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IV. The Bauverband approach

The Bauverband terminology offers a brief description of crystal structures that can be used for explaining relationships between structure types as well as for their classification, recognition of configurational isotypes and for the retrieval of structures with specified relationships.

A crystallographic Bauverband may be defined as a three-dimensionally periodic arrangement of points occupied by atoms or polyhedra of atoms with definite geometric properties: it represents a connectivity pattern typical for a given structure type and it represents, or approximates, a sphere packing with typical self-coordination and with several types of voids for interstitial atoms. The arrangement of such points in a unit cell with specified space group may be described by the parameters of one point position (homogeneous Bauverband) or by the parameters of two or more independent point positions (heterogeneous Bauverband). A Bauverband is described by a combination of applicable original or transformed invariant lattice complexes [henceforth denoted as invariant LC] (IT $\S 14.5.1$), together with a symbol for the circumscribed coordination polyhedra and their orientation (when applicable).

The Bauverband describes the essential part of the structure type. Positions of atoms not involved in the Bauverband are described using the same symbolism as used for the Bauverband. To explore different structure-type relationships, different Bauverband symbols or even choices can be selected in the same structure. The Bauverband is designated by bold-face characters and bold-face square brackets in the structure-type symbol.

The invariant LC in the characteristic Wyckoff positions are represented mainly by capital letters ; LC with equipoints at the nodes of the Bravais lattice are designated by their appropriate lattice symbols: P, A, B, C, I, F, R (Hermann, 1960). The other invariant complexes are designated by letters that recall some structural features of a given complex, for example: D (derived from the diamond structure), E (hexagonal close packing), Q (Si atoms in high-temperature quartz), W ($\beta$-W structure), as well as the two-dimensional complexes G (graphite layer) and N (Kagomé net). Invariant LC in their standard descriptions are given in Table 14.3 of IT.

Those special point configurations of variant LC which have higher symmetry than the symmetry of the variant LC itself, i.e. the so-called limiting complexes (Grenzkomplexe) (IT $\S$ 14.3.2) can also be employed in Bauverband symbolism. Such point configurations are designated by the applicable invariant LC symbols plus small letters showing the direction of positional freedom, e.g. Fxxx in site 4(a) of space group P2₁3.

Some of the non-standard settings of an invariant LC can be described by a shifting vector added in front of the symbol. The shifting vector is given in terms of fractional coordinates; for example, in the CsCl structure type, Cl is considered the Bauverband which is expressed as CsCl, and the entire structure type is described as $\mathbi{P} + \frac{1}{2}\frac{1}{2}\frac{1}{2}P$. The most common shifting vectors have been abbreviated: $P^{\prime}$ represents $\frac{1}{2}\frac{1}{2}\frac{1}{2} P, F^{\prime\prime}$ represents $\frac{1}{4}\frac{1}{4}\frac{1}{4} F$ and $F^{\prime\prime\prime}$ represents $\frac{3}{4}\frac{3}{4}\frac{3}{4} F$; for example $\mathbi{F} + F^{\prime}$ represents NaCl whereas $\mathbi{F} + F^{\prime\prime}$ represents sphalerite, ZnS.

In many structures, the unit cell represents for certain atoms a multiple of the basic cell in standard setting, e.g. the unit cell of spinel represents for oxygen atoms a 2 $\times$ 2 $\times$ 2 multiple of the standard F-centred cell. In general, a transformation matrix which expresses the new basis vectors in terms of the standard ones must be given. The order of the transformation matrix is given by the value of the determinant of this matrix, e.g. 8 in the above case. Thus, the Bauverband description for spinel is MgAl₂O₄ and $\mathbi{F^{\prime\prime\prime}_\mathbf{222}xxx}+D,T^{\prime}$. Fxxx in this formula expresses the fact that in the new cell oxygen is described by a limiting complex with $x \sim \frac{3}{8}$. Mg forms the D and Al the $\frac{1}{2}\frac{1}{2}\frac{1}{2}T$ point configuration.

For complicated structures, efficient description can be achieved if complicated atomic configurations are understood as polyhedra - centred or not, connected or isolated - which are circumscribed about the points of an invariant LC. The same symbolism was chosen for these polyhedra as is used for coordination polyhedra in Table 2. However, centred polyhedra are specified by a dot, e.g. [.4t], the empty ones by a small square, e.g. [$\square$4t]. The linkedness of these polyhedra, i.e. corner, edge and face sharing is described by subscripts c, e and f, respectively. These are equivalent to L = 1, 2 and 3 or more in $\S$ III.2.2. The mutual orientation of these polyhedra is expressed by selected symmetry operations of the space group which are given as a symbol preceding the capital letter.

Examples are: $\mathbi{I[.\mathbf{4}t]}$ for SiF₄ in $I\bar{4}3m$; $\mathbi{I[.\mathbf{12}i]}$(or $\mathbi{I[\mathbf{12}i]}$ + I for W$\mathbf{Al_{12}}$ in $Im\bar{3}$ and $\mathbi{..nI[.\mathbf{12}i]}$ + W in $\mathbf{U_{0.25}H_3}\mathrm{U}_{0.75}$ (i.e. U$\mathbf{H_3}$ in $Pm\bar{3}n$ with the diagonal glide plane ..n generating the second icosahedron [12i] at $\frac{1}{2}\frac{1}{2}\frac{1}{2}$ in contrast to the translation symmetry in the previous case.

If the Bauverbände in a set of structures belong to the same point configuration (the same LC), they form a family. Families are divided into main classes according to the transformation matrix and/or according to splitting of the `parent' LC (Hellner, 1986). Structure types which can be described by the subtraction of one or more LC from the `parent' LC of a main class form subclasses. Examples selected from the F family are: $\mathbi{F}$ (1st order) describes Cu, ZnS, PbS etc. whereas $\mathbi{[F - P] = J}$ describes Re$\mathbf{O_3}$, CaTi$\mathbf{O_3}$ etc.; $\mathbi{F_\mathbf{222}}$ (8th order) describes spinel, MgAl$_2{\mathbf{O}}_4$, and pentlandite, (Ni, Fe, Co)$_9\mathbf{S_8}$, whereas the subclass $\mathbi{[F^{\prime\prime\prime}_\mathbf{222}-I(\mathbf{4}t)]}$ describes tetrahedrite, Cu₁₂Sb$_4\mathbf{S_{13}}$.

For structures in which layer description is essential, two-dimensional LC have been used together with corresponding layer-stacking sequences (Hellner, 1986).

Copyright © 1990, 1998 International Union of Crystallography