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OD Structures: fundamentals of OD theory; examples and applications. (1) Stefano Merlino Department of Earth Sciences, University of Pisa, via S. Maria 53, I-56126 Pisa, Italy. merlino@dst.unipi.it

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Outline of the lecture (1) Introduction Discussion of the OD character of wollastonite –Symmetry of the single layer ( -operations) and operations relating adjacent layers ( -operations) –Infinite possible polytypes; two main polytypes Generalization: main theoretical concepts (for structures built up with one type of OD layer) –Partial operations POs ( - and -operations) –Categories of OD structures and relative schemes –Notations for the partial operations and OD groupoid family symbols –The Z possible position of a layer with respect to the previous one

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Outline of the lecture (2) MDO structures Exercises Application of OD theory to structures built up with one type of OD layer –Xonotlite: OD character; derivation of MDO polytypes –Brochantite: OD character; derivation of MDO polytypes Diffraction features of OD structures –Family reflections and family structure; characteristic reflections Derivation of the OD groupoid family from the diffraction features –Clinotobermorite: OD character; family structure; derivation of the OD groupoid family; structural model; MDO polytypes –Tobermorite 11Å: OD character; derivation of the OD groupoid family; structural model; MDO polytypes Exercises

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Outline of the lecture (3) Other examples of OD structures built up with one kind of OD layers. OD structures built up with two or more types of layers. Discussion of the OD character of stibivanite The four categories and their symbols Examples and applications Exercises

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Introduction OD structures and the corresponding OD theory are interesting from two distinct point of view. A first reason of interest is of general crystallographic character: in fact OD structures are one of the several examples of aperiodic structures; we shall observe that – in general – OD structures are aperiodic in at least one direction. We shall also realize that – notwithstanding the absence, in general, of a full three-dimensional periodicity – the concepts of symmetry, properly re- formulated, are still valid and of great practical usefulness. The second reason of interest is just in this usefulness : OD theory develops a system of theoretical concepts which consent to deal – in a systematic and effective way - with the problems related to the structural study of compounds presenting one-dimensional disorder. It was just in the attempts to deal with the problems occurring in the study of one-dimensionally disordered compounds that the development of OD theory had its beginning. The connection between the two aspects has been described in the most effective way by K. Dornberger-Schiff herself.

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‘The introduction of the notion of OD structures was, in the first place, motivated by the task of determining the structure of a crystal exhibiting one-dimensional stacking disorder Considerations of the causes leading to the tendency to disorder of some crystalline substances led, furthermore, to the conviction that the classical definition of a crystal as a body with three-dimensional periodicity may have its shortcomings; properly speaking, it is not truly atomistic, although it refers to atoms: periodicity, by its very nature, makes statements on parts of the body which are far apart, whereas the forces between atoms are short range forces, falling off rapidly with distance. A truly atomistic crystal definition should, therefore, refer only to neighbouring parts of the structure, and contain conditions understandable from the point of view of atomic theory’ (Dornberger-Schiff, 1979).

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Fully ordered structure of Au 4 Cl 8 Actually, each fully ordered structure – a crystal in the classical sense – may be described in terms of two-dimensional layers [a,c layers of Au 4 Cl 8 molecules in the example]: if a definite arrangement is energetically favored (short range forces of interaction), this arrangement will occur throughout, which eventually results in periodicity (long range property of the structure). Pairs of adjacent layers – wherever taken in the structure – are obviously geometrically equivalent. ‘This equivalence… seems the most important feature of crystals, and may be used for a revision of crystal definition’ (Dornberger- Schiff, 1979). Actually, the equivalence of pairs of adjacent layers is a necessary condition for periodicity.

