1. OD Structures: fundamentals of OD theory; examples and applications. (2) Stefano Merlino Department of Earth Sciences, University of Pisa, via S. Maria 53, I-56126 Pisa, Italy.

2. Polytypism in xonotlite (1) Xonotlite Ca 6 Si 6 O 17 (OH) 2 is generally a product of Ca-metasomatism and found, together with other Ca silicates, at the contact of Ca bearing rocks with igneous rocks. At 775-800 °C it dehydrates to wollastonite by topochemical transformation (Dent & Taylor, 1956). The first reliable model for the structure of xonotlite has been proposed by Mamedov and Belov (1955). On the basis of that structural model, polytypic variants were proposed by Gard (1966) and largely confirmed by Gard himself (1966) and by Chisholm (1980) through electron diffraction studies on natural samples. A comprehensive presentation and discussion of polytypism in xonotlite has been recently given by Hejny & Armbruster (2001). It deals with all the aspects of xonotlite polytypism, from the modeling and identification of the various polytypes to the experimental study of xonotlite samples (Kalahari manganese field, South Africa) to the outline of the OD character of xonotlite. We shall now present the polytypism in xonotlite just through the OD approach and we shall realize – I hope – how the OD concepts may be extremely helpful in describing and discussing the polytypic aspects in natural and synthetic compounds.

3. Polytypism in xonotlite (2) Xonotlite as seen down c Xonotlite as seen down a, with b vertical Xonotlite as seen down b Structural data from Kudoh and Takeuchi (1979)

4. Polytypism in xonotlite (3) In the case of xonotlite we may easily realize its OD character by looking at the results of the first accurate refinement carried by Kudoh & Takeuchi (1979) on crystals from Heguri (Chiba Prefecture, Japan). They found a sp. gr. A-1, with a = 8.712, b = 7.363, c = 14.023 Å,  = 89.99°,  = 90.36°,  = 102.18° The same authors have also sketched the structure of a possible polytypic variant, monoclinic s.g. A2/a, a = 17.032, b = 7.363, c = 14.023 Å,  = 90.36°,  = 102.18°. Building OD layer. Translation vectors b, c, third vector (not a translation vector) a 0 = a/2 Layer symmetry A(1)2/m1

5. OD character of xonotlite The symmetry properties of the whole OD family are presented by listing the symmetry operations of the single layer ( -POs) and the operations relating adjacent layers (  - operations). In xonotlite the set of -POs corresponds to the layer group A(1)2/m1. Adjacent layers may be related through a twofold screw axis with translation component b/4 (2 1/2 ) and a glide normal to b with translation component a 0 (a 2 ). The symbol which completely describes the symmetry properties of the family is: A (1) 2/m 1 (1) 2 1/2 /a 2 1 L 0 L 1

6. MDO polytypes in xonotlite (layer A2/m) Regular alternation of 2 1/2 and 2 -1/2 operations. Space group A12/a1. Constant application of 2 -1/2 operation. Space group A-1. L 0 L 1 L 2 L 0 L 1 L 2

7. The electron diffraction studies of xonotlite crystals from different sources have shown that other main polytypes are possible besides the two we have just derived. In fact there is a different possible way to connect along c the structural modules running along b, thus giving rise to a distinct OD layer. and. Whereas in the layer with A2/m symmetry successive blocks were stacked along c with displacement by b/2, in the new OD layer the blocks are stacked along c without any displacement, thus giving rise to an OD layer with P(1)2/m1 symmetry, b and c translation (b = 7.363, c = 7.012 Å), third vector a 0 (a 0 = 8.516 Å,  = 90.36°). Two possible OD layers in xonotlite down c, b vertical down a, b vertical

8. MDO polytypes in xonotlite (layer P2/m) As already said the new OD layer has P(1)2/m1 layer group symmetry b and c translations (b = 7.363, c = 7.012 Å) third vector a 0 (a 0 = 8.516 Å,  = 90.36°). OD theory indicates which  -POs are compatible with the -POs of the layer group P(1)2/m1, thus obtaining the symbol: P (1) 2/m 1 (1) 2 1/2 /a 2 1 Two more MDO structures: P2/a corresponding to the regular alternation of 2 1/2 and 2 -1/2  -POs P-1 corresponding to the constant application of 2 1/2

