Authors: Y Le Page & H D
Flack
Contact: H D Flack,
Laboratoire de Cristallographie, University of Geneva, CH121
1 Genéve 4, Switzerland.
CREDUC undertakes the calculation of cell
reduction using the algorithm of Le Page (1982). The
essential objective of cell reduction is the recognition of
the metric (lattice) symmetry of the input unit cell. Once
this metric symmetry has been identified, the basis vectors
of the lattice may be chosen to lie along the main symmetry
direction in the conventional way.
The Le Page algorithm is based on the search and
identification of twofold axes of the translation lattice.
The search is carried out by systematically generating
pairs of direct and reciprocal lattice rows. In the normal
case, the maximum values of the indices of these direct and
reciprocal lattice rows may be limited to a value of two by
working from a primitive Burger reduced cell generated from
the input cell. For each directreciprocal row pair, the
vector product is used to calculate the angle between the
rows and the scalar product is also evaluated. The two rows
are parallel or nearly so when the angle between them is
zero or close to zero, and parallel rows are a twofold axis
of the whole translation lattice if their scalar product
has a value of 1 or 2. Directreciprocal row pairs
satisfying these conditions (i.e. having an interrow angle
less than a specified value) are stored in a list
containing possible twofold axes.
The point symmetries of the seven crystal systems
have characteristic distributions of twofold axes in space.
Hence comparison of the distribution in space of the
twofold axes found in the systematic search with the
wellknown distribution of the twofold axes of the seven
crystal systems allows the lattice symmetry to be
identified. This process places the lattice in a
conventional orientation and the choice of rows to be
selected as edges of the conventional cell is guided by the
coincidence of lattice rows with predetermined symmetry
axes. The search for the crystal system takes place in the
order cubic, hexagonal, rhombohedral, tetragonal,
orthorhombic, monoclinic and triclinic. Each time a
solution is found, all the twofold axes with an interrow
angle greater than or equal to the largest one used in the
match are deleted from the list of twofold axes and then
another solution is sought.
The first input parameter allows the maximum
acceptable angular deviation between the direct and
reciprocal lattice rows defining a twofold axis to be set.
The angular deviation of each directreciprocal row pair as
calculated from its vector product is compared against the
maximumacceptable value and the pair is retained if the
angle is less than the input value. True twofold axes have
values of the angular deviation very close to zero. Hence
large values of the angular deviation will find cells
having pseudosymmetric lattices. The second input
parameter fixes both the maximum acceptable value of the
scalar product of the direct and reciprocal row pairs and
the maximum value of the indices in the directreciprocal
row search. Scalar products of 1 and 2 identify twofold
axes causing all lattice points to find a rotationrelated
partner. Values of the scalar product larger than two
identify twofold axes which cause only subsets of the
lattice nodes to be superimposed. Consequently supercells
of higher lattice symmetry can be identified. This is an
undocumented and very interesting option which existed in
the original Le Page programme.
The (pseudo)merohedral twin laws are evaluated by
the algorithm of Flack (1987). The basis of this method is
a coset decomposition of the point symmetry of the declared
crystal space group with respect to the lattice symmetry
determined from the cell reduction process. The programme
produces a list of twinning operations (described both in
direct and reciprocal space) which make the lattices nodes
of the twinrelated lattices superimpose. The algorithm is
arranged in such a way that for a centrosymmetric space
group all of the twin laws are generated, whereas for a
noncentrosymmetric space group each twin law listed (and
saved in the
ltrwin:
)
corresponds to two twin laws i.e. the one listed and one
related to it through an inversion in the origin.
The output will be as follows:
In the above example a structure in space group
R 3 is input on hexagonal axes. The
input cell is first reduced using the Burger algorithm.
Next a systematic search is taken over all pairs of direct
and reciprocal rows up to a maximum index of two. Those
pairs which both have an angular deviation of less than
3° and a scalar product of 1 or 2 are retained. These
directions are (pseudo) twofold axes. The distribution of
the twofold axes in space allows the metric (lattice)
symmetry to be identified. The 9 (pseudo) twofold axes have
the correct distribution for a lattice of cubic symmetry
but as the angle between some of the direct and reciprocal
rows is as large as 0.788°, this is only a
pseudocubic primitive cell. Having eliminated the 6 pseudo
twofold axes with an angular deviation of 0.788°, the
programmeconsiders the remaining three pure twofold axes
and finds that they have the correct spatial distribution
for the trigonal system on rhombohedral axes. As a
conventional choice a triplyprimitive cell (in the obverse
setting) on hexagonal axes is generated. In all cases of
pseudo or metricallyexact cell, the matrices describing
the relation between the input and reduced direct and
reciprocal cells are output. Finally the twin laws are
calculated and the twinlaw transformation matrices are
output both for direct and reciprocal space. In the
example, the first twin law is the identity operation, the
second being a twofold rotation about [2 1 0]. As the
declared space group is noncentrosymmetric, these two twin
operations imply two further inversionrelated ones:
i.e. inversion in the origin and
reflection in (1 0 0).

M.J., Burger (1960).
Z. Kristallogr.
113, 5256.

H.D. Flack (1987).
Acta Cryst.
A43,564568.

Y. Le Page (1982).
J. Appl. Cryst.
15, 255259.