Unit cell - Revision history

KevinYager: /* Vectors */

2022-11-14T14:33:08Z

‎Vectors

KevinYager: /* See Also */

2022-11-14T14:29:30Z

‎See Also

KevinYager: /* Reciprocal vectors */

2014-10-16T13:41:22Z

‎Reciprocal vectors

KevinYager at 13:47, 14 October 2014

2014-10-14T13:47:58Z

KevinYager: /* Cubic */

2014-06-18T15:23:18Z

‎Cubic

KevinYager at 14:52, 18 June 2014

2014-06-18T14:52:52Z

68.194.136.6: /* Angles */

2014-06-04T00:25:10Z

‎Angles

68.194.136.6: /* Reciprocal Vectors */

2014-06-04T00:21:33Z

‎Reciprocal Vectors

68.194.136.6 at 00:17, 4 June 2014

2014-06-04T00:17:22Z

68.194.136.6: /* Notation */

2014-06-04T00:16:00Z

‎Notation

@@ Line 11: / Line 11: @@
 ==Mathematical description==
 ===Vectors===
 :<math>\begin{array}{l}
 \mathbf{a} = \begin{bmatrix}
@@ Line 31: / Line 32: @@
 c \frac{ \cos{\alpha} - \cos{\beta}\cos{\gamma} }{\sin{\gamma}} \\
 c \sqrt{  1 - \cos^2{\beta} - \left( \frac{\cos{\alpha} - \cos{\beta}\cos{\gamma}}{\sin{\gamma}} \right)^2 }
 \end{bmatrix}
 \end{array}

← Older revision		Revision as of 14:29, 14 November 2022
Line 180:		Line 180:
	==See Also==		==See Also==
	* [[Lattices]]		* [[Lattices]]
		+	* K. N. Trueblood, H.-B. Bürgi, H. Burzlaff, J. D. Dunitz, C. M. Gramaccioli, H. H. Schulz, U. Shmueli and S. C. Abrahams [https://scripts.iucr.org/cgi-bin/paper?S0108767396005697 Atomic Dispacement Parameter Nomenclature. Report of a Subcommittee on Atomic Displacement Parameter Nomenclature] ''Acta Cryst'' '''1996''', A52, 770-781. [https://doi.org/10.1107/S0108767396005697 doi: 10.1107/S0108767396005697]

@@ Line 88: / Line 88: @@
 \end{alignat}
 </math>
-Where <math>\mathbf{H}_{hkl}</math> is a vector that defines the position of Bragg reflection <math>hkl</math> for the reciprocal-lattice.
+Where <math>\mathbf{H}_{hkl}</math> is a vector that defines the position of [[Bragg's law|Bragg reflection]] <math>hkl</math> for the reciprocal-lattice.
 ==Examples==

← Older revision		Revision as of 13:47, 14 October 2014
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	\end{alignat}		\end{alignat}
	</math>		</math>
		+
		+	==See Also==
		+	* [[Lattices]]

@@ Line 130: / Line 130: @@
    & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi k}{b} \\ \frac{2 \pi l}{c} \end{bmatrix}
 \end{alignat}
 </math>

← Older revision		Revision as of 14:52, 18 June 2014
Line 1:		Line 1:
		+	[[Image:Bcc02-unit cell.png\|thumb\|right\|300px\|Example of the [[Lattice:BCC\|BCC]] unit cell.]]
		+
	The '''unit cell''' is the basic building block of a crystal [[lattice]] (whether an atomic crystal or a nanoscale [[superlattice]]). Crystalline materials have a periodic structure, with the unit cell being the minimal volume necessary to fully describe the repeating structure. There are a finite number of possible symmetries for the repeating unit cell.		The '''unit cell''' is the basic building block of a crystal [[lattice]] (whether an atomic crystal or a nanoscale [[superlattice]]). Crystalline materials have a periodic structure, with the unit cell being the minimal volume necessary to fully describe the repeating structure. There are a finite number of possible symmetries for the repeating unit cell.

