Difference between revisions of "Reduced cell"
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| − | A primitive basis a, b, c is called a | + | A [[primitive basis]] '''a''', '''b''', '''c''' is called a '''reduced basis''' if it is right-handed and if the components of the [[metric tensor]] satisfy the conditions below. Because of [[lattice]] [[symmetry operation|symmetry]] there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique. |
The type of a cell depends on the sign of | The type of a cell depends on the sign of | ||
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*if <math>\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{c} = 0</math> | *if <math>\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{c} = 0</math> | ||
*if <math>(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2</math> then <math>\mathbf{a}\cdot\mathbf{a}</math> ≤ <math>2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|</math> | *if <math>(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2</math> then <math>\mathbf{a}\cdot\mathbf{a}</math> ≤ <math>2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|</math> | ||
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| + | == See also == | ||
| + | *[[Conventional cell]] | ||
| + | *[[Crystallographic basis]] | ||
| + | *[[Direct lattice]] | ||
| + | *[[Unit cell]] | ||
| + | *Chapter 3.1.3. of ''International Tables for Crystallography, Volume A'', 6th edition | ||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 15:52, 18 December 2017
Maille réduite (Fr). Cella ridotta (It). 規約単位胞 (Ja).
A primitive basis a, b, c is called a reduced basis if it is right-handed and if the components of the metric tensor satisfy the conditions below. Because of lattice symmetry there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique.
The type of a cell depends on the sign of
[math]T = (\mathbf{a}\cdot\mathbf{b})(\mathbf{b}\cdot\mathbf{c})(\mathbf{c}\cdot\mathbf{a})[/math].
If T > 0, the cell is of type I, if T ≤ 0 it is of type II.
The conditions for a primitive cell to be a reduced cell can all be stated analytically as follows.
Contents
Type-I cell
Main conditions
- [math]\mathbf{a}\cdot\mathbf{a}[/math] ≤ [math]\mathbf{b}\cdot\mathbf{b}[/math] ≤ [math]\mathbf{c}\cdot\mathbf{c}[/math]
- [math]|\mathbf{b}\cdot\mathbf{c}|[/math] ≤ [math](\mathbf{b}\cdot\mathbf{b})/2[/math]
- [math]|\mathbf{a}\cdot\mathbf{c}|[/math] ≤ [math](\mathbf{a}\cdot\mathbf{a})/2[/math]
- [math]|\mathbf{a}\cdot\mathbf{b}|[/math] ≤ [math](\mathbf{a}\cdot\mathbf{a})/2[/math]
- [math]\mathbf{b}\cdot\mathbf{c} \gt 0[/math]
- [math]\mathbf{a}\cdot\mathbf{c} \gt 0[/math]
- [math]\mathbf{a}\cdot\mathbf{b} \gt 0[/math]
Special conditions
- if [math]\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}[/math] then [math]\mathbf{b}\cdot\mathbf{c}[/math] ≤ [math]\mathbf{a}\cdot\mathbf{c}[/math]
- if [math]\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}[/math] then [math]\mathbf{a}\cdot\mathbf{c}[/math] ≤ [math]\mathbf{a}\cdot\mathbf{b}[/math]
- if [math]\mathbf{b}\cdot\mathbf{c} = (\mathbf{b}\cdot\mathbf{b})[/math]/2 then [math]\mathbf{a}\cdot\mathbf{b}[/math] ≤ [math]2\mathbf{a}\cdot\mathbf{c}[/math]
- if [math] \mathbf{a}\cdot\mathbf{c} = (\mathbf{a}\cdot\mathbf{a})[/math]/2 then [math]\mathbf{a}\cdot\mathbf{b}[/math] ≤ [math]2\mathbf{b}\cdot\mathbf{c}[/math]
- if [math] \mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})[/math]/2 then [math]\mathbf{a}\cdot\mathbf{c}[/math] ≤ [math]2\mathbf{b}\cdot\mathbf{c}[/math]
Type-II cell
Main conditions
- [math]\mathbf{a}\cdot\mathbf{a}[/math] ≤ [math]\mathbf{b}\cdot\mathbf{b}[/math] ≤ [math]\mathbf{c}\cdot\mathbf{c}[/math]
- [math]|\mathbf{b}\cdot\mathbf{c}|[/math] ≤ [math](\mathbf{b}\cdot\mathbf{b})/2[/math]
- [math]|\mathbf{a}\cdot\mathbf{c}|[/math] ≤ [math](\mathbf{a}\cdot\mathbf{a})/2[/math]
- [math]|\mathbf{a}\cdot\mathbf{b}|[/math] ≤ [math](\mathbf{a}\cdot\mathbf{a})/2[/math]
- [math](|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|)[/math] ≤ [math](\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2[/math]
- [math]\mathbf{b}\cdot\mathbf{c}[/math] ≤ 0
- [math]\mathbf{a}\cdot\mathbf{c}[/math] ≤ 0
- [math]\mathbf{a}\cdot\mathbf{b}[/math] ≤ 0
Special conditions
- if [math]\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}[/math] then [math]|\mathbf{b}\cdot\mathbf{c}|[/math] ≤ [math]|\mathbf{a}\cdot\mathbf{c}|[/math]
- if [math]\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}[/math] then [math]|\mathbf{a}\cdot\mathbf{c}|[/math] ≤ [math]|\mathbf{a}\cdot\mathbf{b}|[/math]
- if [math]|\mathbf{b}\cdot\mathbf{c}| = (\mathbf{b}\cdot\mathbf{b})/2[/math] then [math]\mathbf{a}\cdot\mathbf{b} = 0[/math]
- if [math]|\mathbf{a}\cdot\mathbf{c}| = (\mathbf{a}\cdot\mathbf{a})[/math]/2 then [math]\mathbf{a}\cdot\mathbf{b} = 0[/math]
- if [math]\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})[/math]/2 then [math]\mathbf{a}\cdot\mathbf{c} = 0[/math]
- if [math](|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2[/math] then [math]\mathbf{a}\cdot\mathbf{a}[/math] ≤ [math]2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|[/math]
See also
- Conventional cell
- Crystallographic basis
- Direct lattice
- Unit cell
- Chapter 3.1.3. of International Tables for Crystallography, Volume A, 6th edition