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Ambiguity in the stacking of the layers However the equivalence of pairs of adjacent layers is not a sufficient condition for periodicity. When adjacent layers may follow each to the other in two (or more) geometrically equivalent ways, then infinite possible sequences – ordered as well as disordered sequences – may result. The whole set builds up a family of OD structures; it seems appropriate to call each of them ‘crystal’, within a wider definition of this concept. The close packed sequences of close packed layers of spheres present the most simple example. The AB pair of layers is geometrically equivalent to the AC pair (and similarly for BC and BA, etc…). Infinite ordered (ABAB….; ABACABAC….; ABCABC..) or disordered (ABABCBAC…) sequences may occur.

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Possible peculiar features in diffraction patterns of OD structures OD structures may often be recognized through some peculiar features displayed by their diffraction patterns: presence of sharp spots and diffuse streaks Weissenberg photograph of the h0l diffraction pattern of molybdophyllite: - sharp spots for h = 3n - the other reflections present diffuseness along c*.

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Possible peculiar features in diffraction patterns of OD structures a set of common reflections in diffraction patterns from different crystals – with different structure – in a family (family reflections); diffraction symmetry higher than that corresponding to the point group and the Friedel law (diffraction enhancement of symmetry); non-space group absences; evidences for twinning, often polysynthetic twinning; polytypism. Diffraction features in pectolite crystals

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Wollastonite, an example of OD structure (1) To introduce the audience to the terminology and the procedures of OD theory we shall firstly discuss a single example, namely wollastonite, CaSiO 3, which appears the most appropriate for historical and didactical reasons. Wollastonite displays the whole set of peculiar features of an OD structure: Polytypism: Wollastonite 1A Wollastonite 2M a(Å) 7.926 15.424 b 7.302 7.324 c 7.065 7.069 ( ) 90.06 90.00 95.22 95.37 103.43 90.00 Space group P-1 P2 1 /a Various other polytypes have been found and studied with different techniques (X-ray diffraction, electron diffraction, HRTEM). For example a paper by Henmi et al. (1983) in Am. Mineralogist, 68, 156, has the title: ‘The 3T, 4T and 5T polytypes of wollastonite from Kushiro, Hiroshima Prefecture, Japan’

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Wollastonite, an example of OD structure (2) Diffuse reflections A paper by J.W. Jeffery (1953) [Acta Cryst., 6, 821] was just devoted to ‘Unusual diffraction effects from a crystal of wollastonite’ The figure [1st layer Weissenberg photograph, rotation axis b] shows the streaking which characterizes the reciprocal lattice planes corresponding to odd k values, in a crystal of wollastonite from Devon, with diffuseness along a*. Family reflections The reflections corresponding to even k values are sharp and occur in the same position and with the same intensity in both the triclinic and monoclinic forms (as well as in all the other polytypes. The pattern of these reflections display a monoclinic symmetry (partial diffraction enhancement of symmetry) and are present only for 2h + k = 4n (non-space group absences) (indices are referred to the unit cell of the monoclinic form)

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a = 7.926 Å b = 7.302 c = 7.065 = 90.06° = 95.22 = 103.43 a = 15.424 Å b = 7.324 c = 7.069 = 95.37° P2 1 /a P -1 The single layer has translation vectors b, c (the common vectors of the various polytypes), third basic vector a 0 = a m /2. b 7.32, c 7.07, a 0 7.71 Å, 95.4° Symmetry of the layer P2 1 /m or P 1 2 1 /m 1 0 b

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OD structures in wollastonite Adjacent layers may be related through a screw axis operation, with translational component +b/4 [it is denoted 2 1/2 in keeping with the notation used in the ‘International Tables’ for the usual symmetry operations]. The layers are also related through inversion centers and also through a glide reflection normal to b, with translation component a 0 [it is denoted a 2, once again in keeping with the ‘International Tables’ notation]. Because of the mirror plane m in the layer, adjacent layers may be related also through a screw rotation with translation component –b/4 ( 2 –1/2 ) Pairs of layers related in both ways are geometrically equivalent.