9. MDO polytypes in xonotlite Layer symmetry A (1) 2/m 1 a 0 = 8.516, b = 7.363, c = 14.023 Å,  = 90.36° MDO1 (regular alternation of 2 1/2 and 2 -1/2 ) space group A 1 2/a 1 a = 17.032, b = 7.363, c = 14.023 Å,  = 90.36° MDO2 (constant application of 2 1/2 ) space group A -1 a = 8.712, b = 7.363, c = 14.023 Å,  = 89.99°,  = 90.36°,  = 102.18° Layer symmetry P (1) 2/m 1 a 0 = 8.516, b = 7.363, c = 7.012 Å,  = 90.36° MDO1 (regular alternation of 2 1/2 and 2 -1/2 ) space group P 1 2/a 1 a = 17.032, b = 7.363, c = 7.012 Å,  = 90.36° MDO2 (constant application of 2 1/2 ) space group P -1 a = 8.712, b = 7.363, c = 7.012 Å,  = 89.99°,  = 90.36°,  = 102.18°

10. Brochantite (1) Brochantite, Cu 4 SO 4 (OH) 6 is a widespread phase derived from alteration of copper sulfides. Its crystal structure was firstly investigated (Tsumeb material) by Cocco and Mazzi (1959) who indicated the space group P2 1 /a and cell parameters a 13.08, b 9.85, c 6.02 Å,  103.4°. Crystal structure of brochantite as seen down b, c vertical. The coordination around copper atoms is represented as squares and squeezed tetrahedra (green colour), disregarding the two longest bonds of the octahedral coordination. The sulphate tetrahedra are drawn with red colour.

11. Brochantite (2) constant (100) twinning and diffraction pattern displaying orthorhombic symmetry (C centred pseudo-rhombic cell with A = 2a + c, B = b, C = c) reflections with l = 2n were sharp, whereas reflections with l = 2n+1 appeared diffuse (along a*) non-space-group absences: 0KL (indices referred to the pseudo-rhombic cell) reflections were absent for K = 2n. Cocco and Mazzi also noticed:

12. All those features (twinning, enhancement of symmetry, diffuse streaks and non-space-group absences) pointed to an OD structure and in fact the crystal structure of brochantite may be described as built up by OD layers with Pn2 1 m symmetry, connected to the adjacent layers through the  -operations indicated in the Figure. P (n) 2 1 m {(2 2 ) n 1/2,2 2 -1/2 } Structural layers with P(n)2 1 m symmetry and basis vectors b, c (translation vectors of the layer; b = 9.85,c = 6.02 Å) and a 0 [a 0 = (a sin  )/2 = 6.37 Å].

13. P (n) 2 1 m {(2 2 ) n 1/2,2 2 -1/2 } Category 1a -  -POs: E, [- - m] Z = N/F = 2/1 = 2

14. Two MDO structures are possible, one corresponding to a regular sequence of 2 -1/2 operations [MDO 1 (a)], the other corresponding to the regular alternation of 2 -1/2 and 2 1/2 operations [MDO 2 (b)] MDO 1 has space group symmetry P12 1 /a1 and a 13.07, b 9.85, c 6.02 Å,  103.3°, corresponding to the polytype studied by Cocco and Mazzi (1955) and afterwards by Helliwell and Smith (1997). MDO 2 has space group symmetry P2 1 /n11 and a 12.72, b 9.85, c 6.02 Å,  = 90° These results persuaded us to examine samples of brochantite from various localities and the new polytype was soon found in the sample from Capo Calamita (Elba Island, Italy).

15. We have derived the whole set of - and  -POs of the bronchantite family just looking at the known structural arrangement. That set of POs indicated the MDO structures and prompted us to look for the ‘missing’ polytype. It seems proper to remark that the OD groupoid family symbol of brochantite could obtained also by looking at its diffraction pattern, without any previous knowledge of the structural arrangement. To this aim it is necessary: - to have a complete list of the possible OD groupoid families - to have a procedure which consents the derivation of the OD-groupoid family from the diffractional features. The 400 possible OD-groupoid families (1)

