Difference between revisions of "Unit cell"

Latest revision as of 09:33, 14 November 2022

Example of the 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.

A unit cell can be defined by three vectors that lie along the edges of the enclosing parallelepped. We denote the vectors as $\mathbf {a}$ , $\mathbf {b}$ , and $\mathbf {c}$ ; alternately the unit cell can be described by the lengths of these vectors ( $a$ , $b$ , $c$ ), and the angles between them:

\alpha

, the angle between

b

and

c

\beta

, the angle between

a

and

c

\gamma

, the angle between

a

and

b

Mathematical description

Vectors

There are many ways to define the Cartesian basis for the unit cell in real-space. A typical definition is:

{\begin{array}{l}\mathbf {a} ={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} ={\begin{bmatrix}b\cos {\gamma }\\b\sin {\gamma }\\0\end{bmatrix}}\\\mathbf {c} ={\begin{bmatrix}c\sin {\theta _{c}}\cos {\phi _{c}}\\c\sin {\theta _{c}}\sin {\phi _{c}}\\c\cos {\theta _{c}}\end{bmatrix}}={\begin{bmatrix}c\cos {\beta }\\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}}

There are many mathematically equivalent ways to express a given definition. For instance, the vector $\mathbf {c}$ can also be written as (c.f. these notes and Trueblood et al. Acta Cryst 1996, A52, 770-781):

{\begin{array}{l}\mathbf {c} ={\begin{bmatrix}c\cos {\beta }\\-c\sin \beta \cos \alpha ^{*}\\{\frac {1}{c^{*}}}\\\end{bmatrix}}={\begin{bmatrix}c\cos {\beta }\\c{\frac {\cos \alpha -\cos \beta \cos \gamma }{\sin \gamma }}\\{\frac {c}{\sin \gamma }}{\sqrt {1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma }}\\\end{bmatrix}}\end{array}}

Relations

\mathbf {a} \cdot \mathbf {b} =ab\cos {\gamma }

\mathbf {a} \cdot \mathbf {c} =ac\cos {\beta }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{b} \cdot \mathbf{c} = b c \cos{\alpha}}

Volume

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = |\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})| = |\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})|}

If a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges, the volume is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = a b c \sqrt{1+2\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)}. }

The volume of a unit cell with all edge-length equal to unity is:

v={\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}

Angles

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is the angle between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}} and $\mathbf {b}$
$\beta$ is the angle between $\mathbf {a}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{c}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is the angle between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{b}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{c}}

Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α,β,γ (from Wikipedia fractional coordinates).

Reciprocal vectors

The repeating structure of a unit cell creates peaks in reciprocal space. In particular, we observe maxima (constructive interference) when:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{q} \cdot \mathbf{a} & = 2 \pi h \\ \mathbf{q} \cdot \mathbf{b} & = 2 \pi k \\ \mathbf{q} \cdot \mathbf{c} & = 2 \pi l \\ \end{alignat} }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} are integers. We define reciprocal-space vectors:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{u} & = \frac{\mathbf{b}\times\mathbf{c}}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c}) } = \frac{1}{V} \mathbf{b}\times\mathbf{c} \\ \mathbf{v} & = \frac{\mathbf{c}\times\mathbf{a}}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c}) } =\frac{1}{V} \mathbf{c}\times\mathbf{a} \\ \mathbf{w} & = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c}) } =\frac{1}{V} \mathbf{a}\times\mathbf{b} \\ \end{alignat} }

And we can then express the momentum transfer (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}} ) in terms of these reciprocal vectors:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \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} }

Combining with the three Laue equations yields:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{q}_{hkl} & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\ & = 2 \pi(h\mathbf{u} + k \mathbf{v} + l \mathbf{w}) \\ & = 2 \pi \mathbf{H}_{hkl} \end{alignat} }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{H}_{hkl}} is a vector that defines the position of Bragg reflection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle hkl} for the reciprocal-lattice.

