Authors: Y Le Page & H D Flack
Contact: H D Flack, Laboratoire de Cristallographie, University of Geneva, CH-121 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 direct-reciprocal 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. Direct-reciprocal row pairs satisfying these conditions (i.e. having an inter-row 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 well-known 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 pre-determined 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 inter-row 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 direct-reciprocal row pair as calculated from its vector product is compared against the maximum-acceptable 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 pseudo-symmetric 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 direct-reciprocal row search. Scalar products of 1 and 2 identify twofold axes causing all lattice points to find a rotation-related 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 twin-related 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
non-centrosymmetric space group each twin law listed (and
saved in the
STARTX cell 5.657 5.657 6.828 90. 90. 120. sgname R 3 celcon C 8 CREDUC
The output will be as follows:
Maximum acceptable obliquity: 3.0 degrees Maximum multiple cell index: 2 Input cell: 5.657 5.657 6.828 90.000 90.000 120.0 Lattice mode: R A Buerger cell: 3.98l 3.981 3.981 90.555 90.555 90.555 The input-to-Buerger cell matrix: -.3333 .3333 .3333 -.3333 -.6667 .3333 .6667 .3333 3333 Possible 2-fold axes: Rows Products Direct Reciprocal Dot Vector O 0 1 0 0 1 1 .788 0 1 -1 O 1 -1 2 .0OO 0 1 0 0 1 0 1 .788 0 1 1 0 1 1 2 .788 1 -1 0 1 -1 0 2 .000 1 0 -1 1 0 -1 2 .000 1 0 0 1 0 0 1 .788 1 0 1 1 0 1 2 .788 1 1 0 1 1 0 2 .788 Pseudo cubic P Max delta .788 A = -.3 -.7 .3 3.9809 Alpha = 90.555 A* = .000 -1.000 1.000 B = -.7 -.3 -.3 3.9809 Beta = 89.445 B* = 1.000 .000 -1.000 C = .3 -.3 -.3 3.9809 Gamma = 89.445 C* = 1.000 -1.000 -1.000 Metrically hexagonal R Max delta .000 A = -1.0 -1.0 .0 5.6570 Alpha = 90.000 A* = .000 -1.000 .000 B = 1.0 .0 .0 5.6570 Beta = 90 . 000 B* = 1.000 -1.000 .000 C = .0 .0 1.0 6.8280 Gamma = 120.000 C* = .000 .000 1.000 (Pseudo-)Merohedral Twinning Operations ======================================= If the space group is non-centrosymmetric each operation below impliestwo - i.e. the given operation multiplied by +1 and -1 Twin x = ( 1.00 .00 .00)x : Twin h = ( 1.00 .00 .00)h Twin y = ( .00 1.00 .00)y ; Twin k = ( .00 1.00 .00)k Twin z = ( .00 .00 l.00)z ; Twin l = ( -.00 .00 1.00)l Twin x = (-1.00 .00 .00)x ; Twin h = (-1.00 -1.00 .00)h Twin y = (-1.00 1.00 .00)y ; Twin k = ( .00 1.00 .00)k Twin z = ( .00 .00 -l.00)z ; Twin l = ( .00 .00 -1.00)l
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 pseudo-cubic 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 triply-primitive cell (in the obverse setting) on hexagonal axes is generated. In all cases of pseudo or metrically-exact 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 twin-law 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 non-centrosymmetric, these two twin operations imply two further inversion-related ones: i.e. inversion in the origin and reflection in (1 0 0).