CHOICE AND ORIENTATION OF THE CELL

General Rules -- The cell chosen to define the lattice should obey the follow-
                 ing rules (Donnay, 1943), to be applied in the sequence given:
(1)  Whenever possible the cell edges should coincide with symmetry directions
     of the lattice and the cell should have the same symmetry as the lattice.
(2)  Symmetry-equivalent edges should be chosen, if possible.
(3)  The cell should be the smallest possible cell that obeys conditions (1) and
     (2).
(4)  Cell edges that cannot be symmetry directions should be the shortest 
     possible lattice translations.
(5)  A cell edge parallel to a single 2-fold axis of the lattice is called b.  A
     cell edge parallel to a principal unique axis (n>=3) is called c. Cell 
     edges not governed by the lattice symmetry should obey the inequalities 
     c<a<b.
(6)  The axes of coordinates should be directed, along the cell edges, so as to
     form a right-handed system with nonacute interedge angles alpha and gamma.
     (Angle gamma can vary from 60 degrees to 120 degrees, both values included;
     angles alpha and beta cannot exceed 120 degrees.)

     These rules hold for all crystal systems, whenever applicable.
In the anorthic (triclinic) system Rule 1 does not apply as symmetry directions 
do not exist.  The point-group symmetry of the lattice is 1; every cell is 
centrosymmetric.  Rule 2 does not apply.  The cell chosen should be primitive
(P) to comply with Rule 3.  Its edges must be the shortest three noncoplanar
lattice translations (Rule 4) that define the reduced cell (Niggli, 1928; 
Santoro and Mighell, 1970).  The cell edges are labeled so as to have c<a<b
(Rule 5) and oriented so that the angles alpha and beta are nonacute (Rule 6).
The point-group symmetry of a monoclinic lattice is 2/m.  There is one symmetry
direction, which (rule 1) fixes one cell edge, conventionally labeled b. Rule 2
does not apply.  The shortest two translations in the net perpendicular to b,
together with b, define a cell that obeys Rule 3.  Take angle beta nonacute 
(Rule 4) and c<a (Rule 5).  The resulting cell may be primitive or centered 
(C, A, or I). 
The point-group symmetry of an orthorhombic lattice is 2/m 2/m 2/m.  The three
symmetry directions fix the three cell edges (Rule 1).  Rule 2 does not apply. 
The cell, in accordance with Rule 3, is the smallest possible one that obeys
the first rule.  All interedge angles are 90 degrees, fixed by symmetry, so 
that Rule 4 does not apply.  Take c<a<b (Rule 5). The resulting cell may be
primitive (P), one-face centered (A, B, or C), body centered (I), or all-face
centered (F).  Note also that point group mm2 may have to be oriented as 2mm 
or as m2m.  
The point-group symmetry of a tetragonal lattice is 4/m 2/m 2/m.  Of the five
symmetry directions, one is perpendicular to a plane containing the other four.
This unique direction must be taken as a cell edge (Rule 1); it is convention-
ally labeled c.  The remaining cell edges must (Rules 1 and 2) coincide with 
two equivalent 2-fold axes, either those of the first or those of the second
kind, so that all angles are right angles by symmetry.  The axes of the first 
kind must be chosen, as they lead to the smaller cell (Rule 3).  Rules 4 and 5
do not apply.  The resulting cell may be primitive (P) or body centered (I).
The point-group symmetry of the hexagonal lattice is 6/m 2/m 2/m.  Of the seven
symmetry directions, one is perpendicular to a plane containing the other six.  
This unique direction must be taken as a cell edge (Rule 1); it is convention-
ally labeled c.  The remaining cell edges must (Rules 1 and 2) coincide with two
equivalent 2-fold axes, of either the first or second kind.  Two 2-fold axes of
the first kind are chosen, as they give the smaller cell (Rule 3).  Angles alpha
and beta, fixed by symmetry, are equal to 90 degrees; angle gamma is taken equal
to 120 degrees in preference to 60 degrees.  Rules 4 and 5 do not apply.  The 
resulting cell is primitive (P, formerly designated C).
The rhombohedral lattice, which has point-group symmetry 3 bar 2/m, is here des-
cribed by a cell referred to hexagonal axes (Bravais axes).  Of the four 
symmetry directions, one is perpendicular to a plane containing the other three.
This unique direction must be taken as a cell edge (Rule 1); it is convention-
ally labeled c.  The remaining cell edges must (Rules 1 and 2) coincide with 
two of the three 2-fold axes, so that all angles are fixed by symmetry 
(gamma=120 degrees, alpha=beta=90 degrees).  The resulting cell satisfies 
Rule 3.  Rules 4 and 5 do not apply. The R cell is a triple cell; the additional
nodes are chosen at 1/3 2/3 2/3 and 2/3 1/3 1/3.
The point-group symmetry of a cubic lattice is 4/m 3 bar 2/m.  The three 4-fold
axes are the symmetry directions taken as edges of the cubic cell.  The cell may
be primitive (P), body centered (I), or face centered (F).