III. Nomenclature for crystal-chemical formulae

An acceptable nomenclature for crystal-chemical formulae should exhibit the following general characteristics:

(2) It should retain the chemical symbols of the elements and, whenever possible, follow the normal rules of chemical formulae.

(3) It should retain other widely used symbols (e.g. coordination number, dimensionality etc.) as far as possible.

(4) It should not introduce symbols which are already widely used but with a different meaning.

(5) It should be flexible, allowing symbols to be eliminated for simplification, or permitting the inclusion of extra symbols for additional information.

The proposed nomenclature for crystal-chemical formulae is based on the distribution of bond strengths. The spatial distribution of bond strengths in a structure can be either homogeneous or heterogeneous. If the distribution is heterogeneous, certain atoms are more tightly bonded together than others, resulting in finite groups or in assemblages that are infinite in one, two or three dimensions. These assemblages are considered as structural units and the remaining atoms as interstitial atoms.

If the spatial bond-strength distribution is homogeneous, two limiting situations may be discerned: either the structure is based on a three-dimensional framework (examples are diamond or cristobalite with directional bonds), or it is simply a packing of individual atoms (examples are helium, copper or sodium chloride with non-directional bonds). The corresponding sructural units are thus either a framework or the individual atoms, respectively.

There are five main categories of structural units, according to the kind of bond-strength distribution:

A structural unit may be considered to consist of subunits such as single atoms, polyhedra, single rings, single chains or single layers.

A structure can be considered to consist of structural units packed together, with interstitial atoms located between them. If the structural unit is a framework, the interstitial atoms or groups of atoms occupy holes within the framework.

Since the strengths of bonds cannot always be accurately quantified, some ambiguity may exist in assigning a structure to a given category.

III.2.1. General crystal-chemical formulae. Crystal-chemical formulae give detailed structural information on the structural unit(s), their constitution, the packing scheme, the interstitial atoms, and the coordination of the atoms (both interstitial and those contained in the structural units).

Symbols for atoms belonging to the structural unit(s) are placed between square brackets, [ ], and the packing information between angle brackets, $\langle\,\,\, \rangle$ . The information on constitution which relates to the structural unit as a whole is placed within curly brackets, { }. However, the constitutional information which relates to subunits of the structural unit(s) may be expressed either within curly brackets or as trailing superscripts to the chemical elements or subunits inside the structural unit.

Curly brackets with constitutional information precede and angle brackets for packing information immediately follow the structural unit to which they refer.

Information concerning interstitial atoms and/or groups of atoms should generally be placed before or after that on the structural unit(s) in the sequence that chemical formulae are usually written.

In accordance with IUPAC (1990) rules, the valency state of each atom is expressed immediately after its chemical symbol by a Roman numeral in parentheses [e.g. Fe(III)], a superscripted Roman numeral (e.g. Fe $^\textrm{III}$ ), or by a superscripted Arabic numeral followed by the sign + or - (e.g. Fe³⁺).

The coordination of each atom, either interstitial or in the structural unit, is expressed within square brackets as a trailing superscript to the chemical symbol. If additional constitutional information related to subunits is given within the square brackets for the structural unit, then it should be placed between Japanese quotation marks (called `brackets') below), $\lceil\,\,\, \rfloor$ , as an additional trailing superscript: $A^{n+[\,\,]\lceil \,\rfloor}$ .