Examples

Cubic

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=\beta=\gamma=90^{\circ}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=abc} , and:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{a} & = \begin{bmatrix} a \\ 0 \\ 0 \end{bmatrix} \\ \mathbf{b} & = \begin{bmatrix} 0 \\ b \\ 0 \end{bmatrix} \\ \mathbf{c} & = \begin{bmatrix} 0 \\ 0 \\ c \end{bmatrix} \\ \end{alignat} }

And in reciprocal-space:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{u} & =\frac{1}{V} \mathbf{b}\times\mathbf{c} & =\frac{1}{V} \begin{bmatrix} b c \\ 0 \\ 0 \end{bmatrix} & = \begin{bmatrix} \frac{1}{a} \\ 0 \\ 0 \end{bmatrix}\\ \mathbf{v} & =\frac{1}{V} \mathbf{c}\times\mathbf{a} & =\frac{1}{V} \begin{bmatrix} 0 \\ a c \\ 0 \end{bmatrix} & = \begin{bmatrix} 0 \\ \frac{1}{b} \\ 0 \end{bmatrix}\\ \mathbf{w} & =\frac{1}{V} \mathbf{a}\times\mathbf{b} & =\frac{1}{V} \begin{bmatrix} 0 \\ 0 \\ a b \end{bmatrix} & = \begin{bmatrix} 0 \\ 0 \\ \frac{1}{c} \end{bmatrix}\\ \end{alignat} }

So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{q}_{hkl} & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\ & = (2 \pi h)\begin{bmatrix} \frac{1}{a} \\ 0 \\ 0 \end{bmatrix} + (2 \pi k)\begin{bmatrix} 0 \\ \frac{1}{b} \\ 0 \end{bmatrix} + (2 \pi l)\begin{bmatrix} 0 \\ 0 \\ \frac{1}{c} \end{bmatrix} \\ & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi k}{b} \\ \frac{2 \pi l}{c} \end{bmatrix} \end{alignat} }

And:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{hkl} = 2\pi \sqrt{ \left( \frac{h}{a} \right)^2 + \left( \frac{k}{b} \right)^2 + \left( \frac{l}{c} \right)^2 } }

Hexagonal

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=\beta=90^{\circ}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=60^{\circ}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\frac{\sqrt{3}}{2}abc} , and:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{a} & = \begin{bmatrix} a \\ 0 \\ 0 \end{bmatrix} \\ \mathbf{b} & = \begin{bmatrix} \frac{1}{2}b \\ \frac{\sqrt{3}}{2} b \\ 0 \end{bmatrix} \\ \mathbf{c} & = \begin{bmatrix} 0 \\ 0 \\ c \end{bmatrix} \\ \end{alignat} }

And in reciprocal-space:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{u} & =\frac{1}{V} \mathbf{b}\times\mathbf{c} & =\frac{1}{V} \begin{bmatrix} \frac{\sqrt{3}}{2} b c \\ -\frac{1}{2} b c \\ 0 \end{bmatrix} & = \begin{bmatrix} \frac{1}{a} \\ \frac{1}{\sqrt{3}a} \\ 0 \end{bmatrix}\\ \mathbf{v} & =\frac{1}{V} \mathbf{c}\times\mathbf{a} & =\frac{1}{V} \begin{bmatrix} 0 \\ a c \\ 0 \end{bmatrix} & = \begin{bmatrix} 0 \\ \frac{2}{\sqrt{3}b} \\ 0 \end{bmatrix}\\ \mathbf{w} & =\frac{1}{V} \mathbf{a}\times\mathbf{b} & =\frac{1}{V} \begin{bmatrix} 0 \\ 0 \\ \frac{\sqrt{3}}{2} a b \end{bmatrix} & = \begin{bmatrix} 0 \\ 0 \\ \frac{1}{c} \end{bmatrix}\\ \end{alignat} }

So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{q}_{hkl} & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\ & = (2 \pi h)\begin{bmatrix} \frac{1}{a} \\ \frac{1}{\sqrt{3}a} \\ 0 \end{bmatrix} + (2 \pi k)\begin{bmatrix} 0 \\ \frac{2}{\sqrt{3}b} \\ 0 \end{bmatrix} + (2 \pi l)\begin{bmatrix} 0 \\ 0 \\ \frac{1}{c} \end{bmatrix} \\ & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi h}{\sqrt{3}a} + \frac{4 \pi k}{\sqrt{3}b} \\ \frac{2 \pi l}{c} \end{bmatrix} \\ & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi (h + 2 k)}{\sqrt{3}a} \\ \frac{2 \pi l}{c} \end{bmatrix} \end{alignat} }

