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)
    • $\left\langle hkl\right\rangle$ 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
    • $\left\langle hkl\right\rangle$ 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

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 $\alpha =\beta =\gamma =90^{\circ }$, $V=abc$, 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 $\alpha =\beta =90^{\circ }$ and $\gamma =60^{\circ }$, $V={\frac {\sqrt {3}}{2}}abc$, and:

${\begin{alignedat}{2}\mathbf {a} &={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} &={\begin{bmatrix}{\frac {1}{2}}b\\{\frac {\sqrt {3}}{2}}b\\0\end{bmatrix}}\\\mathbf {c} &={\begin{bmatrix}0\\0\\c\end{bmatrix}}\\\end{alignedat}}$

And in reciprocal-space:

${\begin{alignedat}{2}\mathbf {u} &={\frac {1}{V}}\mathbf {b} \times \mathbf {c} &={\frac {1}{V}}{\begin{bmatrix}{\frac {\sqrt {3}}{2}}bc\\-{\frac {1}{2}}bc\\0\end{bmatrix}}&={\begin{bmatrix}{\frac {1}{a}}\\{\frac {1}{{\sqrt {3}}a}}\\0\end{bmatrix}}\\\mathbf {v} &={\frac {1}{V}}\mathbf {c} \times \mathbf {a} &={\frac {1}{V}}{\begin{bmatrix}0\\ac\\0\end{bmatrix}}&={\begin{bmatrix}0\\{\frac {2}{{\sqrt {3}}b}}\\0\end{bmatrix}}\\\mathbf {w} &={\frac {1}{V}}\mathbf {a} \times \mathbf {b} &={\frac {1}{V}}{\begin{bmatrix}0\\0\\{\frac {\sqrt {3}}{2}}ab\end{bmatrix}}&={\begin{bmatrix}0\\0\\{\frac {1}{c}}\end{bmatrix}}\\\end{alignedat}}$

So:

${\begin{alignedat}{2}\mathbf {q} _{hkl}&=(2\pi h)\mathbf {u} +(2\pi k)\mathbf {v} +(2\pi l)\mathbf {w} \\&=(2\pi h){\begin{bmatrix}{\frac {1}{a}}\\{\frac {1}{{\sqrt {3}}a}}\\0\end{bmatrix}}+(2\pi k){\begin{bmatrix}0\\{\frac {2}{{\sqrt {3}}b}}\\0\end{bmatrix}}+(2\pi l){\begin{bmatrix}0\\0\\{\frac {1}{c}}\end{bmatrix}}\\&={\begin{bmatrix}{\frac {2\pi h}{a}}\\{\frac {2\pi h}{{\sqrt {3}}a}}+{\frac {4\pi k}{{\sqrt {3}}b}}\\{\frac {2\pi l}{c}}\end{bmatrix}}\\&={\begin{bmatrix}{\frac {2\pi h}{a}}\\{\frac {2\pi (h+2k)}{{\sqrt {3}}a}}\\{\frac {2\pi l}{c}}\end{bmatrix}}\end{alignedat}}$