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When the operations 2 1/2 and 2 –1/2 regularly alternate, the first and third layers are at the same level and 2 1 element of each layer is valid for the whole structure; moreover the glide operation a 2 is continuing through the layers, becoming a true a glide valid for the whole structure, which has parameter a = 2a 0. The overall space group symmetry of the resulting structure is therefore P2 1 /a, which is just the space group of the polytype 2M. Polytype 2M of wollastonite

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When the operation 2 –1/2 is constantly applied the elements 2 1 of the single layers are no more total symmetry elements; moreover the glide operation a 2 is not continued from one layer to the other, but systematically transposed by b/4; only inversion centers now act as total symmetry elements and the overall space group is P –1, which is just the space group of the polytype 1A. It seems proper to recall that the structure obtained through the constant application of 2 1/2 corresponds to the twinned counterpart of the previous one. Polytype 1A of wollastonite

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The symmetry of any other polytype in the family may be derived in a similar way from the symmetry properties of the single layer and the knowledge of the stacking sequence. We have considered only the structures corresponding to the polytypes 2M and 1A, because of their significance and widespread occurrence. In all the families a small number of ‘main’ polytypes exists, generally the most frequently found, which are called ‘simple’, ‘standard’, ‘regular’. OD theory presents neatly defined geometrical criteria by which the particular staus of a small number of polytypes, called polytypes with maximum degre of order (MDO structures), may be recognized. These are the members in which not only pairs, but also triples, quadruples,…, n-tuples of consecutive layers are geometrically equivalent, as far as possible. In the present family only two MDO structures exist: MDO1 corresponding to wollastonite 1A MDO2 corresponding to wollastonite 2M MDO structures in wollastonite (1)

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The special stability observed for MDO structures is in keeping with the general Assumptions in OD theory. As it is maintained by Ďurovič and Weiss (1986): ‘…Even when the longer-range forces are much weaker than those between adjacent layers, they may not be negligible and, therefore, under given crystallization conditions either the one or the other kind of triples becomes energetically more favorable, it will occur again and again in the polytype thus formed, and not intermixed with the other kind. Needless to emphasize, however, that such structures are – as a rule – very sensitive to crystallization conditions; small fluctuations of these may reverse the energetic preferences, and this results in the occurrence of stacking faults, twinning, and other kinds of disorder’. This considerations – applied to wollastonite - explain not only why the most frequent polytypes are just 1A and 2M, but also the occurrence of multiple twinning [twin plan (100)] in the case of the triclinic polytype 1A. MDO structures in wollastonite (2)

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In fact layers related by the operations 2 1/2 and 2 –1/2 are translationally equivalent and related by stacking vectors t 1 = a 0 – b/4 and t 2 = a 0 + b/4, respectively. Therefore any adjoining layer may assume two possible positions, with displacements at left or right, according to the simple scheme illustrated in the Figure. Only two kinds of triples exist, ‘stretched’ and ‘bent’, corresponding to MDO1 and MDO2, respectively. As indicated in the Figure, the sequences t 1 t 1 t 1 ….. and t 2 t 2 t 2 …correspond to twin related structures. If the ‘stretched’ triples are favored in the crystallization process, the polytype could form as a polysynthetic twin with MDO1 and MDO1’ domains due to the possible occurrence of stacking faults with local production of ‘bent’ triples. [It seems fair to recall that the stretched and bent triples and the polysynthetic twinning closely correspond to the ‘echelon’, ‘alternate’, and ‘complex’ gliding described by Ito (1950)]. Triples of layers in wollastonite

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On the basis of the knowledge acquired by discussing the OD character of wollastonite, we may now proceed to a more general approach to OD structures (built up with one kind of layer). It is useful to introduce the concept of partial operations (POs), namely operations which transform any layer into another one (including the layer itself). They are called partial as they are not generally valid for the whole structure. It is important to observe that the whole set of POs does not form a group, as two POs p,q a and r,s b cannot be combined in this order unless q = r, namely unless the resulting layer of the first transformation is identical with the starting layer of the second transformation. The POs do form a Brandt groupoid (we shall obviously not develop the mathematical aspects of this matter). The complete set of partial operations of an OD structure may be generated from the set of POs which transform a layer into itself ( -POs)and the set of POs transforming a layer into the adjacent one ( -POs). Partial operations in OD structures (1)