@@ Line 53: / Line 53: @@
 * <math>\alpha</math> is the angle between <math>\mathbf{b}</math> and <math>\mathbf{c}</math>
-[[Image:Unit cell01.png|thumb|center|300px|From [http://en.wikipedia.org/wiki/Fractional_coordinates Fractional Coordinates (Wikipedia)]]]
+[[Image:Unit cell01.png|center|thumb|300px|Unit cell definition using parallelepiped with lengths ''a'', ''b'', ''c'' and angles between the sides given by α,β,γ (from Wikipedia [http://en.wikipedia.org/wiki/Fractional_coordinates fractional coordinates]). ]]
 ===Reciprocal vectors===

@@ Line 55: / Line 55: @@
 [[Image:Unit cell01.png|thumb|center|300px|From [http://en.wikipedia.org/wiki/Fractional_coordinates Fractional Coordinates (Wikipedia)]]]
-===Reciprocal Vectors===
+===Reciprocal vectors===
 :<math>
 \begin{alignat}{2}
@@ Line 63: / Line 72: @@
 \end{alignat}
 </math>
+And we can then express the [[momentum transfer]] (<math>\mathbf{q}</math>) in terms of these reciprocal vectors:
 :<math>
 \begin{alignat}{2}
-\mathbf{q}_{hkl} & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\
+\mathbf{q} & = (\mathbf{q}\cdot\mathbf{a})\mathbf{u} + (\mathbf{q}\cdot\mathbf{b})\mathbf{v} + (\mathbf{q}\cdot\mathbf{c})\mathbf{w}
 \end{alignat}
 </math>
-With components:
+Combining with the three Laue equations yields:
 :<math>
 \begin{alignat}{2}
-\mathbf{u} & = ... \\
+\mathbf{q}_{hkl} & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\
-\mathbf{v} & = ... \\
+  & = 2 \pi(h\mathbf{u} + k \mathbf{v} + l \mathbf{w}) \\
-\mathbf{w}
+  & = 2 \pi \mathbf{H}_{hkl}
 \end{alignat}
 </math>
 ==Examples==

@@ Line 1: / Line 1: @@
 The '''unit cell''' is the basic building block of a crystal [[lattice]] (whether an atomic crystal or a nanoscale [[superlattice]]). Crystalline materials have a periodic structure, with the unit cell being the minimal volume necessary to fully describe the repeating structure. There are a finite number of possible symmetries for the repeating unit cell.
-==Math==
+A unit cell can be defined by three vectors that lie along the edges of the enclosing [http://en.wikipedia.org/wiki/Parallelepiped parallelepped]. We denote the vectors as <math>\mathbf{a}</math>, <math>\mathbf{b}</math>, and <math>\mathbf{c}</math>; alternately the unit cell can be described by the lengths of these vectors (<math>a</math>, <math>b</math>, <math>c</math>), and the angles between them:
 ===Vectors===
 :<math>\begin{array}{l}

← Older revision		Revision as of 00:16, 4 June 2014
Line 1:		Line 1:
	The '''unit cell''' is the basic building block of a crystal [[lattice]] (whether an atomic crystal or a nanoscale [[superlattice]]). Crystalline materials have a periodic structure, with the unit cell being the minimal volume necessary to fully describe the repeating structure. There are a finite number of possible symmetries for the repeating unit cell.		The '''unit cell''' is the basic building block of a crystal [[lattice]] (whether an atomic crystal or a nanoscale [[superlattice]]). Crystalline materials have a periodic structure, with the unit cell being the minimal volume necessary to fully describe the repeating structure. There are a finite number of possible symmetries for the repeating unit cell.
−
−	~~==Notation==~~
−	* '''Real space''':
−	** Crystal ''planes'':
−	*** (hkl) denotes a plane of the crystal structure (and repetitions of that plane, with the given spacing). In cubic systems (but not others), the normal to the plane is [hkl]
−	*** {hkl} denotes the set of all planes that are equivalent to (hkl) by the symmetry of the lattice
−	** Crystal ''directions'':
−	*** [hkl] denotes a direction of a vector (in the basis of the direct lattice vectors)
−	*** <math>\left\langle hkl\right\rangle</math> denotes the set of all directions that are equivalent to [hkl] by symmetry (e.g. in cubic system <100> means [100, [010], [001], [-100], [0-10], [00-1])
−	** hkl denotes a diffracting plane
−
−
−	* '''[[Reciprocal space]]''':
−	** Reciprocal ''planes'':
−	*** [hkl] denotes a plane
−	*** <math>\left\langle hkl\right\rangle</math> denotes the set of all planes that are equivalent to [hkl]
−	** Reciprocal ''directions'':
−	*** (hkl) denotes a particular direction (normal to plane (hkl) in real space)
−	*** {hkl} denotes the set of all directions that are equivalent to (hkl)
−	** hkl denotes an indexed reflection

	==Math==		==Math==