I. Terms that define different degrees of similarity between inorganic structures

The following hierarchy of terms is considered based on the degree of structural similarity: isopointal, isoconfigurational, crystal-chemically isotypic, and homeotypic structures. Other relationships of interest are also listed.

(i) they have the same space-group type (as defined in IT , $\S$ 8.2.1) or belong to a pair of enantiomorphic space-group types; and

(ii) the atomic positions, occupied either fully or partially at random, are the same in both structures, i.e . the complete sequence of the occupied Wyckoff positions (including the number of times each Wyckoff position is occupied) is the same for both structures when the structural data have been standardized.

As there are no limitations on the values of the adjustable parameters of the Wyckoff positions (as used in IT ) or on the cell parameters, isopointal structures may have different geometric arrangements and atomic coordinations.

Two structures may be shown to be isopointal if they can be described in such a way that corresponding occupied Wyckoff positions have the same Wyckoff letters; to achieve this, it may be necessary to shift the origin and/or rotate/permute the coordinate system. This can be done by application of the affine normalizer (IT $\S\S$ 8.3.2 and 14.1) or by standardization procedures (Parthé & Gelato, 1984; Gelato & Parthé, 1987).

I.2. Two structures are defined as isoconfigurational (configurationally isotypic ) if:

(ii) for all corresponding Wyckoff positions, both the crystallographic point configurations (crystallographic orbits) and their geometrical interrelationships are similar.

These conditions require the entire configurations of the two structures to be similar. Consequently, all geometrical properties, such as axial ratios, angles between crystallographic axes, values of corresponding adjustable positional parameters (x, y, z), and coordinations of corresponding atoms are similar. The term `similar' used in this definition is discussed below.

Standardization procedures may be necessary to test whether two structures are isoconfigurational.

(ii) the corresponding atoms and corresponding bonds (interactions) have similar physical/chemical characteristics.

Crystal-chemical isotypism may be defined in different ways, depending on the number and nature of the physical/chemical characteristics that are taken into consideration, such as bond strength distribution, bond character, electronegativities assigned to atoms, radius ratios assigned to pairs of atoms, and electronic states.

The use of the term `similar', in the definitions of configurational and crystal-chemical isotypism, arises from the inherent difficulty in defining a priori limits on the similarity of geometrical configurations or physical/chemical characteristics. These limits may differ between different categories of structures; they may also differ according to the purpose of a particular investigation or the physical or chemical property studied. Several approaches are possible in defining limits, such as a priori considerations (modelling) or statistical studies of known structure categories. Different approaches may yield different results. Examples of crystal-chemical isotypes are given in Table 1. Another example is the low-pressure and high-pressure forms of Ce which are crystal-chemically isotypic with regard to most crystal-chemical criteria; they are, however, not crystal-chemical isotypes with regard to electronic states.

**Table 1:** Examples of pairs of structures with different degrees of similarity
	Pnma	$Pa\bar{3}$	I4/mmm	$Ia\bar{3}d$	$Fm\bar{3}m$	$Fm\bar{3}m$	$Fm\bar{3}m$	$Pa\bar{3}$
	$2{\times}4(c),8(d)$ - $4{\times}4(c)$	4(a), 8(c)xxx	2(a),4(e) c/a=2.52-c/a=3.30	16(a),24(c), 24(d),96(h)xyz	4(a)	4(a),4(e)	4(a),8(c)	4(a),8(c)xxx
	LuRuB₂-ScRhSi₂	FeS₂(pyrite)-CO₂	ZrPd₂-Zr₂Pd	Ca₃Al₂Si₃O₁₂-Y₃Fe³⁺₅O₁₂	Cu-Ne	NaCl-PbS	CaF₂-Li₂O	FeS₂(pyrite)-PtP₂
Isopointal	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Isoconfigurational	-	No	*	Yes	Yes	Yes	Yes	Yes
Crystal-chemically isotypic according to
bond-strength distribution	-	-	Yes	No	Yes	Yes	Yes	Yes
bond character	-	-	Yes	No	No	No	Yes	Yes
radius ratios assigned to pairs of corresponding sites	-	-	No	Yes	Yes	Yes	No	Yes
electronegativities assigned to sites	-	-	Nearly yes	Yes	No	Yes	No	Yes

*Notice that in this case the two structures which have c/a ratios differing by 30% are isoconfigurational if the limit of similarity is set above 30% and non-isoconfigurational if it is set below 30%.

(ii) some important physical/chemical characteristics of corresponding atoms are interchanged (reversed).

