# Reduced cell

### From Online Dictionary of Crystallography

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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

.

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

- ≤ ≤
- ≤
- ≤
- ≤

### Special conditions

- if then ≤
- if then ≤
- if /2 then ≤
- if /2 then ≤
- if /2 then ≤

## Type-II cell

### Main conditions

- ≤ ≤
- ≤
- ≤
- ≤
- ≤
- ≤ 0
- ≤ 0
- ≤ 0

### Special conditions

- if then ≤
- if then ≤
- if then
- if /2 then
- if /2 then
- if then ≤

## See also

- Conventional cell
- Crystallographic basis
- Direct lattice
- Unit cell
- Chapter 3.1.3. of
*International Tables for Crystallography, Volume A*, 6th edition