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 $\mathbf {c}$; 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 $\mathbf {a}$ and $\mathbf {c}$
• $\alpha$ is the angle between $\mathbf {b}$ and $\mathbf {c}$

From Fractional Coordinates (Wikipedia)

Reciprocal Vectors

${\begin{alignedat}{2}\mathbf {u} &={\frac {\mathbf {b} \times \mathbf {c} }{\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}={\frac {1}{V}}\mathbf {b} \times \mathbf {c} \\\mathbf {v} &={\frac {\mathbf {c} \times \mathbf {a} }{\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}={\frac {1}{V}}\mathbf {c} \times \mathbf {a} \\\mathbf {w} &={\frac {\mathbf {a} \times \mathbf {b} }{\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}={\frac {1}{V}}\mathbf {a} \times \mathbf {b} \\\end{alignedat}}$

Vector components

Generally:

${\begin{alignedat}{2}\mathbf {q} _{hkl}&=(2\pi h)\mathbf {u} +(2\pi k)\mathbf {v} +(2\pi l)\mathbf {w} \\&=2\pi (h\mathbf {u} +k\mathbf {v} +l\mathbf {w} )\\&=2\pi \mathbf {H} _{hkl}\end{alignedat}}$

With components:

${\begin{alignedat}{2}\mathbf {u} &=...\\\mathbf {v} &=...\\\mathbf {w} &={\frac {\mathbf {a} \times \mathbf {b} }{\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}\\&={\frac {1}{V}}\mathbf {a} \times \mathbf {b} \\&={\frac {1}{V}}{\begin{bmatrix}0\\0\\ab\sin {\gamma }\end{bmatrix}}\\\end{alignedat}}$

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:

${\begin{alignedat}{2}\mathbf {a} &={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} &={\begin{bmatrix}0\\b\\0\end{bmatrix}}\\\mathbf {c} &={\begin{bmatrix}0\\0\\c\end{bmatrix}}\\\end{alignedat}}$

And in reciprocal-space:

${\begin{alignedat}{2}\mathbf {u} &={\frac {1}{V}}\mathbf {b} \times \mathbf {c} &={\frac {1}{V}}{\begin{bmatrix}bc\\0\\0\end{bmatrix}}&={\begin{bmatrix}{\frac {1}{a}}\\0\\0\end{bmatrix}}\\\mathbf {v} &={\frac {1}{V}}\mathbf {c} \times \mathbf {a} &={\frac {1}{V}}{\begin{bmatrix}0\\ac\\0\end{bmatrix}}&={\begin{bmatrix}0\\{\frac {1}{b}}\\0\end{bmatrix}}\\\mathbf {w} &={\frac {1}{V}}\mathbf {a} \times \mathbf {b} &={\frac {1}{V}}{\begin{bmatrix}0\\0\\ab\end{bmatrix}}&={\begin{bmatrix}0\\0\\{\frac {1}{c}}\end{bmatrix}}\\\end{alignedat}}$

So:

${\begin{alignedat}{2}\mathbf {q} _{hkl}&=(2\pi h)\mathbf {u} +(2\pi k)\mathbf {v} +(2\pi l)\mathbf {w} \\&=(2\pi h){\begin{bmatrix}{\frac {1}{a}}\\0\\0\end{bmatrix}}+(2\pi k){\begin{bmatrix}0\\{\frac {1}{b}}\\0\end{bmatrix}}+(2\pi l){\begin{bmatrix}0\\0\\{\frac {1}{c}}\end{bmatrix}}\\&={\begin{bmatrix}{\frac {2\pi h}{a}}\\{\frac {2\pi k}{b}}\\{\frac {2\pi l}{c}}\end{bmatrix}}\end{alignedat}}$

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 $\gamma =60^{\circ }$, 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} }