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The complete set of partial operations of an OD structure may be generated from the set of POs which transform a layer into itself ( -POs)and the set of POs transforming a layer into the adjacent one ( -POs). The set of -POs - the symmetry operations of the layer – form a group: it is one of the 80 layer groups (or two-sided plane groups), which are the groups of symmetry operations of a structure built up with three-dimensional objects, but with a two-dimensional lattice. The OD notation for the layer groups follows the international notation, using a four-entry symbol [six-entry and eight-entry symbols for square and hexagonal nets, respectively] giving the type of lattice and the symmetry operations corresponding to x, y, z directions and indicating (symm. oper. in parentheses) the direction of missing periodicity For example the symbol for the layer group in wollastonite is P (1) 2 1 /m 1 Partial operations in OD structures (2)

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Polar and non polar layers Among the 80 layer groups (or two-sided plane groups, or black-white groups), 17 are polar (obviously the polarity is with respect to the normal to the layer) and are represented with figures all looking white (they would look black if viewed on the other side). The other 63 layer groups are non-polar. Each of them is represented with a sheet of figures, half presenting the white side up, half presenting the black side up; 17 of these non- polar groups, corresponding to the 17 polar groups, present a mirror plane parallel to the layer plane; both sides are equivalent and the figures are now ‘grey’. In the next two Tables the 80 layer groups are represented with sheets of white-black triangles (J.V. Smith – Geometrical and structural crystallography). Polar layers 17 drawings present only black figures (they would look black if viewed on the other side). Non-polar layers 17 are similar to the previous ones, but are identical on both sides (the figures present dots) 46 present half triangles white and half triangles black

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It is useful to distinguish the POs – both and – in two types: a) the so called -POs, which do not turn the layer upside down, with reference to the direction in which the layer stack each after the other, and b) the so called -POs which do not turn the layer upside down. In the case of wollastonite the -POs 2 1 and inversion centers are of type (with respect to the direction of missing periodicity; whereas m is of type . The multiplication rules for - and -POs may be written as: = = = = Partial operations of type and 0 b

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It is useful to introduce the concept of continuation: if p,q a and r,s a are characterized by the same transformation applied to different layers, we call p,q a continuation of r,s a and viceversa. This concept will be useful later, when we shall discuss the possible equivalence of layer pairs L 0 L 1 and L 1 L 2 in a sequence L 0 L 1 L 2 …. of OD layers: the two pairs L 0 L 1 and L 1 L 2 are equivalent if there exists either a PO 0,1 with continuation 1,2 or a PO 1,1 with continuation 0,2 Reverse continuation A - PO converting any layer into the adjacent one and viceversa is said to have a reverse continuation. This role is generally played by an inversion center of -type. Continuation and reverse continuation

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As the OD layers are equivalent, they are all polar or all non-polar, with reference to the stacking direction. The Figure gives a schematic representation of the various possible sequences. I) non-polar layers (double arrow): they present both - -POs and - -POs and are repeated through - - and - -POs; II) polar layers (single arrow): they present only - -POs and are repeated through - -POs (preserving the arrow orientation); III) polar layers (single arrow): they present only - -POs and are repeated through - -POs (inverting the arrow orientation at each step). Different categories of OD structures with equivalent layers

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.Accordingly there are three categories of OD structures of equivalent layers: Character of -POs Character of -POs Category I and and Category II Category III In the Figure the a,b layers are normal to the plane of drawing The three categories of OD structures with equivalent layers

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.A simple description of the sequences in the three categories can be made by denoting the layers as follows: - a non polar layer is denoted with the symmetric letter A - polar layers are denoted with letters b, d The sequence in Category I is A A A A A A….. The sequence in category II is b b b b b b…. The sequence in Category III is b d b d b d…. The three categories of OD structures with equivalent layers By looking at the Figure we realize that, whereas in the first two categories all the pairs of adjacent layers are equivalent, this is not so for the third category. The OD definition has to be more precisely stated. Pairs of layers L p,L p+1 and L q, L q+1 are equivalent for arbitrary values of p,q in the case of OD structures of Cat. I an II, and for values of p,q both even or both odd in the case of OD structures of Cat. III (vicinity condition).

