OD approach to polytypism: examples, problems, indications

Stefano Merlino*

Dipartimento di Scienze della Terra, Universita di Pisa, Via S. Maria 53, 56126 Pisa, Italy

Received October 3, 2008; accepted November 25, 2008

Polytypism / OD structures / MDO structures

Abstract. The relationships between polytypism and ODstructures are discussed, with presentation of some exam-ples in different classes of compounds: polymorphism andpolytypism in 9,10 phenantrenequinone; OD structures inlatiumite and tuscanite family; polytypism in kaolinite fa-mily.

The basic elements of OD theory are given, both forstructures consisting of OD layers all of the same kindand for structures consisting of two or more distinct kindsof OD layers, and the advantages of an OD approach topolytypism, together with its limitations, are presented anddiscussed.

Introduction

There is still an uncomfortably widespread belief thatpolytypism is a phenomenon occurring only in layeredstructures, such as the hydrous phyllosilicates (kaolinites,pyrophyllite and talc, chlorites, micas), the layered doublehydroxides, etc. . . ., or in the so called ‘classical poly-types’, such as the structure-types of SiC, CdI2, character-ized by the ‘A B C’ stacking of close-packed layers.

Actually polytypism is a phenomenon which is widelydisplayed by compounds in all classes, including organiccompounds, as it will be shown in one of the case studiesin this paper.

An important problem in the study of the polytypicaspect of a structural family is the definition of an appro-priate method to derive all the main polytypes in the fa-mily and to neatly and carefully define the metrics andsymmetry of each one, as well as the relationships amongall of them.

It is proper to recall the definition of polytypism pre-sented by the IUCr Ad-hoc Committee on the Nomencla-ture of Disordered, Modulated and Polytypic Structures(Guinier et al., 1984):

“An element or compound is polytypic if it occurs inseveral different structural modifications, each of whichmay be regarded as built up by stacking layers of (nearly)identical structure and composition, and if the modifica-tions differ only in their stacking sequence. Polytypism is

a special case of polymorphism: the two-dimensionaltranslations within the layers are (essentially) preservedwhereas the lattice spacings normal to the layers vary be-tween polytypes and are indicative of the stacking period.No such restrictions apply to polymorphism.

Comment: The above definition is designed to be suffi-ciently general to make polytypism a useful concept.There is increasing evidence that some polytypic struc-tures are characterized either by small deviations fromstoichiometry or by small amounts of impurities. (In thecase of certain minerals like clays, micas, ferrites, devia-tions in composition up to 0.25 atoms per formula unit arepermitted within the same polytypic series: two layerstructures that differ by more than this amount should notbe called polytypic.) Likewise, layers in different polyty-pic structures may exhibit slight structural differences andmay not be isomorphic in the strict crystallographic sense.

The Ad-hoc Committee is aware that the definition ofpolytypism above is probably too wide since it includes,for example, the turbostratic form of graphite as well asmixed-layer phyllosilicates. However, the sequence andstacking of layers in a polytype are always subject towell-defined limitations. On the other hand, a more gener-al definition of ‘polytypism’ that includes ‘rods’ and‘block’ polytypes may become necessary in the future.”

An implicit assumption in the preceding definition isthat the structural layer which in its different stacking se-quences gives rise to the distinct polytypes does not re-quire to be precisely defined, being evident on the basis ofits crystal chemical characters. Critics on this point havebeen raised by Durovic (1992), who in his thorough dis-cussion of polytypism takes the following position on the‘official definition’ of polytypism: “it specifies neither the‘layers’ (except for their two-dimensional periodicity), northe limitation concerning their sequence and stackingmode, and it does not state the conditions under which apolytype belongs to a family” (Durovic, 1992). The re-quired specifications, limitations and conditions are clearlyindicated in the treatment of polytypic features on the ba-sis of the OD (order-disorder) theory.

The theory of OD structures was worked out by Dorn-berger-Schiff (1956, 1964a, 1966); a simple presentationmay be found in Durovic (1997) and Merlino (1997); themost recent account is given in Ferraris et al. (2008) (inthis last publication a full chapter is devoted to ‘Polytypesand polytype categories’ with discussion of a series of im-

Z. Kristallogr. 224 (2009) 251–260 / DOI 10.1524/zkri.2009.1133 251# by Oldenbourg Wissenschaftsverlag, Munchen

* e-mail: merlino@dst.unipi.it

portant problems which are not dealt with in this review,that mainly concentrates on the relationships between ODtheory and polytypism).

Actually OD theory presents a neat definition of poly-types, a definition which relies on precise symmetry con-cepts, and allows a rigorous discussion of the structuralrelations in polytypic families (and more generally of awhole class of disorder phenomena). In the following,after presentation of the basic elements of OD theory, weshall present some examples of an OD approach to polyty-pic phenomena.

OD structures built up with equivalent layers

OD structures built up by equivalent layers occur whenthe layers may stack according to two (or more) geometri-cally equivalent ways. The various disordered or orderedsequences of the two (or more, say Z) stacking ways giverise to a family of infinite disordered or ordered structures(polytypes): pairs of adjacent layers are geometricallyequivalent (namely congruent or enantiomorphous) in allthe structures of the family (OD principle).

The common symmetry properties of the whole familyare fully described by giving the symmetry of the layer,expressed by one of the 80 layer groups [consisting of theintralayer (or l-) Partial Operations (POs); the partialcharacter of the operations means that they are not neces-sarily valid for the whole structure], as well as the set ofoperations (the interlayer s-POs) converting a layer intothe succeeding one. Both l- and s-POs do not form asymmetry group; they form a Brandt ‘groupoid’ (Brandt,1926) and are represented through an ‘OD groupoid fa-mily symbol’ like:

l operations C m 2 (m)s operations {ns,2 2s (nr,s)}

where, as usual in the OD theory, the parentheses in thethird position of each line indicate that only the basic vec-tors a and b are translation vectors corresponding to theperiodicities of the single layer, whereas the third vectorc0, spanning the whole width of the layer, is not a transla-tion vector. In the example the single layer has C centredlattice, mirror planes normal to a and c0 and a twofoldaxis parallel to b. The s-POs presented in the second lineare: n glide normal to a with translational component sb/2þ c0; n glide normal to c0 with translational componentra/2 + sb/2; a twofold screw axis with translational com-ponent sb/2.

