Unit cell

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

${\begin{array}{l}\mathbf {a} ={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} ={\begin{bmatrix}b\cos {\gamma }\\b\sin {\gamma }\\0\end{bmatrix}}\\\mathbf {c} ={\begin{bmatrix}c\sin {\theta _{c}}\cos {\phi _{c}}\\c\sin {\theta _{c}}\sin {\phi _{c}}\\c\cos {\theta _{c}}\end{bmatrix}}={\begin{bmatrix}c\cos {\beta }\\c{\frac {\cos {\alpha }-\cos {\beta }\cos {\gamma }}{\sin {\gamma }}}\\c{\sqrt {1-\cos ^{2}{\beta }-\left({\frac {\cos {\alpha }-\cos {\beta }\cos {\gamma }}{\sin {\gamma }}}\right)^{2}}}\end{bmatrix}}\end{array}}$

Relations

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

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

$\mathbf {b} \cdot \mathbf {c} =bc\cos {\alpha }$

Volume

$V=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|=|\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )|=|\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )|$

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

$V=abc{\sqrt {1+2\cos(\alpha )\cos(\beta )\cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}.$

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

• $\gamma$ is the angle between $\mathbf {a}$ and $\mathbf {b}$
• $\beta$ 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 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 $\mathbf {c}$

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 $h$, 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} }

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 Atomic Dispacement Parameter Nomenclature. Report of a Subcommittee on Atomic Displacement Parameter Nomenclature Acta Cryst 1996, A52, 770-781. doi: 10.1107/S0108767396005697