@@ 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.
-==Notation==
+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:
-* '''Real space''':
+: <math>\alpha</math>, the angle between <math>b</math> and <math>c</math>
-** Crystal ''planes'':
+: <math>\beta</math>, the angle between <math>a</math> and <math>c</math>
-*** (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]
+: <math>\gamma</math>, the angle between <math>a</math> and <math>b</math>
-*** {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==
+==Mathematical description==
 ===Vectors===
+There are many ways to define the Cartesian basis for the unit cell in [[real-space]]. A typical definition is:
 :<math>\begin{array}{l}
 \mathbf{a} = \begin{bmatrix}
@@ Line 43: / 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}
+</math>
+There are many mathematically equivalent ways to express a given definition. For instance, the vector <math>\mathbf{c}</math> can also be written as (c.f. [https://www.ocf.berkeley.edu/~rfu/notes/vect_unit_cells.pdf these notes] and [https://doi.org/10.1107/S0108767396005697 Trueblood et al. ''Acta Cryst'' '''1996''', A52, 770-781]):
+:<math>\begin{array}{l}
+\mathbf{c} = \begin{bmatrix}
+c \cos{\beta} \\
+-c \sin \beta \cos \alpha^{*} \\
+\frac{1}{c^{*}} \\
+\end{bmatrix}
+= \begin{bmatrix}
+c \cos{\beta} \\
+c \frac{\cos\alpha -\cos\beta \cos\gamma  }{\sin\gamma} \\
+\frac{c}{\sin\gamma} \sqrt{1 + 2 \cos\alpha\cos\beta\cos\gamma - \cos^2 \alpha - \cos^2\beta -\cos^2\gamma}\\
 \end{bmatrix}
 \end{array}
@@ Line 67: / Line 70: @@
 * <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===
+===Reciprocal vectors===
+The repeating structure of a unit cell creates peaks in [[reciprocal space]]. In particular, we observe maxima (constructive interference) when:
+:<math>
+\begin{alignat}{2}
+\mathbf{q} \cdot \mathbf{a} & = 2 \pi h \\
+\mathbf{q} \cdot \mathbf{b} & = 2 \pi k \\
+\mathbf{q} \cdot \mathbf{c} & = 2 \pi l \\
+\end{alignat}
+</math>
+Where <math>h</math>, <math>k</math>, and <math>l</math> are integers. We define reciprocal-space vectors:
 :<math>
 \begin{alignat}{2}
@@ Line 77: / Line 89: @@
 \end{alignat}
 </math>
+And we can then express the [[momentum transfer]] (<math>\mathbf{q}</math>) in terms of these reciprocal vectors:
-====Vector components====
-Generally:
 :<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}
-  & = 2 \pi(h\mathbf{u} + k \mathbf{v} + l \mathbf{w}) \\
-  & = 2 \pi \mathbf{H}_{hkl}
 \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}
- & = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c}) }  \\
- & =\frac{1}{V} \mathbf{a}\times\mathbf{b} \\
- & =\frac{1}{V} \begin{bmatrix} 0 \\ 0 \\ a b \sin{\gamma} \end{bmatrix} \\
 \end{alignat}
 </math>
+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==
@@ Line 139: / Line 145: @@
    & = \begin{bmatrix} \frac{2 \pi h}{a} \\ \frac{2 \pi k}{b} \\ \frac{2 \pi l}{c} \end{bmatrix}
 \end{alignat}
+</math>
+And:
+:<math>
+ q_{hkl} = 2\pi \sqrt{ \left( \frac{h}{a} \right)^2 + \left( \frac{k}{b} \right)^2 + \left( \frac{l}{c} \right)^2 }
 </math>
@@ Line 182: / Line 192: @@
 \end{alignat}
 </math>
+==See Also==
+* [[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]

Difference between revisions of "Unit cell"

Latest revision as of 09:33, 14 November 2022

Contents

Mathematical description

Vectors

Relations

Volume

Angles

Reciprocal vectors

Examples

Cubic

Hexagonal

See Also

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