Special conditions

Wolff, P. M. de

doi:10.1107/97809553602060000518

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 9.2, pp. 751-754

Section 9.2.4. Special conditions

P. M. de Wolff^a ^‡

^a Laboratorium voor Technische Natuurkunde, Technische Hogeschool, Delft, The Netherlands

9.2.4. Special conditions

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For a given lattice, the main condition (i) defines not only the lengths a, b, c of the reduced basis vectors but also the plane containing a and b, in the sense that departures from special conditions can be repaired by transformations which do not change this plane. An exception can occur when [b = c] ; then such transformations must be supplemented by interchange(s) of b and c whenever either (9.2.2.3b) or (9.2.2.5b) is not fulfilled. All the other conditions can be conveniently illustrated by projections of part of the lattice onto the ab plane as shown in Figs. 9.2.4.1 to 9.2.4.5. Let us represent the vector lattice by a point lattice. In Fig. 9.2.4.1, the net in the ab plane (of which OBAD is a primitive mesh; [OA = a] , [OB = b] ) is shown as well as the projection (normal to that plane) of the adjoining layer which is assumed to lie above the paper. In general, just one lattice node $[P_{0}]$ of that layer, projected in Fig. 9.2.4.1 as P, will be closer to the origin than all others. Then the vector $[\overrightarrow{OP_{0}}]$ is $[\pm {\bf c}]$ according to condition (i). It should be stressed that, though the ab plane is most often (see above) correctly established by (i), the vectors a, b and c still have to be chosen so as to comply with (ii), with the special conditions, and with right-handedness. The result will depend on the position of P with respect to the net. This dependence will now be investigated.

Figure 9.2.4.1| top | pdf |

The net of lattice points in the plane of the reduced basis vectors a and b; OBAD is a primitive mesh. The actual choice of a and b depends on the position of the point P, which is the projection of the point $[P_{0}]$ in the next layer (supposed to lie above the paper, thin dashed lines) closest to O. Hence, P is confined to the Voronoi domain (dashed hexagon) around O. For a given interlayer distance, P defines the complete lattice. In that sense, P and [P'] represent identical lattices; so do Q, [Q'] and [Q''] , and also R and [R'] . When P lies in a region marked $[-c^{\rm II}]$ (hatched), the reduced type-II basis is formed by $[{\bf a}^{\rm II}]$ , $[{\bf b}^{\rm II}]$ and $[{\bf c} = -\overrightarrow{OP}_{0}]$ . Regions marked $[c^{\rm I}]$ (cross-hatched) have the reduced type-I basis $[{\bf a}^{\rm I},{\bf b}^{\rm I}]$ and $[{\bf c} = + \overrightarrow{OP}_{0}]$ . Small circles in O, M etc. indicate twofold rotation points lying on the region borders (see text).

Figure 9.2.4.2| top | pdf |

The effect of the special conditions. Border lines of type-I and type-II regions are drawn as heavy lines if included. The type-I and type-II regions are marked as in Fig. 9.2.4.1. A heavy border line of a region stops short of an end point if the latter is not included in the region to which the border belongs. a, b net primitive orthogonal; special conditions (9.2.2.5c), (9.2.2.5d).

Figure 9.2.4.3| top | pdf |

The effect of the special conditions. Border lines of type-I and type-II regions are drawn as heavy lines if included. Type-I and type-II regions are marked as in Fig. 9.2.4.1. $[n_{b}]$ belongs to the type-II region. A heavy border line of a region stops short of an end point if the latter is not included in the region to which the border belongs. a, b net oblique; special conditions (9.2.2.3c), (9.2.2.3d), (9.2.2.5f).

Figure 9.2.4.4| top | pdf |

The effect of the special conditions. Border lines of type-I and type-II regions are drawn as heavy lines if included. The type-I region is cross-hatched; the type-II region is a mere line. A heavy border line of a region stops short of an end point if the latter is not included in the region to which the border belongs. a, b net centred orthogonal (elongated); special conditions (9.2.2.3e), (9.2.2.5e).

Figure 9.2.4.5| top | pdf |

The effect of the special conditions. Border lines of type-I and type-II regions are drawn as heavy lines if included. Type-I and type-II regions are marked as in Fig. 9.2.4.1. $[n_{b}]$ belongs to the type-II region. A heavy border line of a region stops short of an end point if the latter is not included in the region to which the border belongs. a, b net centred orthogonal (compressed); special conditions (9.2.2.3a), (9.2.2.5a).