The characteristics usually considered in this context are electronegativities (positive versus negative charges) and larger versus smaller radii.

I.5. Two structures are defined as homeotypic if one or more of the following conditions required for isotypism are relaxed:

(i) identical or enantiomorphic space-group types, allowing for group-subgroup and group-supergroup relationships;

(ii) limitations imposed on the similarity of geometric properties, i.e. axial ratios, interaxial angles, values of adjustable positional parameters, and the coordination of corresponding atoms;

(iii) site occupancy limits, allowing given sites to be occupied by different atomic species.

The present definition narrows the original concept of Laves (1980). Two structures are considered as homeotypic if all essential features of topology are preserved between them. Relaxation of the geometric limits as in I.5(ii) without relaxation of the isosymmetry condition leads to structure type branches.

Distortion variants (or distortion derivatives): CaTiO₃ (ideal perovskite) $\rightarrow$ KCuF₃, BaTiO₃, GdFeO₃, etc . with subgroup symmetries.

Site-ordering variants (or substitution derivatives): C (diamond) $\rightarrow$ ZnS (sphalerite) $\rightarrow$ Cu₃SbS₄ (famatinite) $\rightarrow$ Cu₂FeSnS₄ (stannite); K(Al_0.25Si_0.75)₄O₈ (sanidine) $\rightarrow$ KAlSi₃O₈ (microcline).

Examples are: closest-packed structures of chemical elements, SiC polytypes, Friauf-Laves phases, micas.

I.7. Interstitial (or `stuffed') derivatives represent compounds in which unoccupied `interstitial' sites (voids) of the basic structure are (progressively) filled by atoms in the derivative structure. In general, the relationship between the unfilled parent (basic) structure and the derivatives based on filling one specific interstitial site approaches homeotypism.

Examples are: ReO₃ $\rightarrow$ CaTiO₃; Mg $\rightarrow$ Mo₂C; SiO₂ (`ideal' tridymite) $\rightarrow$ KNa₃(AlSiO₄)₄ (nepheline).

I.8. 'Recombination' structures are formed when topologically simple parent structures are periodically divided into blocks, rods or slabs which in turn are recombined into derivative structures by means of one or more structure building operations.

The most important of these operations are (1) unit-cell twinning, (2) crystallographic shear planes, (3) intergrowth of blocks, rods or slabs of different structure types, (4) periodic out-of-phase or antiphase boundaries, (5) rotation of rods (blocks) and (6) the vernier principle (Makovicky & Hyde, 1981). For most of these operations, not only parallel but also cyclic examples have been observed. In general, the sites on block, rod or slab interfaces differ in coordination and chemistry from those in the interior. This interface modulation can be accompanied or substituted by an overall long-range compositional modulation.

In cases (1) to (5), the frequency of structure building operators (or, conversely, the size of `undisturbed' structure blocks, rods or slabs between two consecutive operators) can vary by well defined increments so that these phases often occur as members of homologous series.

Twinning on unit-cell scale: PbS $\rightarrow$ lillianite homologous series (Pb_n-1Bi₂S_n+2), e.g. Pb₃Bi₂S₆ (lillianite).

Shear derivatives: TiO₂ (rutile) $\rightarrow$ Ti_nO_2n-1 (Magnéli phases) or TiO₂ (rutile) $\rightarrow$ Nb₂O₅.

Intergrowth on unit-cell scale: olivine-norbergite homologous (polysomatic) series n(Mg,Fe)₂SiO $_4{\cdot}$ Mg(OH,F)₂; biopyriboles, e.g. Mg_3n+1Si_4nO_10n+2(OH)_2n-2; MgCu₂-type and CaCu₅-type slabs $\rightarrow$ Ce₂Ni₇.

Block (rod) rotation: ReO $_3 \rightarrow$ Nb₈W₉O₄₇ or ReO $_3 \rightarrow$ Mo₅O₁₄.

Vernier principle: slabs PbS (100) and PbS (111) $\rightarrow$ Pb₄₆Bi₅₄S₁₂₇ (cannizzarite), Y₆O₅F₈, Pb₂Bi₂S₅ (cosalite), NdCo₄B $_4 \rightarrow$ Sm₁₇(Fe₄B₄)₁₅.

Compositional modulation: Pb₄₆Bi₅₄S₁₂₇ (cannizzarite), FePb₃Sn₄Sb₂S₁₄ (cylindrite).

Note : In each example the sequence indicated is: parent(s) $\rightarrow$ derivative(s).