**Table 3:** Examples of crystal-chemical formulae and *Bauverband* description of inorganic structures
Compound		Crystal-chemical formulae with different structural interpretations and degrees of simplification			Bauverband description (lattice-complex notation)
He (hex.)	${}^0_\infty[\textrm{He}]\langle h\rangle$	$[\textrm{He}]^h$	${}^3_\infty[\textrm{He}^{[aco]}]$	$\mathbi{E}$ or $(\mathbi{h})\mathbi{H}$	He	P6₃/mmc
Cu	${}^0_\infty[\textrm{Cu}]\langle c\rangle$	$[\textrm{Cu}]^c$	${}^3_\infty[\textrm{Cu}^{[co]}]$	$\mathbi{F}$ or $(\mathbi{c})\mathbi{H}$	Cu	Fm3m
C (diamond)	${}^3_\infty[\textrm{C}^{[4t]}]$	${}^3_\infty[\textrm{C}]^{[4]}$	${}^3_\infty[\textrm{C}^{t}]$	$\mathbi{D}$	C	$Fd\bar 3m$
NaCl	$\textbf{Na}^{[6]}{ }^{\,0}_\infty[\textrm{Cl}^{[6]}]$	$\textrm{Na}^o[\textrm{Cl}]^{c}$	${}^3_\infty[\textrm{Na}^{[6o]}\textrm{Cl}^{[6o]}]$	$\mathbi{F} + F'$	NaCl	Fm3m
SiO₂ (quartz)	${}^3_\infty[\textrm{Si}^{[4t]}\textrm{O}_2]$	${}^3_\infty[\textrm{Si}^{[4t][1;4]}\textrm{O}_2]$	${}^3_\infty[\textrm{Si}^{t}\textrm{O}_{2}]$	${}^+\mathbi{Q[.\mathrm{4}t_{\textit{c}}]}$	SiO $_\mathbf{2}$	P6₂22
SiO₂ (cristobalite)	${}^3_\infty[\textrm{Si}^{[4t]}\textrm{O}_2]$	${}^3_\infty[\textrm{Si}^{[4t][1;4]}\textrm{O}_2]$	${}^3_\infty[\textrm{Si}^{t}\textrm{O}_{2}]$	$\mathbi{T} + D$	SiO $_\mathbf{2}$	$Fd\bar 3m$
FeS₂ (pyrite)	$\textrm{Fe}^{[6o]}\{^0_\infty\}[\textrm{S}_2^{[3;(1+2)]}]$	$\textrm{Fe}^{[6o]}\wedge[\textrm{S}_2^{[(3;1)t]}]$	$\textrm{Fe}^{o}\{g\}[\textrm{S}_{2}]^c$	$\mathbi{F}\mathrm{(\square2\mathbi{l})} + F'$	FeS $_\mathbf{2}$	$Pa\bar{3}$
FeS₂ (marcasite)	$\textrm{Fe}^{[6o]}\{^0_\infty\}[\textrm{S}_2^{[3;(1+2)]}]$	$\textrm{Fe}^{[6o]}\wedge[\textrm{S}_2^{[(3;1)t]}]$	$\textrm{Fe}^{o}\{g\}[\textrm{S}_{2}]$	$\mathbi{..nI}\mathrm{[.6\mathbi{o}_{2\mathbi e,c}]}$	FeS $_\mathbf{2}$	Pmnn
(Mg,Fe)₂SiO₄ (olivine)	$\textrm{(Mg,Fe)}_2^{[6o]}\{^0_\infty\}[\textrm{Si}^{[4t]}\textrm{O}_4]$	$\textrm{(Mg,Fe)}_2^{[6o]}\{^0_\infty\}[\textrm{Si}^{[,4t]}\textrm{O}_4]$	$\textrm{(Mg,Fe)}_2^{o}\textrm{Si}^{t}[\textrm{O}]_4^h$	$\mathrm{(\mathbi{h})\mathbi{nC}_{22}} + 00{1\over2}I2_{xy}, A_{211}{1\over4}{1\over4}$	$F(\mathrm{Mg,Fe})_2\textrm{Si}\mathbf{O_4}$	Pmcn
MgAl₂O₄ (spinel)	${}^3_\infty[\textrm{Mg}^{[4t]}Al_2^{[6o]}\textrm{O}_4^{[1,3;12co]}]$	$\textrm{Mg}^{[,4t]}Al_2^{[,6o]}\textrm{O}_4$	$\textrm{Mg}^{t}\textrm{Al}_2^{o}[\textrm{O}]_4^c$	$\mathrm{\mathbi{F}_{222}^{\prime\prime\prime}} + D, T'$	$\mathrm{MgAl}_2\mathbf{O_4}$	$Fd\bar 3m$
CaMgSi₂O₆ (diopside)	$\textrm{Ca}^{[8]}\textrm{Mg}^{[6o]}\{^1_\infty\}[\textrm{Si}_2^{[4t][1;2]}\textrm{O}_6]$		$\textrm{Ca}^{[8]}\textrm{Mg}^o\,{}^1_\infty[\textrm{Si}^t_2\textrm{O}_6]$
KAl₃Si₃O₁₀(OH)₂ (muscovite)	$\textrm{K}^{[6+6]}\{^2_\infty\}[\textrm{Al}_2^{[6o]}\{^2_\infty\}[(\textrm{Al}_{0.5}\textrm{Si}_{1.5})^{[4t][1;3]}\textrm{O}_5]_2(\textrm{OH})_2]$		$\textrm{K}^{[6]}\textrm{Al}_2^o\,{}^2_\infty[\textrm{Al}^t\textrm{Si}_3^t\textrm{O}_{10}](\textrm{OH})_2$
LaP₂ (HT form)	$\textrm{La}_{4}\{^0_\infty\}[\textrm{P}_2^{[;1]}\textrm{P}^{[;2]}]\{^0_\infty\}[\textrm{P}^{[;1]}_2\textrm{P}_3^{[;2]}]$		$\textrm{La} \wedge [\textrm{P}_3] \wedge [\textrm{P}_5]$
Ba₃AlSb₃	$\textrm{Ba}_{6}\{^0_\infty\}[\textrm{Al}_2^{[;4t][2;1]}\textrm{Sb}_6]$
Ca₃AlAs₃	$\textrm{Ca}_{3}\{^1_\infty\}[\textrm{Al}^{[;4t][1;2]}\textrm{As}_3]$
(Mn,Fe)AlPO₄(OH)₂H₂O (eosphorite)	$\textrm{(Mn,Fe)}^{[6o][2;2]}\textrm{Al}^{[6o][1;2]}\{^0_\infty\}[\textrm{P}^{[4t]}\textrm{O}_4](\textrm{OH})_2.\textrm{H}_2\textrm{O}$
Na₃AlF₆ (cryolite)	$\textrm{Na}_3\{^3_\infty\}[\textrm{Al}^{[6o][1;6]}\textrm{F}_6]$
Ca₃Si₂O₇ (rankinite)	$\textrm{Ca}_3\{^0_\infty\}[\textrm{Si}_2^{[4t][1;1]}\textrm{O}_7]$
Ca₃Si₂O₇ (kilchoanite)	$\textrm{Ca}_6\{^0_\infty\}[\textrm{Si}^{[4t][0;0]}\textrm{O}_4] \{^0_\infty\}[\textrm{Si}_2^{[4t][1;1]}\textrm{Si}^{[4t][1;2]}\textrm{O}_{10}]$

