},\frac{\sqrt{3}b}{2},\frac{5c}{8}\right),\left(a+\frac{b}{2},\frac{\sqrt{3}b}{2},\frac{5c}{8}\right)</math>
* <math> 2 \, \mathrm{internal\,\, strut}: \, 1 + 1 = 2</math>
** <math>\left(\frac{a}{2},\frac{b}{2\sqrt{3}},\frac{c}{8}\right),\left(\frac{a}{2},\frac{b}{2\sqrt{3}},\frac{4c}{8}\right)</math>
* <math> 4 \, \mathrm{top\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{1}{2}</math>
** <math>\left(0,0,c\right),(a,0,c),\left(\frac{b}{2},\frac{\sqrt{3}b}{2},c\right),\left(a+\frac{b}{2},\frac{\sqrt{3}b}{2},c\right)</math>

====Distances====
For a particle-particle bond-length of <math>l</math>:
:<math>a=\frac{2\sqrt{6}}{3}l</math>
:<math>c=\frac{8}{3}l</math>
:<math>\frac{a}{c}=\frac{\sqrt{6}}{4}\approx 0.61237</math>
:<math>\frac{c}{a}=\frac{4}{\sqrt{6}}\approx 1.63299</math>

===Examples===
====Atomics====
* [http://en.wikipedia.org/wiki/Lonsdaleite Lonsdaleite] form of [http://en.wikipedia.org/wiki/Carbon carbon] (C), also known as '''hexagonal diamond''', 2H diamond, or 'sp<sup>3</sup> diamond' (a = 2.51 Å, c = 4.12 Å)

==Alternating Hexagonal Diamond==
This is the [http://en.wikipedia.org/wiki/Wurtzite_%28crystal_structure%29 Wurtzite] crystal structure, a hexgonal unit cell with alternating species.
===Examples===
====Atomics====
* [http://en.wikipedia.org/wiki/Wurtzite Wurtzite] (Zn,Fe)S (a = b = 3.82 Å, c = 6.26 Å)

==Along Connections==
This lattice can be thought of as the hexagonal-dimaond analog of the cubic-diamond [[Lattice:Diamond#Cristobalite|cristobalite]]. Here, a four-bonded species occupies all the sites of the canonical hexagonal diamond lattice, and a two-bonded species sits along each of the connections between these tetrahedral sites.

===Particle Positions===

====Particle Type A (bond tetrahedrally)====
These are the same positions as the canonical hexagonal diamond.

====Particle Type B (two-fold bonded)====
There are 11 positions. In total there are 8 particles of this type in the unit cell.
====Fractional====
Positions are given in terms of fractional coordinates relative to the unit-cell edge-vectors:
::: <math>\mathbf{a}=(a,0,0)</math>
::: <math>\mathbf{b}=(b/2,\sqrt{3}b/2,0)</math>
::: <math>\mathbf{c}=(0,0,c)</math>
* <math> 3 \, \mathrm{lower\,\,tripod}: \, 1 + 1 + 1 = 3</math>
** <math>\left(\frac{1}{6},\frac{1}{6},\frac{1}{16}\right),\left(\frac{4}{6},\frac{1}{6},\frac{1}{16}\right),\left(\frac{1}{6},\frac{4}{6},\frac{1}{16}\right)</math>
* <math> 1 \, \mathrm{mid\,\,strut}: \, 1 = 1</math>
** <math>\left(\frac{1}{3},\frac{1}{3},\frac{5}{16}\right)</math>
* <math> 3 \, \mathrm{midlevel\,\,tripod}: \, 1 + 1 + 1 = 3</math>
** <math>\left(\frac{1}{6},\frac{1}{6},\frac{9}{16}\right),\left(\frac{4}{6},\frac{1}{6},\frac{9}{16}\right),\left(\frac{1}{6},\frac{4}{6},\frac{9}{16}\right)</math>
* <math> 4 \, \mathrm{upper\,\,connections}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 1</math>
** <math>\left(0,0,\frac{13}{16}\right),\left(1,0,\frac{13}{16}\right),\left(0,1,\frac{13}{16}\right),\left(1,1,\frac{13}{16}\right)</math>

==See Also==
* [http://en.wikipedia.org/wiki/Lonsdaleite Lonsdaleite (Wikipedia)]
* [[Lattice:Diamond| Cubic Diamond Lattice]]