16. The derivation of all the OD-groupoid families has been one of the most important results of OD theory. A first derivation of OD-groupoid families has been carried on by Dornberger- Schiff (1964). Subsequently a new procedure for listing all the OD-groupoid families has been presented (Dornberger-Schiff and Fichtner, 1972; Fichtner, 1977). The availability of a complete Table of those families, which includes 400 hundred members, is extremely useful in describing and discussing OD structures. They provide for each layer group symmetry the sets of  -POs compatible with it, which allows us to derive the MDO structures, to sketch the corresponding diffraction patterns and to decipher the generally complex patterns of OD crystals. Moreover they sometimes provide – when supported by proper crystal chemical reasoning – a formidable help in guessing reliable structural arrangements. On the other hand, the OD-groupoid family may be derived, in many cases, from the peculiarities of the diffraction patterns (streaks, diffuseness of reflections, non-space-group absences,….), just as the possible space group of ‘fully ordered structures’ may be derived from the ‘normal’ systematic absences. The 400 possible OD-groupoid families (2)

17. We have already explained that disordered or ordered structures in an OD family display diffraction patterns with common reflections (family reflections) [they are independent on the particular sequence and are always sharp]; the various members of the family may be distinguished for the positions and intensities of the characteristic reflections [they may be sharp, as well as more or less diffuse, sometimes appearing as continuous streaks running in the direction normal to the layers]. The family reflections correspond to a fictitious structure, periodic in 3-D, closely related to the structures of the family and called family structure. It may be obtained from a general polytype of the family, by superposing Z copies of it translated by the vector (or vectors) corresponding to the possible Z positions of each OD layer. In the case of wollastonite, the vector relating the two possible positions of the OD layer is b/2; consequently, the family structure may be easily obtained from any polytype of the family by applying the superposition vector b/2. Diffractional features of OD structures

18. Two basis vectors of the family structure are always chosen collinear with the translation vectors of the single layer. If, as in the case of wollastonite, the vectors defining the single layer are b, c (translation vectors) and a 0 (not a translation vector), the vectors A, B, C of the family structure are such that: B = b/q C = c/t A = pa 0 with q, t and p integer numbers. The results are reported in the Figure, which presents schematic drawings corresponding to both MDO structures, to the ‘family structure’ and their diffraction patterns: the family structure of wollastonite, which may be derived from either MDO polytypes by applying the superposition vector b/2, has space group symmetry C12/m1; q = 2, t = 1, p = 2. Diffractional features of wollastonite The values of q and t may be easily obtained by looking at the unit cell of the ‘family structure’, which presents a halved B vector with respect to the b vector of the single layer, whereas C = c. The number p of layers for each A translation [in general for the translation in the direction of the layer stacking] in the ‘family structure’ may be derived also without any previous knowledge of the structure.

19. Category p 1 Isogonality of the corresponding p 2 Bravais symbol of the p 3 and  operations ‘family structure’ I, II 1 yes 1 P, C 1 III 2 no 2 A, B, I, F 2 A general procedure for determining the number p of layers in each translation C of the family structure [the standard orientation assumes a and b as translation vectors of the single layer and c 0 in the direction of ‘missing periodicity’] has been presented by Dornberger- Schiff and Fichtner (1973), who indicate how to obtain the values of p 1, p 2 e p 3 (p=p 1 p 2 p 3 ). - The OD family of wollastonite correspond to category I (p 1 =1). - The corresponding and  operations are isogonal (p 2 =1). - The lattice of the ‘family structure’ is C centered, whereas the translation vectors of the OD layers are b and c. Consequently is p 3 would be 1 for P,A lattices and 2 for B, C, I, and F lattices. Therefore in this family p 3 = 2 and p = p 1 p 2 p 3 = 2. Derivation of the parameter p

20. OD-groupoid family  Diffraction aspects We are generally faced with the opposite problem, namely - to realize whether the crystal has OD character and - to derive the OD-groupoid family from observations of the diffraction features of that crystal. OD character: clearly evident through one-dimensional streaking or diffuseness; or hidden in more subtle features, as partial enhancement of symmetry, non-space-group absences, polysynthetic twinning. Derivation of OD-groupoid family: necessary step in obtaining the structural arrangement of the crystal, just as the derivation of the space group is the first step in solving ‘fully ordered structures’. Once again we shall use the example of wollastonite. Derivation of OD-groupoid family from the diffraction features