If several distinct structural units are present, each is considered separately with its information in curly brackets followed by that in square brackets, for example:

$\begin{displaymath} A_a^{[\alpha]}\{\, \}[B_b^{[\beta]}C_c^{[\gamma]}]\{ \, \}[D... ...^{[\varepsilon]}]\langle \,\,\rangle F_f^{[\zeta]}G_g^{[\eta]}.\end{displaymath}$

The packing information within angle brackets describes the way the two different structural units pack together.

The hierarchy of bonds leads to a hierarchy of structural units when several degrees of bond strengths may be discerned in a structure. This often leads to weaker bond-strength units incorporating previous more strongly bonded units, and can be expressed by multiple brackets, with the central brackets referring to the structural unit having the strongest bonds:

$\begin{displaymath} A_a^{[\alpha]}\{\}[B_b^{[\beta]}\{\}[C_c^{[\gamma]}D_d^{[\de... ...gle\,\, \rangle]\langle\,\, \rangle F_f^{[\zeta]} G_g^{[\eta]}.\end{displaymath}$

The proposed formula can be used with any amount of any selection of structural information depending on the purpose of the study; see below.

III.2.2. Constitution of structural units. The constitution of a structural unit expresses its extensional and geometrical `structural'. i.e. the way the structural unit is built from its subunits, which may be polygons, polyhedra or any other clusters.

Some of the constitutional aspects are concerned with the structural unit as a whole, whereas other aspects are only concerned with the way each subunit is linked to other subunits. The former include dimensionality, multiplicity, branchedness and periodicity.

(i) The dimensionality is the number of dimensions in which a structural unit has infinite extension. It is zero for individual atoms and finite groups and one, two or three for infinite chains, sheets and frameworks, respectively. The corresponding symbols to be used in a crystal-chemical formula are ${}_\infty^0, {}_\infty^1, {}_\infty^2$ and ${}_\infty^3$ .

The following specific symbols may be used instead of ${}_\infty^0$ for 0-dimensional structural units:

Examples are: Cs $_2{\wedge}[$ S₆], Na $_4{\bigcirc\kern-9pt\vee}\,\,[$ Si₄], Cu₆{r}[Si₆O₁₈].6H₂O.

If dimensionality is the only information expressed, the ${}_\infty^n$ and the pictorial symbols may be used without curly brackets. Otherwise, curly brackets are compulsory in order to avoid ambiguity.

The symbol {a} is not needed when several individual atoms, $A, B, C,\dots$ , considered as structural units, are written $[A] [B] [C]\dots$ . When only one atom symbol is placed within square brackets, it means that the structural unit is reduced to an individual atom. However, if the same atom symbols are written [ABC], then it is necessary to add {a} in front of the square brackets.

In the case of group structures, e.g. ring, chain fragment, and cage structures, the number of atoms of each chemical element within square brackets must be equal to the number of atoms of each chemical element in the finite group.

(ii) The multiplicity of a structural unit is the number of single subunits, e.g. polyhedra, single rings, single chains or single layers which are linked to form a complex structural unit of the same dimensionality.

(iii) With regard to branchedness, finite structural units and single chains are called unbranched if they contain no subunits that are linked to more than two other units. They are called branched if they do. In addition, complex structural units, which can be considered as formed by linking unbranched (branched) finite structural units or single chains, are described as unbranched (branched).

(iv) The periodicity of a structural unit of infinite extension is the number of subunits, excluding branches, within one repeat unit of the chain from which the structural unit can be generated by successive linking.