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Both - and -POs fully describe the common symmetry properties of a whole OD family. This is done through a symbol such as: P m m (2) -symmetry {n s,2 n 2,r ( 2 2 )} -symmetry In it the first line presents the -operations, namely the symmetry operations of the single layer. A convenient notation to indicate the various elements of symmetry and their relations with the direction x, y, z is: [ m - -] [ - m -] [- - 2] The second line presents the -operations (their symbols are obtained as generalization of the international space group symbols). They are: n glide normal to a with translation component sb/2 + c 0 [n s,2 - -] n glide normal to b with translation component c 0 + ra/2 [- n 2,r -] twofold screw axis with translation component corresponding to c 0 [- - 2 2 ] There is also a translation t = ra + sb + c 0 [not explicitly indicated in the symbol] among the -POs. OD family symbols (1)

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P m m (2) -symmetry {n s,2 n 2,r ( 2 2 )} -symmetry The most general glide operation is denoted n r,s : the order of the indices in the symbol is chosen in such way that the direction to which n and the two indices refer follow each to the other in a cyclic way. Important point: the indices r and s are only of interest modulo 2, namely r may be replaced by r+2 or r-2, and similarly s by s+2 and s-2. It may be observed that, whereas two-line symbols occur for OD structures of Categories I and II, a three-line symbol is necessary for OD families of Category III: the line corresponding to the -POs of the single OD layer (e.g. layer b 0 ); the line of the -POs transforming b 0 into d 1 ; the line of the -POs transforming d 1 into b 2. OD family symbols (2) P m m (2) -symmetry {2 r 2 s (n r,s )} -symmetry {2 r’ 2 s’ (n r’,s’ )} -symmetry

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OD family symbols (3) Whereas only three positions are needed to indicate the operations related to the various directions in the case of oblique and rectangular lattices, more than three positions are necessary in the case of the square and hexagonal lattices, More precisely five positions for square net, seven positions for the hexagonal net. With the square lattice, the operations corresponding to the directions a, b, c, a+b, a-b have to be specified. With the trigonal/hexagonal lattices the operations corresponding to the directions a 1, a 2, a 3, c, a 2 -a 3, a 3 -a 1, a 1 -a 2 have to be specified. Vector directions in the square lattice, with c normal to the drawing.

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P m m (-4) 2 2 {2 2 2/3 4 4 /n 0,2/3 n –1/3, 2 n 1/3,2 } Zeolite beta: an example of an OD family with OD layer presenting square lattice. The family is built up with layers presenting P –4m2 layer group symmetry, represented at left in the Figure. The layers follow each to the other through the -operations listed in the second row and indicated at right in the Figure.

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OD family symbol in wollastonite We may now more precisely define the family symbol for the OD structure in wollastonite. P (1) 2 1 /m 1 -operations { (1) 2 1/2 /a 2 1} -operations The layers are also related through inversion centers. (as well as through a translation a 0 – b/4). How is it possible to evaluate the number Z of possible ‘equivalent’ ways to stack an adjacent layer? In the case of wollastonite it is evident – just by looking at the Figure – that Z = 2. OD theory has derived a simple formula by which we may calculate Z on the basis of the knowledge of the - and -POs.