One of the most important results of the OD theorywas just the systematic derivation of the sets of s opera-tions compatible with the symmetry of each layer group(Dornberger-Schiff, 1964a). The knowledge of these setsis extremely helpful in deriving the symmetry of particularordered sequences and in dealing with the diffraction as-pects of the whole family. In fact, all disordered and or-dered members of the OD family present a common set ofreflections (family reflections), whereas they can be distin-guished on the basis of additional ‘characteristic’ reflec-tions. The family reflections correspond to a fictitious

structure, periodic in three dimensions, closely related tothe structures of the family and called ‘family structure’(more loosely indicated as average structure). It may beobtained from a polytype of the family, by superposing Zcopies of it translated by the vector (or vectors) corre-sponding to the possible Z positions of each OD layer.The family reflections are always sharp and their set gen-erally displays diffraction symmetry higher than that corre-sponding to the diffraction symmetry of the various poly-types. The characteristic reflections may be sharp, as wellas more or less diffuse, sometimes appearing as continu-ous streaks running in the direction normal to the layers.Often it is just the presence of diffuse reflections, togetherwith partial diffraction enhancement of symmetry andnon-space-group systematic absences, as well as with thepolytypic aspects, which points to the OD character of thephase under study.

There is another aspect in which OD approach maygive a useful contribution in dealing with polytypic fa-milies. In all the polytypic families a small number of‘main’ polytypes exist, generally those most frequently oc-curring, which are called ‘simple’, ‘standard’ or ‘regular’.OD approach presents a neatly defined geometrical criter-ion by which the particular status of a small number ofpolytypes, named polytypes with Maximum Degree of Or-der (MDO structures), may be recognized. These are themembers in which not only pairs but also triples (andquadruples, . . ., n-tuples) of consecutive layers are geomet-rically equivalent [this definition is strictly valid for theOD structures built up with equivalent layers we shall dis-cuss in the following pages; a more general definition(Dornberger-Schiff, 1964a; Ferraris et al., 2008) will bepresented when we shall deal with OD structures built upwith two (or more) kinds of OD layers].

9,10 Phenatrenequinone: polymorphsand polytypes

The structural features of 9,10 phenantrenequinone (PHQ)have been discussed by Matsuzaki et al. (1987) who de-scribed two different forms of it: a monoclinic a form,prepared by vapour deposition and a triclinic b form ob-tained by crystallization from chloroform. Preparations ofPHQ crystals, followed by structural studies were carriedon by Rae and Willis (2003) who have shown that bothpolymorphs present polytypic forms, which appear as dis-tinct crystals for the b polymorph and as distinct domainsin crystals of the a polymorph.

Recently Marchetti et al. (in preparation) have carriedon new preparations, obtaining crystals of the four distinctforms and the results of their structural studies are used inthe following analysis.

The ‘partial enhancement of symmetry’ and ‘non-space-group absences’ observed by Rae and Willis (2003)in crystals of both polymorphs are clearly pointing to theirOD character and just on the basis of an OD approach weshall discuss the polytypic relationships between the dis-tinct forms of each polymorph.

252 S. Merlino

OD character of 9,10 phenantrenequinone a

The unit cell dimensions and space group symmetry of thea1 and a2 forms of polymorph a, prepared and studied byMarchetti et al. (in preparation), are here reported:

a1 P2/a, a ¼ 7.743, b ¼ 7.989, c ¼ 16.476 �A, b ¼ 103.56�

a2 Pna21, a ¼ 7.732, b ¼ 7.986, c ¼ 16.000 �A

Both forms are built up with layers presenting layer groupsymmetry P12/a1, characterized by translation vectors a,b, third basic vector c0, with values a ¼ 7.73, b ¼ 7.99 �A;c0 ¼ c/2 ¼ 8�A, b ¼ 90� (all the values are given in termsof the parameters found for the a2 form) and representedin Fig. 1.

Layers with layer group symmetry P12/a1 are compati-ble with two possible OD groupoid family symbols (Dorn-berger-Schiff, 1964a; Ferraris et al., 2008). One of them

P 1 2/a (1)

{2r � 1/ns,2 1 (22/nr,s)}(1)

just corresponds to the present case, with the valuesr ¼ 1=2, s ¼ 1 of the parameters in (1).

With these values of the parameters, (1) becomes

P 1 2/a (1)

{2�1/2/n1,2 1 (22/n1/2,1)}

Subsequent layers (along the direction normal to the a, blayer) follow each other related by a glide normal to c0, withtranslation component �a/4 þ b/2, namely [� � n�1/2,1], asindicated in Fig. 2 or a/4 + b/2, namely [� � n1/2,1]. Glides[� � n1/2,1] and [� � n�1/2,1] give rise to pair of layerswhich are geometrically equivalent (principle of OD struc-tures).

The ordered or disordered succession of the two opera-tions give rise to infinite polytypes (ordered sequences) or

disordered arrangements. Particularly interesting are thosestructures in which not only pairs of layers, wherever tak-en, are geometrically equivalent, but also triples, quadru-ples, . . . n-tuples of layers are geometrically equivalent(principle of MDO structures). Those structures corre-spond to the ‘regular’ sequences of operations. Two suchstructures may be found in the present family:

MDO1 /n�1/2,1/n�1/2,1/n�1/2,1/. . . (the sequence /n1/2,1/n1/2,1/n1/2,1/. . . gives rise to the same structure (MDO10) twin-relatedwith MDO1, twin plane (001).

MDO2 /n1/2,1/n�1/2,1/n1/2,1/n�1/2,1/. . .

9,10 Phenantrenequinone a:symmetry of the polytype MDO1

The application of two successive operations [� � n�1/2,1]corresponds to a pure translation c ¼ 2c0 � a/2. The unitcell of this polytype may be referred to the basic vectors:a, b, c ¼ 2c0 � a/2, with a ¼ 7.73, b ¼ 7.99, c ¼ 16.46 �A,b ¼ 103.6�.

The glide a of the single layer is total operation of thewhole structure; the position of the subsequent layers issuch that the symmetry operation 2 of the single layer is atotal operation, valid for the whole structure, which there-fore presents space group P12/a1. The cell parameters andspace group symmetry are just those of the a1 polytype.