For details of concepts under (ii)-(iv) see Liebau (1982, 1985); a publication on their usage in the present formulae is in preparation.

The main constitutional aspects concerned only with the way each subunit is linked to the other subunits are linkedness and connectedness.

(i) The linkedness is the number L of peripheral atoms shared between two subunits. The value of linkedness is zero for an isolated subunit. It is one or two for two subunits sharing a corner or an edge, respectively, and it is three or more for two subunits sharing a face. The average linkedness value of a subunit may be non-integral if the given subunit shares corners plus edges with different adjacent subunits.

(ii) The connectedness of a subunit is the total number s of adjacent subunits with which it shares common atoms, irrespective of its linkedness with a particular adjacent subunit. A subunit may be singular (isolated), primary (linked to only one other subunit), secondary (linked to two others), etc.

The specific values L₁, L₂ etc. of linkedness and/or s of connectedness of a subunit are written within `Japanese brackets' as trailing superscripts to its central atom, by analogy with the coordination symbols. The first entries in the Japanese brackets are the different values of L_n, separated from the value of s by a semicolon. The general formula for a structural unit with only one kind of subunit then reads

For example, SiO₂ exists in a number of polymorphs having different values of linkedness and connectedness of the SiO₄ tetrahedra:

$\begin{displaymath} {}_{\infty}^1[\textrm{Si}^{[4t][2;2]}\textrm{O}_2];\end{displaymath}$

$\begin{displaymath} {}_{\infty}^3[\textrm{Si}^{[4t][1;4]}\textrm{O}_2];\end{displaymath}$

$\begin{displaymath} _{\infty}^3[Si^{[6o][1,1,1,1,1,1,1,1,2,2;10]}O_2],\end{displaymath}$

A structural unit can often be generated from a part of either lower or the same dimensionality by a simple geometrical process that usually represents an infinitely repeated translation. This imaginary geometrical process is called condensation because it emphasizes the way a chain can be generated from a group, a sheet from a chain, and a framework from a sheet. It also reveals certain similarities between different structural units, and a specific composite notation for the structural units has been developed which emphasizes this interrelationship ( $\S$ V).

III.2.3. Packing of structural units. The packing of structural units expresses the three-dimensional arrangement in space. When the structural units are individual atoms, the known nomenclature for describing the packing of atoms (three-dimensional and layer-stacking descriptions) may be used. When the structural units are groups, their centres of gravity may be used with the same nomenclature as for the packing of atoms. However, this will be an incomplete description because of the lack of information on the orientation of the groups.

Packing of structural units in structures based on groups, infinite chains or sheets may be treated by layer description. Such a layer description consists of slicing the structure into layers which, by stacking, completely generate the original crystal structure. Structural units should be preserved intact in the process of slicing. The structure is then described by the packing of structural units in the layer and by a set of stacking operators.

The layer description can also be applied to framework structures taking into consideration the fact that the units operated upon are parts of a single framework.

With respect to the nomenclature for the packing of structural units, only the symbols for cubic closest packing, c, and hexagonal closest packing, h, and their sequential combination are adopted here. When no other packing information is provided these symbols may be given as trailing superscripts to the square brackets which contain the structural unit. In this case, angle brackets are not compulsory. Any other packing information, particularly the packing (or stacking) symbolism used by individual authors should be given in angle brackets on the line.

$\begin{displaymath}[ABC] ^c \hbox{ or } [ABC]{\langle}{\ldots}{\rangle}.\end{displaymath}$

If packing information is to be given for a set of atoms which does not constitute a structural unit, the symbol should be placed within vertical bars followed by the packing information:

$\begin{displaymath} \vert ABC\vert^c \hbox{ or } \vert ABC\vert{\langle}{\ldots}{\rangle}.\end{displaymath}$

Dimensionality of structural unit		Category of structural unit

0-dimensional		individual atoms
0-dimensional		groups (i.e. rings, chain fragments, cages)
1-dimensional		chains
2-dimensional		sheets
3-dimensional		frameworks.

$A_a^{[\alpha]}B_b^{[\beta]}$	$\{\quad \}$	$[C_c^{[\gamma]\lceil{\varphi}\rfloor}D_d^{[\delta]\lceil{\chi}\rfloor} E_e^{{\varepsilon}]}]$	$\langle\,\,\, \rangle$	$F_j^{[\xi]}G_g^{[\eta]}$
interstitial atoms	constitution of structural unit	structural unit	packing of structural units	interstitial atoms.

individual atom: {a}
group: {g}	ring:	$\begin{displaymath} \{r\} \hbox{ or }\bigcirc\end{displaymath}$
	chain fragment:	$\begin{displaymath} \{f\} \hbox{ or }\wedge\end{displaymath}$
	cage:	$\begin{displaymath} \{k\} \hbox{ or }\bigcirc\kern-11pt\vee\end{displaymath}$