The inner hexagon shown, which is the two-dimensional Voronoi domain around O, limits the possible projected positions P of $[P_{0}]$ . Its short edges, normal to OD, result from (9.2.2.4b); the other edges from (9.2.2.2a). If the spacing d between ab net planes is smaller than b, the region allowed for P is moreover limited inwardly by the circle around O with radius $[(b^{2} - d^{2})^{1/2}]$ , corresponding to the projection of points $[P_{0}]$ for which $[OP_{0} = c = b]$ . The case [c = b] has been dealt with, so in order to simplify the drawings we shall assume $[d \;\gt\; b]$ . Then, for a given value of d, each point within the above-mentioned hexagonal domain, regarded as the projection of a lattice node $[P_{0}]$ in the next layer, completely defines a lattice based on $[\overrightarrow{OA}]$ , $[\overrightarrow{OB}]$ and $[\overrightarrow{OP}_{0}]$ . Diametrically opposite points like P and [P'] represent the same lattice in two orientations differing by a rotation of 180° in the plane of the figure. Therefore, the systematics of reduced bases can be shown completely in just half the domain. As a halving line, the $[n_{a}]$ normal to OA is chosen. This is an important boundary in view of condition (ii), since it separates points P for which the angle between $[OP_{0}]$ and OA is acute from those for which it is obtuse.

Similarly, $[n_{b}]$ , normal to OB, separates the sharp and obtuse values of the angles $[P_{0} OB]$ . It follows that if P lies in the obtuse sector (cross-hatched area) between $[n_{a}]$ and $[n_{b}]$ , the reduced cell is of type I, with basis vectors $[{\bf a}^{\rm I}]$ , $[{\bf b}^{\rm I}]$ , and $[OP_{0} = + {\bf c}]$ . Otherwise (hatched area), we have a type-II reduced cell, with $[OP_{0} = - {\bf c}]$ and $[+ {\bf a}]$ and $[+ {\bf b}]$ as shown by $[{\bf a}^{\rm II}]$ and $[{\bf b}^{\rm II}]$ .

Since type II includes the case of right angles, the borders of this region on $[n_{a}]$ and $[n_{b}]$ are inclusive. Other borderline cases are points like R and [R'] , separated by b and thus describing the same lattice. By condition (9.2.2.5c) the reduced cell for such cases is excluded from type II (except for rectangular a, b nets, cf. Fig. 9.2.4.2); so the projection of c points to R, not [R'] . Accordingly, this part of the border is inclusive for the type-I region and exclusive (at [R'] ) for the type-II region as indicated in Fig. 9.2.4.3. Similarly, (9.2.2.5d) defines which part of the border normal to OA is inclusive.

The inclusive border is seen to end where it crosses OA, OB or OD. This is prescribed by the conditions (9.2.2.3d), (9.2.2.3c) and (9.2.2.5f), respectively. The explanation is given in Fig. 9.2.4.1 for (9.2.2.3c): The points Q and [Q''] represent the same lattice because [Q'] (diametrically equivalent to Q as shown before) is separated from [Q''] by the vector b. Hence, the point M halfway between O and B is a twofold rotation point just like O. For a primitive orthogonal a, b net, only type II occurs according to (9.2.2.5c) and (9.2.2.5d), cf. Fig. 9.2.4.2. A centred orthogonal a, b net of elongated character (shortest net vector in a symmetry direction, cf. Section 9.2.5) is depicted in Fig. 9.2.4.4. It yields type-I cells except when $[\beta = 90^{\circ}]$ [condition (9.2.2.5c)]. Moreover, (9.2.2.3c) eliminates part of the type-I region as compared to Fig. 9.2.4.3. Finally, a centred net with compressed character (shortest two net vectors equal in length) requires criteria allowing unambiguous designation of a and b. These are conditions (9.2.2.3a) and (9.2.2.5a), cf. Fig. 9.2.4.5. The simplicity of these bisecting conditions, similar to those for the case [b = c] mentioned initially, is apparent from that figure when compared with Fig. 9.2.4.3. This compressed type of centred orthogonal a, b net is limited by the case of a hexagonal net (where it merges with the elongated type, Fig. 9.2.4.4) and by the centred quadratic net (where it merges with the primitive orthogonal net, Fig. 9.2.4.2). In the limit of the hexagonal net, the triangle Ohh in Figs. 9.2.4.4 and 9.2.4.5 is all that remains, it is of type I except for the point O. For the quadratic net, only the type-II region in Fig. 9.2.4.5, then a triangle with all edges inclusive, is left. It corresponds to the triangle Oqq in Fig. 9.2.4.2.

References

International Tables for Crystallography (2006). Vol. A. ch. 9.2, pp. 751-754