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The group of -POs in the OD family of wollastonite has order 4, with operations: -POs E (identity), [- m -] subgroup of -POs of order N = 2 -POs -1 (inversion), [- 2 1 - ] coset of -POs The subgroup of -POs which are valid for a pair of adjacent layers (we say that these operations have a continuation in the adjacent layer) has only 1 element, the identity E. Its order is therefore F = 1. The number Z of positions of the adjacent layer leading to geometrically equivalent layer pairs is given by Z = N/F. In the present case Z = 2/1 = 2. This is the so-called NFZ relation valid with only minor alterations for all the categories of OD structures. P (1) 2 1 /m 1 NFZ relation in wollastonite

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NFZ relation in the stacking of close packed layers of spheres

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P m m (2) {n s,2 n 2,r (2 2 )} 4 - POs: E, [m - -], [- m -], [- - 2] Only E is valid for a pair of layers. Z = 4/1 = 4 In fact each of the four 0,1 POs [n s,2 - -] [- n 2,r -] [- - 2 2 ] translation t = c 0 + r a/2 + s b/2 has a 1,2 continuation, leading to the four positions of the L 2 layer. The continuation testifies that the four L 1 L 2 pairs are equivalent to L 0 L 1. NFZ relation in OD structures of Category II

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P m m (m) {2 r /n s,2 2 s /n 2,r (2 2 /n r,s )} Also in this case there are 4 - POs: E, [m - -], [- m -], [- - 2] Only E is valid for a pair of layers. Z = 4/1 = 4 As in the previous case each of the four 0,1 POs has a 1,2 continuation, leading to the four positions of the L 2 layer. There are also four 1,1 POs (they are [2 - -], [- 2 -], [- - m] and inversion center) with continuation 0,2 . However they lead to the same four positions already obtained with the continuation of the - POs. The reason for that is dependent on the presence – among the -POs – of a -PO with reverse continuation. This operation is – in the present case - the inversion center, converting any layer into the adjacent one and viceversa. OD structures of Category I with - -PO with reverse continuation

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P 2 m (m) {2 r n 2,r (n r,s )} In this case there are 2 - POs: E, [- m -]. Only E is valid for a pair of layers. Z = 2/1 = 2 However, besides the two positions of the L 2 layer indicated in (a) and resulting from the continuations of 0,1 POs (namely [- n 2,r -] and translation t = c 0 + r a/2) into 1,2 POs, there are two new positions which result from the two 1,1 POs (they are [2 - -], [- - m]) with continuation 0,2 . Therefore the NFZ relation has to be modified as it follows: Z = 2 N/F OD structures of Category I without - -PO with reverse continuation

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Category Z Z 2n,2n+1 Z 2n+1, 2n+2 Ia at least one p,p+1 has a reverse continuation N/F Ib no p,p+1 has a reverse continuation 2N/F II N/F IIIa at least one 2n,2n+1 and one 2n+1, 2n+2 have reverse continuations N/F 2n,2n+1 N/F 2n+1, 2n+2 IIIb no 2n,2n+1 and no 2n+1, 2n+2 have reverse continuations 2N/F 2n,2+1 2N/F 2n+1, 2n+2 IIIc at least one 2n,2n+1 and no 2n+1, 2n+2 have reverse continuations N/F 2n,2n+1 2N/F 2n+1, 2n+2 Z values in the various categories and subcategories Of OD structures

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The results we have obtained are useful in the derivation of the MDO structures in the various categories of OD families. In presenting the OD family of wollastonite we introduced the concept of MDO structures indicating the physical reason of their special stability and widespread occurrence. Actually as maintained by Dornberger-Schiff (1982) ‘behind the OD theory and the concept of MDO polytypes, there is the basic idea of decreasing interatomic forces with increasing distance, thus leading to a preference of polytypes with a minimum number of layer pairs (principle of OD structures) or even a minimum number of kinds of n-tuples of layers, for any number n (principle of MDO structures)’. In the particular example of wollastonite we emphasized the role played by the ‘layer triples’ and we have presented a rough definition of MDO structures as those members of an OD family in which ‘triples (and quadruples,...n-tuples) of consecutive layers are geometrically equivalent as far as possible’. This definition is satisfactory for most of the OD families of structures built with equivalent layers, occurring in natural and synthetic compounds. As a matter of fact the concretely realized examples of OD structures of equivalent layers mostly belong to categories Ia (and also II, IIIa). To include all the categories and subcategories of OD structures built up with equivalent layers, it is necessary to clearly specify the second part of the definition, namely to explain what ‘…as far as possible’ actually means. MDO Structures