9,10 Phenantrenequinone a:symmetry of the polytype MDO2

The application of two successive operations n1/2,1 andn�1/2,1 corresponds to a pure translation c ¼ 2c0. The unitcell of this polytype may be referred to the basic vectors:a, b, c ¼ 2c0. The symmetry [– a –] is valid for all thelayers (total operation). The position of the layers is suchthat the operation (� � 22) is continuing through the ad-

OD approach to polytypism 253

Fig. 1. The OD layer of 9,10 phenantrenequinone a as seen down c0;the position of the symmetry elements is indicated.

Fig. 2. A pair of OD layers in phenantrenequinone a as seen down b,with indication of the l-operations (heavy marks) and s-operations[� � n�1=2; 1] and 22 (light marks).

joining layers, becoming a total operation 21 (� � 21) ina cell with c ¼ 2c0. Finally the operation (n1,2 � �) too iscontinuing through the adjoining layers, becoming a totaloperation (n1,1 � �) ¼ (n � �) in a cell with c ¼ 2c0.Therefore this polytype has space group Pna21 with para-meters: a ¼ 7.73, b ¼ 7.99, c ¼ 16.00 �A, namely the spacegroup and unit cell parameters of the a2 polytype.

The space groups and unit cell parameters of the a1 anda2 forms may be reconciled with those given by Rae andWillis (2003) for the ordered domains within the crystals ofthe a polymorphs, by exchanging the a and c vectors.

OD character of 9,10 phenantrenequinone b

The unit cell dimensions and space group symmetry of theb1 and b2 forms of polymorph b, prepared and studied byMarchetti et al. (in preparation), are here reported:

b1 C-1, a ¼ 12.957, b ¼ 10.442, c ¼ 15.551 �A,a ¼ 109.78, b ¼ 97.27, g ¼ 90.30�

b2 Cc, a ¼ 12.953, b ¼ 10.428, c ¼ 14.671 �A,b ¼ 97.51�

On the basis of the results obtained in the structuralstudy of the b2 form (crystallized from chloroform), thestructure may be described as an OD structure character-ized by structural layers with translation vectors a, b, thirdvector (not a translation vector) c0 (a ¼ 12.953,b ¼ 10.428, c0 ¼ 7.335 �A, b ¼ 97.51�) and layer groupsymmetry C11(�1), as represented in Fig. 3.

In accordance with these l operations, the Table of ODgroupoid family symbols (Dornberger-Schiff, 1964a; Fer-raris et al., 2008) points to the groupoid family symbol:

C 1 1 (�1){1 2s/c2 (1)}

The structural arrangement of b2 suggests the value s ¼ 1=2:

C 1 1 (�1){1 21/2/c2 (1)}

The structural layers a, b succeed each other along thedirection normal to the layers, related by the operation 21/2

or 2�1/2; pairs of layers related by 21/2 are geometricallyequivalent to pairs of layers related by 2�1/2.

The ordered or disordered sequences of 2�1/2 and 21/2

operations give rise to infinite polytypes (ordered se-quences) or disordered structures. Also in this family welook for the MDO polytypes; two such structures may befound in the present family:

MDO1 2�1/2/2�1/2/2�1/2/. . . [the sequence 21/2/21/2/21/2/. . . gives rise to the same structure, twin-related to the pre-ceding one, with twin plane (001)]

MDO2 2�1/2/21/2/2�1/2/21/2/. . .

9,10 Phenantrenequinone b:symmetry of the polytype MDO1

The MDO1 polytype is characterized by the constant ap-plication of 2�1/2 operation. The double application of 2�1/2

operation corresponds to a pure translation c ¼ 2c0 � b/2,outlining a unit cell with basis vectors a, b, c ¼ 2c0 � b/2.

With this structural arrangement only the inversion cen-tres (of l-type) are total symmetry elements. As all thestacked subsequent layers are C centred, the resultingspace group symmetry is C-1, with the following cellparameters (calculated on the basis of the parameters ofMDO2 polytype): a ¼ 12.953, b ¼ 10.428, c ¼ 15.570 �A,a ¼ 109.56, b ¼ 97.07, g ¼ 90�. Through the transforma-tion matrix [100/010/012] we obtain the cell parameters of‘crystal 3’ of Rae and Willis (2003), presenting spacegroup symmetry F-1.

9,10 Phenantrenequinone b:symmetry of the polytype MDO2

The regular alternation of the operations 2�1/2 and 21/2 cor-responds to a translation of 2c0. The unit cell of this poly-type will present basic vectors: a, b, c ¼ 2c0.

Due to the regular alternation of 2�1/2 and 21/2 the firstand third layer are at the same y value and the s-PO c2 iscontinuing, becoming a true c glide, valid for the wholestructure with a parameter c ¼ 2c0 (Fig. 4). As the C cen-tring is valid for all the successive layers, the overall

254 S. Merlino

Fig. 3. The OD layer of 9,10 phenantrenequinone b as seen normalto a, b plane; the position of the inversion centres is indicated.

Fig. 4. The sequence of OD layers in the structure of the MDO2polytype of phenantrenequinone b as seen down a, with indication ofthe s-operations relating adjacent layers.

space group symmetry of the structure is therefore C1c1(Cc). Due to the alternate displacements of the layers inthe b direction, the inversion centres of the layers are nomore valid for the whole structure. The space group sym-metry and the cell parameters closely correspond to thosefound for ‘crystal 1’ by Rae and Willis (2003).

As a general comment we may observe that the specialstability of MDO structures is in keeping with the generalassumptions behind OD theory. “. . . Even when the long-er-range forces are much weaker than those between adja-cent layers, they may not be negligible and, therefore, un-der given crystallization conditions either the one or theother kind of triples becomes energetically more favour-able, it will occur again and again in the polytype thusformed, and not intermixed with the other kind. Needlessto emphasize, however, that such structures are – as a rule– very sensitive to crystallization conditions, small fluctua-tions of these may reverse the energetical preferences, andthis results in the occurrence of stacking faults, twinningand other kinds of disorder” (Durovic and Weiss, 1986).These statements just correspond to what has been ob-served in the crystallization of the two polytypes of eachpolymorph. In fact the form a1 has been obtained by slowcrystallization from the melt (220 �C); the a2 form is ob-tained by sublimation and occurs as large domains in crys-tals containing also quite small domains of a1. The b1

form has been obtained by evaporation of saturated solu-tion of PHQ in acetone or in benzene, whereas the b2

form has been obtained by evaporation of saturated solu-tion in chloroform (Marchetti et al., in preparation).

