International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 709714
Section 3.1.3. Reduced bases
P. M. de Wolff^{c}

Unit cells are usually chosen according to the conventions mentioned in Section 3.1.1, so one might think that there is no need for another standard choice. This is not true, however; conventions based on symmetry do not always permit unambiguous choice of the unit cell, and unconventional descriptions of a lattice do occur. They are often chosen for good reasons or they may arise from experimental limitations such as may occur, for example, in highpressure work. So there is a need for normalized descriptions of crystal lattices.
Accordingly, the reduced basis^{1} (Eisenstein, 1851; Niggli, 1928), which is a primitive basis unique (apart from orientation) for any given lattice, is at present widely used as a means of classifying and identifying crystalline materials. A comprehensive survey of the principles, the techniques and the scope of such applications is given by Mighell (1976). The present contribution merely aims at an exposition of the basic concepts and a brief account of some applications.
The main criterion for the reduced basis is a metric one: choice of the shortest three noncoplanar lattice vectors as basis vectors. Therefore, the resulting bases are, in general, widely different from any symmetrycontrolled basis, cf. Section 3.1.1.
A primitive basis a, b, c is called a `reduced basis' if it is righthanded and if the components of the metric tensor G (cf. Section 3.1.1) satisfy the conditions shown below. The matrix (3.1.3.1) for the reduced basis is called the reduced form.
Because of lattice symmetry there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique.
Any basis, reduced or not, determines a unit cell – that is, the parallelepiped of which the basis vectors are edges. In order to test whether a given basis is the reduced one, it is convenient first to find the `type' of the corresponding unit cell. The type of a cell depends on the sign of If , the cell is of type I, if it is of type II. `Type' is a property of the cell since T keeps its value when a, b or c is inverted. Geometrically speaking, such an inversion corresponds to moving the origin of the basis towards another corner of the cell. Corners with all three angles acute occur for cells of type I (one opposite pair, the remaining six corners having one acute and two obtuse angles). The other alternative, specified by main condition (ii) of Section 3.1.3.3, viz all three angles nonacute, occurs for cells of type II (one or more opposite pairs, the remaining corners having either one or two acute angles).
The conditions can all be stated analytically in terms of the components (3.1.3.1), as follows:
The geometrical interpretation in the following sections is given in order to make the above conditions more explicit rather than to replace them, since the analytical form is obviously the most suitable one for actual verification.
The main conditions^{2} express the following two requirements:
Condition (i) is by far the most essential one. It uniquely defines the lengths a, b and c, and limits the angles to the range . However, there are often different unit cells satisfying (i), cf. Gruber (1973). In order to find the reduced basis, starting from an arbitrary one given by its matrix (3.1.3.1), one can: (a) find some basis satisfying (i) and (ii) and if necessary modify it so as to fulfil the special conditions as well; (b) find all bases satisfying (i) and (ii) and test them one by one with regard to the special conditions until the reduced form is found. Method (a) relies mainly on an algorithm by Buerger (1957, 1960), cf. also Mighell (1976). Method (b) stems from a theorem and an algorithm, both derived by Delaunay (1933); the theorem states that the desired basis vectors a, b and c are among seven (or fewer) vectors – the distance vectors between parallel faces of the Voronoi domain – which follow directly from the algorithm. The method has been established and an example is given by Delaunay et al. (1973), cf. Section 3.1.2.3 where this method is described.
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 ; then such transformations must be supplemented by interchange(s) of b and c whenever either (3.1.3.3b) or (3.1.3.5b) 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. 3.1.3.1 to 3.1.3.5. Let us represent the vector lattice by a point lattice. In Fig. 3.1.3.1, the net in the ab plane (of which OBAD is a primitive mesh; , ) 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 of that layer, projected in Fig. 3.1.3.1 as P, will be closer to the origin than all others. Then the vector is 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 righthandedness. The result will depend on the position of P with respect to the net. This dependence will now be investigated.
The inner hexagon shown, which is the twodimensional Voronoi domain around O, limits the possible projected positions P of . Its short edges, normal to OD, result from (3.1.3.4b); the other edges from (3.1.3.2a). 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 , corresponding to the projection of points for which . The case has been dealt with, so in order to simplify the drawings we shall assume . Then, for a given value of d, each point within the abovementioned hexagonal domain, regarded as the projection of a lattice node in the next layer, completely defines a lattice based on , and . Diametrically opposite points like P and 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 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 and OA is acute from those for which it is obtuse.
Similarly, , normal to OB, separates the sharp and obtuse values of the angles . It follows that if P lies in the obtuse sector (crosshatched area) between and , the reduced cell is of type I, with basis vectors , , and . Otherwise (hatched area), we have a typeII reduced cell, with and and as shown by and .
