Reduced cell

Maille réduite (Fr). Cella ridotta (It). 既約格子 (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.

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]

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