← Older revision		Revision as of 15:44, 14 October 2014
Line 1:		Line 1:
−	The '''hexagonal diamond''' [[~~unit cell~~]] is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional [[Lattice:Diamond\|diamond]] (a.k.a. cubic diamond) exists within a cubic unit cell, hexagonal diamond exists within a hexagonal unit cell. In both cases, elements are bonded tetrahdrally. However, in cubic diamond, the six-membered rings are all in the [http://en.wikipedia.org/wiki/Chair_conformation chair conformation], whereas in hexagonal diamond, some six-membered rings are in the [http://en.wikipedia.org/wiki/Boat_conformation boat conformation].	+	The '''hexagonal diamond''' [[lattice]] is an arrangement of tetrahedrally-bonded elements, within a [[Lattice:Hexagonal\|hexagonal]] [[unit cell]]. Whereas conventional [[Lattice:Diamond\|diamond]] (a.k.a. cubic diamond) exists within a cubic unit cell, hexagonal diamond exists within a hexagonal unit cell. In both cases, elements are bonded tetrahdrally. However, in cubic diamond, the six-membered rings are all in the [http://en.wikipedia.org/wiki/Chair_conformation chair conformation], whereas in hexagonal diamond, some six-membered rings are in the [http://en.wikipedia.org/wiki/Boat_conformation boat conformation].

	[[Image:Lonsdaleite unit cell.png\|thumb\|500px\|center\|The four distinct positions in the unit cell are each given a different color. From: "Structural and vibrational properties of the 6H diamond: First-principles study" [http://dx.doi.org/10.1016/j.diamond.2006.03.013 doi: 10.1016/j.diamond.2006.03.013] ]]		[[Image:Lonsdaleite unit cell.png\|thumb\|500px\|center\|The four distinct positions in the unit cell are each given a different color. From: "Structural and vibrational properties of the 6H diamond: First-principles study" [http://dx.doi.org/10.1016/j.diamond.2006.03.013 doi: 10.1016/j.diamond.2006.03.013] ]]

← Older revision		Revision as of 15:44, 14 October 2014
Line 1:		Line 1:
−	The '''hexagonal diamond''' unit cell is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional [[Lattice:~~diamond~~ diamond]] (a.k.a. cubic diamond) exists within a cubic unit cell, hexagonal diamond exists within a hexagonal unit cell. In both cases, elements are bonded tetrahdrally. However, in cubic diamond, the six-membered rings are all in the [http://en.wikipedia.org/wiki/Chair_conformation chair conformation], whereas in hexagonal diamond, some six-membered rings are in the [http://en.wikipedia.org/wiki/Boat_conformation boat conformation].	+	The '''hexagonal diamond''' [[unit cell]] is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional [[Lattice:Diamond\|diamond]] (a.k.a. cubic diamond) exists within a cubic unit cell, hexagonal diamond exists within a hexagonal unit cell. In both cases, elements are bonded tetrahdrally. However, in cubic diamond, the six-membered rings are all in the [http://en.wikipedia.org/wiki/Chair_conformation chair conformation], whereas in hexagonal diamond, some six-membered rings are in the [http://en.wikipedia.org/wiki/Boat_conformation boat conformation].

	[[Image:Lonsdaleite unit cell.png\|thumb\|500px\|center\|The four distinct positions in the unit cell are each given a different color. From: "Structural and vibrational properties of the 6H diamond: First-principles study" [http://dx.doi.org/10.1016/j.diamond.2006.03.013 doi: 10.1016/j.diamond.2006.03.013] ]]		[[Image:Lonsdaleite unit cell.png\|thumb\|500px\|center\|The four distinct positions in the unit cell are each given a different color. From: "Structural and vibrational properties of the 6H diamond: First-principles study" [http://dx.doi.org/10.1016/j.diamond.2006.03.013 doi: 10.1016/j.diamond.2006.03.013] ]]

@@ Line 52: / Line 52: @@
 :<math>\frac{a}{c}=\frac{\sqrt{6}}{4}\approx 0.61237</math>
 :<math>\frac{c}{a}=\frac{4}{\sqrt{6}}\approx 1.63299</math>
 ===Examples===

@@ Line 48: / Line 48: @@
 For a particle-particle bond-length of <math>l</math>:
 :<math>a=\frac{2\sqrt{6}}{3}l</math>
 :<math>c=\frac{8}{3}l</math>
 :<math>\frac{a}{c}=\frac{\sqrt{6}}{4}\approx 0.61237</math>

@@ Line 57: / Line 57: @@
 ==Alternating Hexagonal Diamond==
-This is the [http://en.wikipedia.org/wiki/Wurtzite_%28crystal_structure%29 Wurtzite] crystal structure, a hexgonal unit cell with alternating species.
+This is the [[Lattice:Wurtzite|Wurtzite]] crystal structure, a hexagonal unit cell with alternating species.
 ===Examples===
 ====Atomics====
 * [http://en.wikipedia.org/wiki/Wurtzite Wurtzite] (Zn,Fe)S (a = b = 3.82 Å, c = 6.26 Å)
 ==Along Connections==

@@ Line 88: / Line 88: @@
 ==See Also==
-* [http://en.wikipedia.org/wiki/Lonsdaleite Lonsdaleite (Wikipedia)]
+* [http://en.wikipedia.org/wiki/Lonsdaleite Wikipedia: Lonsdaleite]
 * [[Lattice:Diamond| Cubic Diamond Lattice]]