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P (1) 2 1 /m 1 {(1) 2 1/2 /a 2 1} Category 1a The value of Z has been already derived Z = 2 There are two - -POs, glide [- a 2 -] and the translation t = a 0 – b/4. Starting with the pair of layers (L0,L1), the layer L2 may assume two postions L2(1) and L2(2). This two positions are obtained when the two - -POs 0,1 t and 0,1 [- a 2 -] have continuations 1,2 t and 1,2 [- a 2 -], respectively. As these two - -POs have general continuations, they become the generating operations of the two MDO polytypes. As we have already seen, the first MDO polytype is triclinic (characterized by stretched triples of layers), with cell vectors a 0 – b/4, b, c and space group symmetry P-1. The second MDO polytype is monoclinic (characterized by bent triples of layers), with cell vectors 2a 0, b, c and space group symmetry P2 1 /a. MDO structures in wollastonite and their generating operations

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Each MDO structure in the OD family of wollastonite is characterized by one kind of layer triples (and one kind of layer quadruples, quintuples,….n-tuples), in keeping with the ‘naive’ definition of MDO structures in OD families built up with equivalent layers. Stretched triples in the triclinic MDO polytype Wollastonite 1A Bent triples in the monoclinic MDO polytype Wollastonite 2M Stretched and bent triples in wollastonite

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P 2 1 (1) {2 r 1 (1)} Category 1b. Non polar layer No -PO with reverse continuation N = 1 (E), F = 1 (E), Z = 2 N/F = 2 Starting with the pair (L 0,L 1 ) the third layer may be placed in two positions L2(1) and L2(2). In the sequence at left (triple T1), the translation 0,1 t has continuation 1,2 t. which guarantees the equivalence of (L 0,L 1 ) and (L 1,L 2 (1)). In the sequence at right the operation [2 - -] in L 1 (1), namely 1,1 [2 - -] has continuation 0,2 [2 - -], which guarantees the equivalence of (L 0,L 1 ) and (L 1,L 2 (2)). The pair of layers L 1,L 2 (1) are related through the - -PO [2 r - -], whereas the pair of layers L 1,L 2 (2) are related through the - -PO [2 -r - -]. MDO structures: general definition (1)

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It is to be noted that whereas in the first sequence [obtained through the constant application of the 2 r operation] all the triples [of type T1] are equivalent, and similarly the quadruples,…n-tuples, in the second sequence [obtained when the 2 r and 2 -r operations regularly alternate] the triples T2 and T3 are not equivalent and simultaneously occur in the structure. It is now important to remark that, in this family, no structure may be built containing only T2 or only T3 triples, whereas an ordered structure where T2 and T3 triples regularly alternate may form. Moreover in this arrangement all the quadruples are equivalent. Although it presents two kind of triples, it is proper to include it among the MDO Structures, within a more general definition of MDO structures. MDO structures: general definition (2)

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We may now better specify, as it was our aim, the definition of structures with Maximum Degree of Order (MDO): A structure containing M distinct kinds of triples (two triples in the example) is defined MDO structure if no other member of the family exists which is built only by a selection of the M types of triples, and if the same is valid for quadruples,…n-tuples. MDO structures: general definition (3)

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Exercise 1 We have presented a schematic drawing corresponding to the OD family P m m (2) n s,2 n 2,r (2 2 ) The single layer has translation vectors a, b and third basic vector (not a translation vector) c 0. The layers follow each other in the direction normal to the drawing. In the Figure the position of L 2 (4) is just over that of L 0 and is not indicated. List all the -POs and all the -POs. The value of Z (possible distinct positions of adjacent layers) we have determined (Z = 4) is valid for generic values of r and s parameters (in the Figure r = s = ½). Determine the values of r and s for which Z = 2. Determine the values of r and s for which Z = 1 (fully ordered structure).

