# Reduced cell

### From Online Dictionary of Crystallography

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(No, it is cell（単位胞）not lattice (格子)) |
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- | <font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple"> | + | <font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">既約単位胞</font> (''Ja''). |

- | 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> | ||

+ | |||

+ | ==Geometrical meaning of the reduced cell== | ||

+ | The main conditions express the following two requirements: | ||

+ | *Of all lattice vectors, none is shorter than '''a'''; of those not directed along '''a''', none is shorter than '''b'''; of those not lying in the '''ab''' plane, none is shorter than '''c'''. | ||

+ | *The three angles between basis vectors are either all acute (type I) or all non-acute (type II). | ||

+ | |||

+ | == 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]] |

## Latest revision as of 08:54, 27 March 2019

Maille réduite (*Fr*). Cella ridotta (*It*). 既約単位胞 (*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 ≤

## Geometrical meaning of the reduced cell

The main conditions express the following two requirements:

- Of all lattice vectors, none is shorter than
**a**; of those not directed along**a**, none is shorter than**b**; of those not lying in the**ab**plane, none is shorter than**c**. - The three angles between basis vectors are either all acute (type I) or all non-acute (type II).

## See also

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