Since type II includes the case of right angles, the borders of this region on and are inclusive. Other borderline cases are points like R and , separated by b and thus describing the same lattice. By condition (3.1.3.5c) the reduced cell for such cases is excluded from type II (except for rectangular a, b nets, cf. Fig. 3.1.3.2); so the projection of c points to R, not . Accordingly, this part of the border is inclusive for the typeI region and exclusive (at ) for the typeII region as indicated in Fig. 3.1.3.3. Similarly, (3.1.3.5d) 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 (3.1.3.3d), (3.1.3.3c) and (3.1.3.5f), respectively. The explanation is given in Fig. 3.1.3.1 for (3.1.3.3c): The points Q and represent the same lattice because (diametrically equivalent to Q as shown before) is separated from 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 (3.1.3.5c) and (3.1.3.5d), cf. Fig. 3.1.3.2. A centred orthogonal a, b net of elongated character (shortest net vector in a symmetry direction, cf. Section 3.1.3.5) is depicted in Fig. 3.1.3.4. It yields typeI cells except when [condition (3.1.3.5c)]. Moreover, (3.1.3.3c) eliminates part of the typeI region as compared to Fig. 3.1.3.3. 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 (3.1.3.3a) and (3.1.3.5a), cf. Fig. 3.1.3.5. The simplicity of these bisecting conditions, similar to those for the case mentioned initially, is apparent from that figure when compared with Fig. 3.1.3.3. 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. 3.1.3.4) and by the centred quadratic net (where it merges with the primitive orthogonal net, Fig. 3.1.3.2). In the limit of the hexagonal net, the triangle Ohh in Figs. 3.1.3.4 and 3.1.3.5 is all that remains, it is of type I except for the point O. For the quadratic net, only the typeII region in Fig. 3.1.3.5, then a triangle with all edges inclusive, is left. It corresponds to the triangle Oqq in Fig. 3.1.3.2.
Apart from being unique, the reduced cell has the further advantage of allowing a much finer differentiation between types of lattices than is given by the Bravais types. For twodimensional lattices, this is apparent already in the last section where the centred orthogonal class is subdivided into nets with elongated character and those with compressed character, depending on whether the shortest net vector is, or is not, a symmetry direction. It is impossible to perform a continuous deformation – within the centred orthogonal type – of an elongated net into a compressed one, since one has to pass through either a hexagonal or a quadratic net.
In three dimensions, lattices are of the same character if, first, a continuous deformation of one into the other is possible without leaving the Bravais type. Secondly, it is required that all matrix elements of the reduced form (3.1.3.1) change continuously during such a deformation. These criteria lead to 44 different lattice characters (Niggli, 1928; Buerger, 1957). Each of them can be recognized easily from the relations between the elements of the reduced form given in Table 3.1.3.1 [adapted from Table 5.1.3.1 in International Tables for Xray Crystallography (1969), which was improved by Mighell & Rodgers (1980)]. The numbers in column 1 of this table are at the same time used as a general notation of the lattice characters themselves. We speak, for example, about the lattice character No. 7 (which is part of the Bravais type tI) etc.
^{†}The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985). The capital letter of the symbols in this column indicates the centring type of the cell as obtained by the transformation in the last column. For this reason, the standard symbols mS and oS are not used here.
^{‡}. ^{§} plus . 
In Table 3.1.3.2, another description of lattice characters is given by grouping together all characters of a given Bravais type and by indicating for each character the corresponding interval of values of a suitable parameter p, expressed in the usual parameters of a conventional cell. In systems where no generally accepted convention exists, the choice of this cell has been made for convenience in the last column of this table.
^{†}The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985).
^{‡}These conventions refer to the cells obtained by the transformations of Table 3.1.3.1. They have been chosen for convenience in this table. ^{§}. ^{¶}. ^{††}Setting with unique axis b; ; for both P and I cells, or for C cells. ^{‡‡}This number specifies the centred net among the three orthogonal nets parallel to the twofold axis and passing through (1) the shortest, (2) the second shortest, and (3) the third shortest lattice vector perpendicular to the axis. For example, `2, 3' means that either net (2) or net (3) is the centred one. 
The subdistinctions `centred net = 1, 2 or 3' for the monoclinic centred type are closely related to the description in other conventions. For instance, they correspond to C, A or Icentred cells, respectively, if b is the unique axis and a and c are the shortest vectors perpendicular to b; note that in Table 3.1.3.2 only C and I, not A, cells are listed. From the multiple entries in Table 3.1.3.2 for this type, it follows that the description in terms of b/a is not exhaustive; the distinctions depend upon rather intricate relations (cf. Mighell et al., 1975; Mighell & Rodgers, 1980).
No attempt has been made in Table 3.1.3.2 to specify whether the end points of p intervals are inclusive or not. For practical purposes, they can always be taken to be noninclusive. Indeed, the end points correspond either to a different Bravais type or to a purely geometric singularity without physical significance. If p is very close to an interval limit of the latter kind, one should be aware of the fact that different measurements of such a lattice may yield different characters, with totally differing aspects of the reduced form.
Classification. The reduced basis can be used to derive the Bravaislattice type and the conventional cell parameters, starting from an arbitrary description of the lattice. For this purpose, the reduced form is first derived from the given description, e.g. by means of the algorithm of Křivý & Gruber (1976). Subsequently it is compared with the reduced forms (Table 3.1.3.1) for the 44 lattice characters and transformed to the appropriate conventional cell. Thus the reduced cell is helpful as an accessory in classifications based on conventional cells.
Alternatively, the parameters of the reduced form itself (either of the direct lattice or of the reciprocal lattice) can be used as a basis for determinative classification.
Comparison of lattices. Two lattices, defined by their reduced cells, can be compared on a rigorous basis to find out whether they are identical lattices or are related by one cell being a subcell of the other (Santoro et al., 1980).
Further properties of lattices are discussed in Section 3.1.4.
P. M. de Wolff wishes to thank Dr B. Gruber (Prague) and Dr A. Santoro (Washington) for reading the manuscript for Section 3.1.3 and for suggesting several improvements as well as pointing out errors, especially in Tables 3.1.3.1 and 3.1.3.2.
References
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