Reduced cell

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

$T = (\mathbf{a}\cdot\mathbf{b})(\mathbf{b}\cdot\mathbf{c})(\mathbf{c}\cdot\mathbf{a})$.

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

• $\mathbf{a}\cdot\mathbf{a}$ ≤ $\mathbf{b}\cdot\mathbf{b}$ ≤ $\mathbf{c}\cdot\mathbf{c}$
• $|\mathbf{b}\cdot\mathbf{c}|$ ≤ $(\mathbf{b}\cdot\mathbf{b})/2$
• $|\mathbf{a}\cdot\mathbf{c}|$ ≤ $(\mathbf{a}\cdot\mathbf{a})/2$
• $|\mathbf{a}\cdot\mathbf{b}|$ ≤ $(\mathbf{a}\cdot\mathbf{a})/2$
• $\mathbf{b}\cdot\mathbf{c} \gt 0$
• $\mathbf{a}\cdot\mathbf{c} \gt 0$
• $\mathbf{a}\cdot\mathbf{b} \gt 0$

Special conditions

• if $\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}$ then $\mathbf{b}\cdot\mathbf{c}$ ≤ $\mathbf{a}\cdot\mathbf{c}$
• if $\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}$ then $\mathbf{a}\cdot\mathbf{c}$ ≤ $\mathbf{a}\cdot\mathbf{b}$
• if $\mathbf{b}\cdot\mathbf{c} = (\mathbf{b}\cdot\mathbf{b})$/2 then $\mathbf{a}\cdot\mathbf{b}$ ≤ $2\mathbf{a}\cdot\mathbf{c}$
• if $\mathbf{a}\cdot\mathbf{c} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{b}$ ≤ $2\mathbf{b}\cdot\mathbf{c}$
• if $\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{c}$ ≤ $2\mathbf{b}\cdot\mathbf{c}$

Type-II cell

Main conditions

• $\mathbf{a}\cdot\mathbf{a}$ ≤ $\mathbf{b}\cdot\mathbf{b}$ ≤ $\mathbf{c}\cdot\mathbf{c}$
• $|\mathbf{b}\cdot\mathbf{c}|$ ≤ $(\mathbf{b}\cdot\mathbf{b})/2$
• $|\mathbf{a}\cdot\mathbf{c}|$ ≤ $(\mathbf{a}\cdot\mathbf{a})/2$
• $|\mathbf{a}\cdot\mathbf{b}|$ ≤ $(\mathbf{a}\cdot\mathbf{a})/2$
• $(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|)$ ≤ $(\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2$
• $\mathbf{b}\cdot\mathbf{c}$ ≤ 0
• $\mathbf{a}\cdot\mathbf{c}$ ≤ 0
• $\mathbf{a}\cdot\mathbf{b}$ ≤ 0

Special conditions

• if $\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}$ then $|\mathbf{b}\cdot\mathbf{c}|$ ≤ $|\mathbf{a}\cdot\mathbf{c}|$
• if $\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}$ then $|\mathbf{a}\cdot\mathbf{c}|$ ≤ $|\mathbf{a}\cdot\mathbf{b}|$
• if $|\mathbf{b}\cdot\mathbf{c}| = (\mathbf{b}\cdot\mathbf{b})/2$ then $\mathbf{a}\cdot\mathbf{b} = 0$
• if $|\mathbf{a}\cdot\mathbf{c}| = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{b} = 0$
• if $\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{c} = 0$
• if $(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2$ then $\mathbf{a}\cdot\mathbf{a}$ ≤ $2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|$

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