OD structures built up with two kinds of ODlayers: latiumite and tuscanite OD family

So far we have dealt with OD structures built up withequivalent layers and the example we have just presentedclearly shows how simply the OD approach may explainthe structural relationships in polytypic families. ActuallyOD theory has been extended to structures built up withtwo or more kinds of OD layers. This extension has beencarried out by Dornberger-Schiff (1964a) on the basis ofthe same principle, namely the geometrical equivalence ofpairs of adjacent layers, and making use of the same con-cepts introduced to deal with OD structures consisting ofequivalent layers: OD layers, l partial operations (POs),namely the group of symmetry operations of the layer, spartial operations, namely operations relating adjacentlayers, and MDO structures. Due to the presence of two

or more distinct type of layers, the definition of structureswith maximum degree of order has to be modified as it fol-lows. A structure containing M distinct types of triples is de-fined as an MDO structure if no other member of the familyexists which is built only by a selection of the M types oftriples, and if the same is valid for quadruples,. . . n-tuples.

Due to the above mentioned extension the close con-nection between OD structures and polytypes may be con-sidered of general type embracing all the possible ‘true’polytypic families. Such has been the position of severalresearchers active in OD field, such as Durovic and Dorn-berger-Schiff herself, who maintained: “There is no poly-typic structure known to us as yet, which could not beinterpreted as an OD structure of layers” (Durovic, 1979);“For all polytypic structures which have come to myknowledge, the layers may be chosen in such way that thecondition for OD structures are fulfilled, and thus the re-sults of OD theory may be applied” (Dornberger-Schiff,1982). The key point is the choice of the number and thekind of the OD layers: critical remarks on this aspect willbe reported in a forthcoming chapter.

It is useful to distinguish, among the POs of a layer,those operations which reverse the layer with respect tothe direction of the layer stacking (q-operations) and thosewhich do not reverse the layer (t-operations) (we shalldenote A the non polar layers, presenting both t- and q-operations, and b, or d, the polar ones, presenting only t-operations). This distinction consents us to determine thenumber Z of equivalent position of an adjacent layer, withrespect to the starting one, making use of the so-calledNFZ relation (Dornberger-Schiff, 1964a) in the form: iftwo adjacent layers are non-equivalent, then Z ¼ N=F,where N is the order of the subgroup of t-operations inthe starting layer and F is the order of the subgroup of t-operations valid for the layer pair.

Latiumite-tuscanite polytypic family

The results of the structural study of the minerals latiumite(Cannillo et al., 1973) and tuscanite (Mellini et al., 1977) –both with ideal formula (K,H2O)2(Ca,Na)6(Si,Al)10O22(SO4)2

and crystallographic data as given in Table 1 – point to closepolytypic relationships between them.

Both are built up with double layers of (Si,Al)O4 tetra-hedra, alternating with sheets of Ca2þ cations and (SO4)2�

anions. In latiumite (Fig. 5a) all the tetrahedra of eachdouble layer point in one direction. In tuscanite (Fig. 5b)the tetrahedra of the two single layers that form the doublelayer point in opposite directions.

OD approach to polytypism 255

Table 1. Crystallographic data for the structure types of latiumite and tuscanite and of the related synthetic compound LaAlSiO5.

Phase Composition Space group a ð�AÞ b ð�AÞ c ð�AÞ b

Latiumite (K,H2O)2(Ca,Na)6(Si,Al)10O22(SO4)2 P21 12.036 5.106 10.952 107.08� a

Tuscanite idem P21=a 24.036 5.110 10.888 106.95� b

LaAlSiO5 La6Si4Al6O22(SiO4)2 P212121 23.705 5.226 11.053c

a: Latiumite from Pitigliano (Leoni et al., 1977)b: Tuscanite from Pitigliano (Mellini et al., 1977)c: Kahlenberg and Kruger (2004); the unit cell has been changed through the transformation matrix [001/0�110/100], to emphasize the relationshipswith the parameters of latiumite and tuscanite.

The polytypic relationships between the two structuralarrangements may be described – in a very simple andclear way – in terms of OD theory. In fact the two struc-tures may be described as distinct members of an OD fa-mily built up with two types of OD layers, indicated inFig. 5 as A2n and A2nþ1. Both layers have translation vec-tors b, c (b � 5.1, c � 10.9 �A) and regularly alternate inthe a0 direction (a0 � 11.5 �A). The proper symbol for thisOD family is:

A2n A2nþ1

P(1) 21 1 P(2/n) 21/m 21/m

[0, s]

where the first line presents the layer group symmetry ofthe two kinds of layers and the second line (withs ¼ �0.162) gives the positional relationships of subse-quent layers by presenting in brackets the components ofthe projection into the layer plane of the vector connectingthe origins [chosen according to the International Tablesfor X-ray Crystallography, vol. I (1952)] of the two subse-quent layers; the values of the components are referred tothe basic vectors b and c of the common translationgroup.

It is proper to remark that OD layers do not generallycorrespond to crystalchemically significant modules; themerely geometrical nature of OD layers in the present fa-mily is clearly indicated by the fact that the crystalchemi-cally significant modules are the double tetrahedral layers,whereas the A2nþ1 OD layers (which include the bridgingoxygens and the four-coordinated cations connected tothem) cut just across the double tetrahedral layers.

The symmetry of the A2nþ1 layers is higher than that ofthe A2n layers, which is the basis for the polytypic char-acter. The NFZ relation indicates that there are fourequivalent positions of the A2nþ2 layer with respect to the

layer A2nþ1. In fact in latiumite structure-type A2n andA2nþ2 are related by the 21 screw axes parallel to b whichare l-q-POs of the A2nþ1 layers and the pair (A2n; A2nþ1)is converted by these operations into the pair (A2nþ2;A2nþ1); however, beside the [� 21 �] screw axes, there arethree other l-q-POs in A2nþ1, namely inversion centres�1, [� � 21] and [n � �]; the pair of layers (A2n; A2nþ1)is converted also by these operations into the pair (A2nþ2;A2nþ1). Therefore pairs of adjacent layers are geometri-cally equivalent in the four resulting arrangements, as areequivalent in any of the infinite possible structures corre-sponding to the various possible sequences of the four l-qoperations which may be active in the A2nþ1 layers (prin-ciple of OD structures).

