Niggli-formula

The composition of a compound is intimately related to the way of linking the polyhedra. An atom X with coordination number c.n.(X) that acts as a common vertex to this number of polyhedra makes a contribution of l/c.n.(X) to every polyhedron. If a polyhedron has n such atoms, this amounts to n/c.n.(X) for this polyhedron. This can be expressed with Niggli formulae, as shown in the following sections. To specify the coordination polyhedra, the formalism presented at the end of Section 2.1 and in Fig. 2.2 (p. 5) is useful. 

If the anions in NaCl (see Fig. 2.18) or in NiAs are approximately allocated to cubic or hexagonal close-packing respectively, then formally the cations occupy all of the octahedral interstices. Niggli formulae provide information concerning the mutual co-ordination numbers Thus, NaCle/e s or NiAsg/e 3 mean that both anions and cations are octahedrally coordinated°1by counterions°° Considering ZnS as an example the zinc-blende (sphalerite) structure (Fig. 2.23) can be viewed as a cubic... 

Fig. 5.3a In the rock salt structure (e.g. white spheres Cl, black spheres Ag) all the octahedral interstices of the close-packed anion sublattice are occupied by cations (filled-in circles). Accordingly, the tetrahedral spaces are interstitial sites (see asterisk). The Niggli-formula (cf. Section 2.2.7) for the cluster that contains the interstitial ion is (Agi(ClAg6/6)4/i) - For the vacancy (remove a black sphere) we have to formulate (VAg(ClAg5/6)6/l) -...

Fig. 5.3b In the CaF2 structure (white spheres Ca, black spheres F) the fluoride ions occupy sdl the tetrahedral interstices of a formedly close-packed sublattice. The octahedral spaces are interstitial sites (see asterisk). As indicated, these are also the centres of cubes formed by regular F ions (filled-in circles). (Niggli-formulae would be (Fi(CaF8/4)6/i) for the interstitial and (VF(CaF7/4)4/i)+ for the vacancy cluster.)...

Another type of notation, introduced by P. Niggli, uses fractional numbers in the chemical formula. The formula Ti06/, for instance means that every titanium atom is surrounded by 6 O atoms, each of which is coordinated to 3 Ti atoms. Another example is NbOCl3 = NbC C C /i which has coordination number 6 for the niobium atom (= 2 -)- 2 + 2 = sum of the numerators), coordination number 2 for the O atom and coordination numbers 2 and 1 for the two different kinds of Cl atoms (cf. Fig. 16.11, p. 176). 

The symbols suggested by Jensen, based on Niggli s proposals, describe the local coordination by means of coordination number ratios. For instance, a formula AEm/n will indicate a binary compound where m is the coordination number (defined as the nearest-neighbour number (NNN)) of atoms E around A and n is the similarly defined coordination number of A around E. 

In 1922 Shubnikov proposed his fundamental law of crystal chemistry which drew attention to the relationship between the frequencies with which atoms appear in the chemical formula and the multiplicities of the Wyckoff positions they occupy. A similar relationship had been pointed out earlier by Niggli (1918). A more powerful version of Shubnikov s law that reflects the role of symmetry as well as multiplicity can be stated as ... 

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