Table 1.

Conditions for a reduced cell^a

The cell is specified by three noncoplanar vectors: a, b, c. The cell matrix (a·a b·b·c·c/b·c a·c a·b) is defined by the dot products between these vectors.

A. Positive Reduced form, Type I cell, all angles < 90°

Main conditions:

a·a≤b·b≤c·c; b·c≤½b·b; a·c≤½a·a; a·b≤ ½ a·a

Special conditions:

(a) if a·a = b·b then b·c≤a·c

(b) if b·b = c·c then a·c≤a·b

(c) if b·c = ½ b·b then a·b ≤ 2 a·c

(d) if a·c = ½ a·a then a·b ≤ 2 b·c

(e) if a·b = ½ a·a then a·c ≤ 2 b·c

B. Negative reduced form, Type II cell, all angles ≥90°

Main conditions:

(a) a·a≤b·b≤c·c; |b·c| ≤½b·b; |a·c| ≤½a·a; |a·b|≤½ a·a

(b) (|b·c| + |a·c| + |a·b|) ≤ ½ (a·a + b·b)

Special conditions:

(a) if a·a = b·b then |b·c| ≤ |a·c|

(b) if b·b = c·c then |a·c| ≤ |a·b|

(c) if |b·c| = ½ b·b then a·b = 0

(d) if |a·c| = ½ a·a then a·b = 0

(e) if |a·b| = ½ a·a then a·c = 0

(f) if (|b·c| + |a·c| + |a·b|) = ½ (a·a + b·b) then a·a ≤ 2 |a·c| + |a·b|

^aTo be reduced the cell must be in normal representation (type I or II) and all the main and special conditions for the given cell type must be satisfied. The main conditions are used to establish that a cell is based on the three shortest lattice translations. The special conditions are used to select a unique cell when two or more cells in the lattice have the same numerical values for the cell edges.