The NFZ relation indicates also that assuming an arbi-trary position of the A2n layer, the positions of the preced-ing and subsequent layers A2n�1 and A2nþ1 are uniquelydetermined (and related by the 21 screw axis parallel to b);consequently only one type of (A2n�1; A2n; A2nþ1) tripleexists. On the contrary there are four kinds of (A2n; A2nþ1;A2nþ2) triples. Evidently the smallest number of differenttriples necessary to build a periodic polytype is two, fromwhich it follows – according to the definition we havegiven above – that four MDO polytypes are possible inthis family. In all the four structures the l-q-PO [� 21 �]is constantly active in the A2n layers, whereas differentoperators are active in the A2nþ1 layers.

MDO1. The first one is obtained when 21 screw axesparallel to b are constantly operating in the A2nþ1 layers.The resulting structure (Fig. 5a) has space group P1211,with basic vectors, a ¼ a0 þ 2sc, b, c. It corresponds tothe structure type of latiumite.

MDO2. The second one is obtained when the inversioncentres are constantly active in the A2nþ1 layers. The re-sulting structure (Fig. 5b) has space group P21/a (the com-bination of [� 21 �] in A2n and inversion centres in A2nþ1

256 S. Merlino

Fig. 5. The crystal structures of latiumite (MDO1) (a) and tuscanite (MDO2) (b) as seen down b, c horizontal. The l-q-operations, active in theA2nþ1 layer, namely screw axes [� 21 �] and inversion centers –1, respectively, are indicated. The small black circles indicate the bridgingoxygen atoms.

generates the [� a �] glide) and basic vectors a ¼ 2a0 þ4sc, b, c. It corresponds to the structure type of tuscanite.

MDO3. The third one is obtained when 21 screw axesparallel to c are constantly operating in the A2nþ1 layers.The resulting structure (Fig. 6a) has space group P212121

(the combination of [� 21 �] in A2n and [� � 21] inA2nþ1 generates the [21 � �] operation) and basic vectorsa ¼ 2a0, b, c.

This structure-type has not been found with the samecomposition as in latiumite and tuscanite. However alanthanum aluminosilicate, closely related to those miner-als from the crystalchemical point of view (Table 1), dis-plays just this structure-type.

MDO4. The last MDO structure in this family is ob-tained when the [n � �] glide is constantly operating inthe A2nþ1 layers. The resulting structure (Fig. 6b) hasspace group Pn21a (the combination of [� 21 �] in A2n

and [n � �] in A2nþ1 generates the [� � a] operation)and basic vectors a ¼ 2a0, b, c. So far this structure-typehas never been found in natural or synthetic compounds.

Polytypism in kaolinite family

The systematic use of the NFZ relationship has been parti-cularly fruitful in the analysis of the polytypic behaviour ofvarious classes of hydrous phyllosilicates (kaolinites, pyro-phyllite and talc, chlorites, and micas). The NFZ relations,which are quite complex for those compounds, due to thenumber of distinct OD layers and to their symmetry rela-tionships, allow calculation of all the stacking possibilitiesand eventually derivation of all the MDO polytypes. Theresults of those investigations may be found in reviews byDurovic (1981, 1992). We here refer only to the OD inter-pretation of the polytypism in kaolinite which has beenpresented by Dornberger-Schiff and Durovic (1975a, b).

To apply the OD procedure, kaolinites have to be con-sidered as consisting of three kinds of polar OD layers, b1,b2 and b3 as represented in Fig. 7.

The OD family symbol (Grell and Dornberger-Schiff,1982) is:

b1 b2 b3

[1=3, 0] [1=3, 0] [1=3, 0]

Through a complex and careful analysis, Dornberger-Schiff and Durovic (1975b) derived 36 non-equivalentMDO polytypes, presenting for each of them the basicvectors and the space group symmetry. For sixteen ofthem an enantiomorphous counterpart exists, leading to atotal of 52 non-congruent MDO-polytypes.

The non-crystalchemical, purely geometrical choice ofthe OD layers, a choice necessary for the application ofthe OD approach to polytypism in kaolinites, has been thetarget of the critical remarks by Zvyagin (2001):

“A complex problem in the nomenclature of phyllosili-cates is the relationships between polytypes and OD struc-tures. The term OD (order-disorder) implies that pairs ofadjacent layers are symmetrically equivalent for some setof layer structures (Dornberger-Schiff, 1964a). However,

OD approach to polytypism 257

Fig. 6. The crystal structures of LaAlSiO5 (MDO3) (a) and the hypothetical new polytype (MDO4) (b) as seen down b, c horizontal. The l-q-operations, active in the A2nþ1 layer, screw axis [� � 21] and glide [n � �], respectively, are indicated. The small black circles indicate thebridging oxygen atoms.

Fig. 7. The structural scheme for kaolinites, with indication of thethree types of OD layers. H indicates that, with reference to the com-mon basic vectors, the cell is triply primitive [Int. Tables for X-rayCrystallography (1952), Vol. I, p. 18].

the phenomenon of polytypism itself does not require sucha condition and no restrictions exist for the layers. Thestructure of the layers should be fixed and several equiva-lent variants of their stacking provide the diversity of layerstructures (polytypes). However, it is just the freedom ofchoice of constituent layers that was the reason for consid-ering polytypism and OD as synonyms (Dornberger-Schiff, 1982). The OD interpretations were given to phyl-losilicate structures with another choice of the so-calledbuilding layers (BL), which are certain parts and evenatomic planes of the nomenclature units, which were ac-companied with increasing numbers of layer types . . . De-pending on the success in finding OD layers, which are,to some extent, artificial and not always obvious, such aninterpretation complicates the understanding of polytypenomenclature.” (Zvyagin, 2001).

