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 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}} , 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}} ; alternately the unit cell can be described by the lengths of these 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 a} , 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 b} , 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 c} ), and the angles between them:

$\alpha$, 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 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 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 \beta} , 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 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 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 \gamma} , 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 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 b}

Mathematical description

Vectors

There are many ways to define the Cartesian basis for the unit cell in real-space. A typical definition 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 \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 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}} can also be written as (c.f. these notes and Trueblood et al. Acta Cryst 1996, A52, 770-781):

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{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

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

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} \cdot \mathbf{c} = a c \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:

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 =\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 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}}
• 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 \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 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} }

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