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Solution of Exercise 1 (a) P m m (2) (s = r = ½) n 1/2,2 n 2,1/2 (2 2 ) Translation vectors a, b and third basic vector (not a translation vector) c 0. The layers follow each other in the direction normal to the drawing. In the Figure the position of L 2 (4) is just over that of L 0 and is not indicated. -POs: E, [m - -], [- m -], [- - 2] -POs: [n 1/2,2 - -], [- n 1/2,2 - -], [- - 2 2 ], t = (a+b)/4 +c 0 N = 4, F = 1, Z = 4 For r = 0 (Figure at right) F = 2, as the -PO [m - -] is valid for the pair of adjacent layers; Z = N/F = 2. r = 0 s = 1/2 r = 1/4 s = 1/4

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Solution of Exercise 1 (b) P m m (2) (s = r = ½) n 1/2,2 n 2,1/2 (2 2 ) . r = 1 s = 1/2 r = 1/4 s = 1/4 The same situation occurs for r = 1. Also in that case the -PO [m - -] is valid for the pair of adjacent layers; Z = N/F = 2. Similar situations occur also for the other parameter s. Namely, when s = 0 or 1, the -PO [- m -] is valid for the pair of adjacent layers. F = 2 and Z = 2. Moreover, when r = 0 or 1 and s = 0 or 1 all the -POs, namely E, [m - -], [- m -], [- - 2], are valid for the pair of adjacent layers. Therefore F = 4 and Z = 1.

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P 1 m (1) {n 1/2,2 1 (2 2 )} The OD layers have translation vectors a, b and stack in the c 0 direction. In the Figure the layers are seen down a. a) Indicate the Category and derive the Z value. b) Derive the MDO structures, indicating their cell parameters and space group symmetry. c) Indicate the generating operation for each MDO structure. Exercise 2

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P 1 m (1) {n 1/2,2 1 (2 2 )} Category 2 -POs E, [- m -] Z = 2 MDO1: generating operation [n 1/2,2 - -] a, b, c = 2c 0 +b/2, space group Pc11 MDO2: generating operation [- - 2 2 ] a, b, c = 2c 0, space group P112 1 Solution of Exercise 2 MDO1 MDO2

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Exercise 3 P m m (2) (s = r = ½) n 1/2,2 n 2,1/2 (2 2 ) . Translation vectors of the layers: a, b. The layers stack in direction normal to the drawing. Distance between adjacent layers c 0. a) Derive the MDO structures, indicating their cell parameters and space group symmetry. b) Indicate the generating operations for each MDO structure.

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Solution of Exercise 3 P m m (2) (s = r = ½) n 1/2,2 n 2,1/2 (2 2 ) The four 0,1 operations, namely E, [n 1/2,2 - - ], [- n 1/2,2 -], [- - 2 2 ] and the translation t = (a+b)/4 + c 0, have continuation 1,2 as well as continuations p, p+1 for any value p. Therefore all these operations are generating operations for MDO structures, characterized by the following lattice vectors, space groups, and characteristic triples of layers. MDO1 t = (a+b)/4 + c 0 a, b, c = c 0 + (a+b)/4 P1 L 0 L 1 L 2 (1) MDO2 [n 1/2,2 - - ] a, b, c = 2c 0 + b/2 Pc11 L 0 L 1 L 2 (2) MDO3 [- n 2,1/2 -] a, b, c = 2c 0 + a/2 P1c1 L 0 L 1 L 2 (3) MDO4 [- - 2 2 ] a, b, c = 2c 0 P112 1 L 0 L 1 L 2 (4) Generating operation cell vectors space group triples

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