In fact Zvyagin (1962) had been able to derive thesame list of 52 non-congruent polytypes of the kaolinitefamily in a more straightforward way on the basis of thecrystalchemically significant TO layer, built up by the tet-rahedral T and octahedral O networks, by looking for allthe ‘regular’ sequences of layers: a sequence is considered‘regular’ if the position of a layer relative to its neigh-bours does not differ in principle from the correspondingposition of any other layer.

To describe the distinct possible structures the displace-ment vectors s (intralayer) and t (interlayer) are indicated;these vectors are given relative to the rectangular cell a, b(b ¼ a

ffiffiffi

3p

) which can always be selected in layer silicates.To fully understand the s, t Zvyagin’s terminology thefollowing points are to be considered:

1) The octahedral network is placed below the tetrahe-dral one in each TO layer.

2) Within each layer s indicates the displacement ne-cessary to bring the centre of the tetrahedral net-work to overlap the ‘vacant octahedron’ in the octa-hedral network (namely s indicates the displacementof the tetrahedral network necessary to bring all theSi and Al atoms in the same layer to overlap eachother); s is given in terms of the vectors a/3 and b/3as indicated in the Table 2.As regards adjacent layers, t indicates the displace-ment necessary to bring the octahedral network tooverlap (in the already indicated way) the tetrahe-dral network of the preceding layer; t is given in

terms of the vectors a/3 and b/3 as indicated in theTable 2.

3) The layers follow each other from the bottom up;the sequence of the s, t displacements is given ac-cordingly.

For example the Zvyagin’s symbols for the structuresof the most common polytypes in the kaolinite family arehere reported (Table 3) and the layer sequence in dickiteis illustrated in Fig. 8, left.

We shall present now an interesting application of theZvyagin’s compilation to indicate how useful may be acomplete and reliable list of all the ‘regular’ polytypes(MDO structures) in a family, accompanied by a highlydescriptive and informative symbolic nomenclature.

Johnston et al. (2002) carried out a single crystal Ra-man study of dickite at pressures up to 6.5 GPa in a dia-mond-anvil cell, finding a ‘dramatic’ and reversible shiftin the n(OH) bands at �2.2 GPa, indicative of a signifi-cant variation in the interlayer hydrogen bonding, withonly minor changes in the lattice mode region, which sug-gested that the individual TO layers did not change signif-icantly with pressure. These results stimulated Dera et al.(2003) to determine the structural aspects of the corre-sponding phase transition by in situ high-pressure single-crystal X-ray diffraction.

The crystallographic parameters of dickite,Al2Si2O5(OH)4 at 0.1 MPa and 4.1 GPa are:

0.1 MPa, monoclinic, space group C1c1, a ¼ 5.161,b ¼ 8.960, c ¼ 14.459 �A, b ¼ 96.77�

4.1 GPa, monoclinic, space group C1c1, a ¼ 5.082,b ¼ 8.757, c ¼ 13.771 �A, b ¼ 89.60�

258 S. Merlino

Table 2. The components of the displacements are expressed in termsof the vectors a/3 and b/3.

Displacements Components Displacements Components

s1 a/3 þ b/3 t1 a/3 þ b/3

s2 �a/3 þ b/3 t2 �a/3 þ b/3

s3 a/3 t3 a/3

s4 �a/3 � b/3 t4 �a/3 � b/3

s5 a/3 � b/3 t5 a/3 � b/3

s6 �a/3 t6 �a/3

t0 0

tþ b/3

t� �b/3

Table 3. Sequences of displacement vectors corresponding to thestructure of kaolinite (both enantiomorphous forms), nacrite and dick-ite; c0 is normal to the layers and corresponds to the distance be-tween successive TO layers. The values of the basic vectors may beeasily derived on the basis of the reported s and t displacements.

Phase Symbol Space group Basic vectors

Caolinite s2t�s2 C1 a, b, c0� a/3

s4tþs4

Nacrite s1t6s2t3s1 Cc11 a, b, 2c0� b/3

Dickite s1tþs5t�s1 C1c1 a, b, 2c0� a/3

Fig. 8. The TO layer stackings in dickite (left) and high pressuredickite (right), with indication of the sequence of displacement vec-tors s and t in the two polytypes.

A reliable guess for the structural arrangement of thehigh pressure polytype can be made on the basis of thecrystallographic data and the following considerations.

a) The intralayer TO connections as expressed by thes values should remain constant during the transfor-mation, as it is also indicated by the absences ofsubstantial changes in the lattice mode region of theRaman spectra.

b) On the contrary, changes should occur in the t val-ues. The HP transformation should give rise to astructural arrangement more stable at high pressureand it was reasonable to presume that the higherstability would be dependent on a more favorablehydrogen bonds distribution, corresponding to a de-finite t value, the same for both TO layers in theunit period along c.

The Zvyagin’s list presents four monoclinic polytypeswith basic vectors corresponding to those of the HP dick-ite and space group C1c1 (namely conventional Cc):

Symbol space group basic vectors

s1t3s5t3s1 C1c1 a, b, 2c0

s1t1s5t5s1 C1c1 a, b, 2c0

s1t5s5t1s1 C1c1 a, b, 2c0

s3t5s3t1s3 C1c1 a, b, 2c0

Among them only the first one is in keeping with theindicated requirements.

The structural study of Dera et al. (2003) not only con-firmed, on the basis of sound single crystal X-ray diffrac-tion data, the indicated sequence, which is illustrated inFig. 8 (right), but also determined the positions of all thehydrogen atoms (in both dickite and HP dickite) by acombination of simulated annealing and energy minimiza-tion. In their discussion of the interlayer bonding in kaoli-nite phases Dera et al. (2003) point out that to optimizeinterlayer hydrogen bonding there are two distinct ways tostack the Al and Si sheets in the interlayer region, namelytype I (half of the Si ions within one layer overlap Al ionsin the next layer, and the other half is placed above thevoids in the Al sheet of the next layer), corresponding tothe stacking types tþ and t�; and type II (all the Si atomshave two equidistant nearest neighbours in the Al sheet ofthe next layer), corresponding to the stacking types t1 tot6. The layer stacking in HP dickite is of type II, to com-pare with the layer stacking of type I in dickite. “Consid-ering the similarities in the structure of kaolin polytypesand the simple mechanism of this transformation in dick-ite, it is tempting to predict whether similar transforma-tions will occur for the other polytypes.” (Dera et al.,2003). The nacrite structure has interlayer connections ofType II under ambient pressure and the assumption of aninverse transformation to Type I stacking at high pressureseems untenable. Dera et al. (2003) suggest that, as nacriteis metastable under ambient conditions (DeLigny and Nav-rostky, 1999), a change in thermodynamic conditions (notnecessarily in pressure conditions) may induce a transitionto a stable and possibly quenchable Type I structure. Onceagain a reliable guess for the corresponding structural ar-

