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| ==See Also== | | ==See Also== |
| * [[Lattices]] | | * [[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] |
Revision as of 10:29, 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
,
, and
; alternately the unit cell can be described by the lengths of these vectors (
,
,
), and the angles between them:
, the angle between
and ![{\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
, the angle between
and ![{\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
, the angle between
and ![{\displaystyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
Mathematical description
Vectors
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67b1dcabbd3d21847468716cb6081ed386a90495)
Relations
![{\displaystyle \mathbf {a} \cdot \mathbf {b} =ab\cos {\gamma }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeaad04dded5fe78d0bccfd44a89602a4747b91f)
![{\displaystyle \mathbf {a} \cdot \mathbf {c} =ac\cos {\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92637b5cf1beefc263ad8994afb815a59a25aa25)
![{\displaystyle \mathbf {b} \cdot \mathbf {c} =bc\cos {\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/020596ac32871bde7c8f5cd54baa4e1a6f4655f5)
Volume
![{\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} )|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51a824dcb18a47e7863bab009127d1551f8728c9)
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
![{\displaystyle V=abc{\sqrt {1+2\cos(\alpha )\cos(\beta )\cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058ce6de0426b5784058ba71c4e790e916a0f368)
The volume of a unit cell with all edge-length equal to unity is:
![{\displaystyle v={\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/957760f5184f67dbcd40e6f90b660786d132d25d)
Angles
is the angle between
and ![{\displaystyle \mathbf {b} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9)
is the angle between
and ![{\displaystyle \mathbf {c} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8798d172f59e21f2ce193a3118d4063d19353ded)
is the angle between
and ![{\displaystyle \mathbf {c} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8798d172f59e21f2ce193a3118d4063d19353ded)
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:
![{\displaystyle {\begin{alignedat}{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{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f64b45378da113df826b9f182a40f92c706eef1)
Where
,
, and
are integers. We define reciprocal-space vectors:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72ef255bcaa65a842ce88e7235673f2d336f88bc)
And we can then express the momentum transfer (
) in terms of these reciprocal vectors:
![{\displaystyle {\begin{alignedat}{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{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0357da4525e9d32785bc9181c50d35edd06954)
Combining with the three Laue equations yields:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ba51fb61d782f41d769c54c70c737c6dd59e2e)
Where
is a vector that defines the position of Bragg reflection
for the reciprocal-lattice.
Examples
Cubic
Since
,
, and:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aec2a1687f622c215c61539d68bfce4055065d28)
And in reciprocal-space:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55a9c3b60154e22d253aff17479d3f73a62e27a2)
So:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ece407b14bf5be8ba45edf9990571d9a70c45be7)
And:
![{\displaystyle q_{hkl}=2\pi {\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ada387945a82a83319cfc369e5489694d87be104)
Hexagonal
Since
and
,
, and:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b02d5e677cc842ea5fd6a776b3cf99d4b1899989)
And in reciprocal-space:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d39c179e79504cd690fb4e825eb3b591684c1e3d)
So:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2a2132c07c6f94da385fca2c80b2d5085e6de5)
See Also