rangement may be found through examination of theZvyagin’s compilation, looking for a sequence presentingthe same s values as found in nacrite (s1 t6 s2 t3 s1, nots1 t2 s2 t1 s1 as assumed by Dera et al., 2003) and tvalues corresponding to Type I connections. There is onlyone sequence presenting these features:

symbol space group basic vectorss1 t� s2 t� s1 Cc11 a, b, 2c0

Conclusions

When the polytypic features are amenable to a descriptionas OD structures consisting of layers of one kind or ofonly two distinct kinds (especially in the more frequentcase of two non-polar layers) the advantage of an OD ap-proach is evident, also at the cost of introducing ‘formal’OD layers, not coincident with crystalchemically signifi-cant modules.

In fact the OD approach permits to clearly describe thesymmetry properties common to the whole polytypic fa-mily, to present an immediate and precise account of me-trics and symmetry of the main polytypes, namely theMDO structures, and to suggest reliable structural modelsfor possible polytypes not yet found in nature or preparedin the laboratory. It has also consented to successfully dealwith structures of compounds affected by OD problems(diffuse reflections, twinning, simultaneous presence ofmore polytypes) and to achieve important structural resultsin the study of the C––S––H (calcium––silicate––hydrates)compounds. As a matter of fact it was just through theapplication of OD theory that the real structures of thevarious phases of the tobermorite group (Merlino et al.,1999, 2000, 2001; Bonaccorsi et al., 2005), as well as ofjennite (Bonaccorsi et al., 2004), which are so relevant tounderstand the hydration processes of the Portland cement(Taylor, 1997), have been defined.

However, the more the number of OD layers necessaryto deal with the polytypic features increases, the more theadvantage of an OD approach decreases, with respect to atreatment based on crystalchemically significant modules,and the critical remarks of Zvyagin (2001) acquire sub-stantial validity. However it seems proper to remark that,also in those cases, the OD treatment, which permits areliable derivation of the MDO structures, may be usefulas a reliable validation tool for the crystalchemical deriva-tion of the ‘main’ or ‘standard’ polytypes.

The case studies presented and discussed in the variouschapters of this review give also the occasion to remarksome important aspects of polytypism. The study of thedifferent phases of 9,10 phenantrenequinone presents aclear example of neat distinction between polymorphismand polytypism, a distinction which not only has relevancefrom a general point of view, but may also present inter-esting practical consequences in the field of the organiccompounds with pharmaceutical properties, suggesting asolution of the problem to properly define ‘commercially’distinct polymorphs.

In the chapter devoted to the OD family of latiumiteand tuscanite an useful extension of the concept of poly-

OD approach to polytypism 259

typism has been tacitly introduced. In fact we have placedin the same OD family not only the two chemically identi-cal natural phases, but also the compound LaAlSiO5 withdistinct chemical composition; that extension has been dis-cussed by Makovicky (1997) and has been called ‘config-urational polytypism’ (Ferraris et al., 2008).

At last it seems useful to go back to the final statementin the definition of polytypism by the IUCr Ad-hoc Com-mittee (Guinier et al., 1984), where the possibility of afuture more general definition, including ‘rods’ (and also‘block’) polytypes, is suggested. Actually there are onlyvery few examples of OD structures built up by rods withambiguity of rod stacking in two directions (Dornberger-Schiff, 1964b; Merlino et al., 2009) and no systematic the-ory for such kind of OD structures has been so far devel-oped. Possibly concepts and procedures to systematicallydeal with ‘rod polytypism’ will gradually emerge as far asthe number of compounds displaying such kind of poly-typism will substantially increase.

Acknowledgments. This work was financially supported by MIURthrough the project PRIN 2007 “Compositional and structural com-plexity in minerals (crystal chemistry, microstructures, modularity,modulations): analysis and applications.” I am grateful to Elena Bo-naccorsi for helpful discussions and assistance in the preparation ofthe text. I am also grateful to Emil Makovicky, a second anonymousreferee and to the Editor Giovanni Ferraris: their comments and sug-gestions were helpful in improving the paper.

References

Bonaccorsi, E.; Merlino, S.; Taylor, H.F.W.: The crystal structure ofjennite, Ca9Si6O18(OH)6 � 8 H2O. Cem. Concr. Res. 34 (2004)1481–1488.

Bonaccorsi, E.; Merlino, S.; Kampf, A.R.: The crystal structure oftobermorite 14 �A (plombierite), a C––S––H phase. J. Am. Ceram.Soc. 88 (2005) 505–512.

Brandt, H.: �ber eine Verallgemeinerung des Gruppenbegriffs. Mathe-matische Annalen 96 (1927) 360–366.

Cannillo, E.; Dal Negro, A.; Rossi, G.: The crystal structure of latiumite,a new type of sheet-silicate. Am. Mineral. 58 (1973) 466–470.

DeLigny, D.; Navrotsky, A.: Energetics of kaolin polymorphs. Am.Mineral. 84 (1999) 149–158.

Dera, P.; Prewitt, C.T.; Japel, S.; Bish, D.L.; Johnston, C.T.: Pressure-controlled polytypism in hydrous layered materials. Am. Mineral.88 (2003) 1428–1435.

Dornberger-Schiff, K.: On the order-disorder (OD-structures). ActaCryst. 9 (1956) 593–601.

Dornberger-Schiff, K.: Grundzuge einer Theorie von OD-Strukturenaus Schichten. Abhandlungen der Deutschen Akademie der Wis-senschaften zu Berlin, Klasse fur Chemie, Geologie und Biologie3 (1964a) 1–107

Dornberger-Schiff, K.: Zur Strukturanalyse zweidimensional fehl-geordneter OD-Strukturen: die Strukturbestimmung des Natriumte-trametaphosphats Na2H2P4O12. Acta Cryst. 17 (1964b) 482–491.

Dornberger-Schiff, K.: Lehrgang uber OD-Strukturen. Akademie-Ver-lag, Berlin 1966.

Dornberger-Schiff: Geometrical properties of MDO polytypes andprocedures for their determination. I. General concept and applica-tions to polytype families consisting of OD layers all of the samekind. Acta Cryst. A38 (1982) 483–491.

Dornberger-Schiff, K.; Durovic, S.: OD-interpretation of kaolinite-type structures – I: Symmetry of kaolinite packets and their stack-ing possibilities. Clays and Clay Minerals 23 (1975a) 219–229.

Dornberger-Schiff, K.; Durovic, S.: OD-interpretation of kaolinite-type structures – II: The regular polytypes (MDO polytypes) andtheir derivation. Clays and Clay Minerals 23 (1975b) 231–246.

Durovic, S.: Desymmetrization of OD structures. Krystall und Tech-nik 14 (1979) 1047–1053.

Durovic, S.: OD-Charakter, Polytypie und Identifikation von Schicht-silikaten. Fortschritte der Mineralogie 59 (1981) 191–226.

Durovic, S.: Layer stacking in general polytypic structures. Interna-tional Tables for X-ray Crystallography, Vol. C, chapter 9.2.2.Kluwer Acad. Publ., Dordrecht 1992.

Durovic, S.: Fundamental of the OD theory. In: Modular Aspects ofMinerals, EMU Notes in Mineralogy, Vol. 1 (Ed. S. Merlino),pp. 3–28. Eotvos University Press, Budapest 1997.

Durovic, S.; Weiss, Z.: OD structures and polytypes. Bulletin deMineralogie 109 (1986) 15–29.

Ferraris, G.; Makovicky E.; Merlino S.: Crystallography of modularmaterials. IUCr Monographs on Crystallography 15, Oxford Uni-versity Press, New York, 2008.

Grell, H.; Dornberger-Schiff, K.: Symbols for OD groupoid familiesreferring to OD structures (polytypes) consisting of more than onekind of layers, Acta Cryst. A38 (1982) 49–54.

Guinier, A.; Bokij, G.B.; Boll-Dornberger, K.; Cowley, J.M.; Durovic,S.; Jagodzinski, H.; Krishna, P.; de Wolff, P.M.; Zvyagin, B.B.;Cox, D.E.; Goodman, P.; Hahn, Th.; Kuchitsu, K.; Abrahams,S.C.: Nomenclature of polytype structures. Report of the Interna-tional Union of Crystallography Ad hoc Committee on the No-menclature of Disordered, Modulated and Polytype Structures.Acta Crystallographica A40 (1984) 399–404

International Tables for X-ray Crystallography, Vol. I (1952). Bir-mingham: Kynoch Press.

Johnston, C.T.; Wang, S.L.; Bish, D.L.; Dera, P.; Agnew, S.F.; Kenne-dy, J.W.II: Novel pressure induced phase transformations in hy-drous layered materials: a comparative single crystal high pressureRaman/XRD study of dickite and gibbsite. Geophysical ResearchLetters 29 (2002) 1029.

Kahlenberg, V., Kruger, H.: LaAlSiO5 and apatite-typeLa9.71(Si0.81Al0.19O4)6O2 – the crystal structures of two syntheticlanthanum alumosilicates. Solid State Sciences 6 (2004) 553–560.

Leoni L.; Mellini, M.; Merlino, S.; Orlandi, P.: Crystal chemistry oflatiumite. Rend. Soc. It. Miner. Petr. 33 (1977) 531–537.

Makovicky, E.: Prediction of modular crystal structures. In: Proceed-ings Symposium on Predictability of Crystal Structures of Inor-ganic Solids, Hunfeld 27.10–30.10.1997 (Ed. H. Burzlaff),pp. 107–124. Erlangen: Universitat Erlangen-Nurnberg, 1997.

Matsuzaki, S.Y.; Gotoh, M.; Kuboyama, A.: Molecular and crystalstructure of a and b forms of 9,10 Phenantrenequinone. Mol.Cryst. Liq. Cryst. 142 (1987) 127–139.

Mellini, M.; Merlino, S.; Rossi, G.: The crystal structure of tuscaniteAm. Mineral. 62 (1977) 1114–1120.

Merlino, S.: OD approach in minerals: examples and applications. In:Modular Aspects of Minerals, EMU Notes in Mineralogy, Vol. 1(Ed. S. Merlino), pp. 29–54. Eotvos University Press, Budapest1997.

Merlino, S.; Bonaccorsi, E.; Armbruster, T.: Tobermorites: Their realstructure and order-disorder (OD) character. Am. Mineral. 84(1999) 1613–1621.

Merlino, S.; Bonaccorsi, E.; Armbruster, T.: The real structures ofclinotobermorite and tobermorite 9 �A: OD character, polytypes,and structural relationships. Eur. J. Mineral. 12 (2000) 411–429.

Merlino, S.; Bonaccorsi, E.; Armbruster, T.: The real structure of to-bermorite 11 �A: normal and anomalous forms, OD character andpolytypic modifications. Eur. J. Mineral. 13 (2001) 577–590.

Merlino, S.; Bonaccorsi, E.; Grabezhev, A. I.; Zadov, A. E.; Pertsev,N. N.; Chukanov, N. V.: Fukalite: an example of OD structurewith two-dimensional disorder. Am. Mineral. 94 (2009) 323–333.

Rae, A.D.; Willis, A.C.: 9,10 Phenantrenequinone, not your averagestructure. Z. Kristallogr. 218 (2003) 221–230.

Taylor, H.F.W.: Cement Chemistry, 2nd edition, Thomas Telford Pub-lishing, London 1997.

Zvyagin, B.B.: Polymorphism of double-layer minerals of the kaoli-nite type. Soviet Physics Crystallography 7 (1962) 38–51.

Zvyagin, B.B.: Current problems with the nomenclature of phyllosili-cates. Clays and Clay Minerals 49 (2001) 492–499.

260 S. Merlino