21. The diffraction pattern of the monoclinic polytype of wollastonite shows a peculiar distribution of the reciprocal lattice points with: a) systematic absences limited to the reflections with k = 2n, which are absent for h + 2k = 4n b) systematic absences valid for the whole pattern: reflections 0k0 are absent for k = 2n + 1 Rule (a) is not required by any monoclinic space group. However it becomes an ordinary rule for reflections with even k values considered by themselves. These reflections, the ‘family reflections’, correspond to a reciprocal lattice with vectors A* B* C* related to the vectors a* b* c* of the monoclinic polytype as here indicated: A* = a* B* = 2b* C* = c* (1) For these reflections rule (a) becomes: HKL are absent for H+K=2n+1, which points to the following space groups for the family structure: C 1 2/m 1 C 1 2 1 C 1 m 1 We already know – in keeping with the relations (1) - that b = 2B and c = C; therefore q=2 and t=1; moreover we have just shown that A = 2a 0. OD-family from diffractional features in wollastonite (1)

22. Possible space groups of the family structure: C 1 2/m 1 C 1 2 1 C 1 m 1 The systematic absences valid for the whole pattern [reflections 0k0 are absent for k = 2n + 1] can only be due to operations of the single layer ( -POs) and point to the presence of 2 1 operation.The possible symmetries of the single layer are: P (1) 2 1 1 P (1) 2 1 /m 1 (a) The Table presenting all the OD-groupoid families of monoclinic and orthorhombic symmetries is extremely helpful at this point. We may look at that Table, searching for OD-groupoid families presenting a first line of type (a) and we may find: We may search in that Table for OD groupoid families presenting a first line of the type (a). We easily find two of them: P (1) 2 1 1 P (1) 2 1 /m 1 {(1) 2 s-1 1} {(1) 2 s-1 /a 2 1} (b) OD-family from diffractional features in wollastonite (2)

23. Possible space groups of the family structure: C 1 2/m 1 C 1 2 1 C 1 m 1 P (1) 2 1 1 P (1) 2 1 /m 1 {(1) 2 s-1 1} {(1) 2 s-1 /a 2 1} (b) To obtain the value of s (r and s in more general cases), we proceed in this way. I) Operations which are present in the possible space groups of the ‘family structure’: C121 C12/m1 2 2 2 1 1 1 1 ____ 1 2 1 m a II) The translation components of - and  -POs in (b) have to be modified in accordance with the passing from the dimensions of the single layer to those of the family structure, namely the translation components which refer to b have to be doubled and those which refer to a have to be halved. We obtain: 1 2 2 1 1 2 2 /m 1 1 2 2s 1 1 2 2s /a 1 III) With s = ½ the operations reproduce those found in the s.g. of the ‘family structure’ and we obtain the two possible OD-groupoid families for wollastonite: P (1) 2 1 1 P (1) 2 1 /m 1 {(1) 2 1/2 1} {(1) 2 1/2 /a 2 1} OD-family from diffractional features in wollastonite (3)

24. I have described and discussed the OD character of brochantite Cu 4 (SO 4 )(OH) 6 and I have shown how the symmetry of the single layer, as well as the  -POs which characterize that family, may be obtained just by looking at the structural arrangement obtained by Cocco and Mazzi (1959). Obviously the  -POs could also be derived on the basis of the layer group symmetry by looking at the Table compiled by Dornberger- Schiff and Fichtner (1972) [the Table is enclosed as file Tabgroupoids.doc], searching for a set of  -POs compatible with that layer group symmetry. However it is possible to obtain the OD groupoid family without any previous knowledge of the structure, through a careful scrutiny of the diffraction pattern, and appropriate use of the Table compiled by Dornberger-Schiff and Fichtner (1972). Exercise 4

25. Diffraction pattern of brochantite Additional observation: the sharp diffraction pattern displays orthorhombic symmetry and the statistical distribution of the intensities points to a point group symmetry 2/m 2/m 2/m.

26. Fig. 1 represents the diffraction pattern of brochantite from Tsumeb (and from other various localities), as seen down b; a* and c* vectors correspond to the monoclinic cell given by Cocco e Mazzi (1959) in their structural study, with: a=13.08, b=9.85, c=6.02 A,  =103°22’, space group P2 1 /a. The OD features of this mineral are clearly displayed in the diffraction pattern; in particular: a) streaks parallel to a* for reflections with l=2n+1; b) non space group absences The sharp reflections (l=2n) may be referred to a reciprocal cell with vectors A* = a* (‘diffusion’ direction; therefore direction of ‘missing periodicity’) B* = b* C* = 2c*-a* A, B, C are the corresponding direct vectors (basis vectors of the ‘family structure’). Solution of exercise 4 (a)

27. We assume that brochantite has OD character and its structure consists of equivalent layers. We shall try to derive, under such assumption, the symmetry properties of the whole OD family, on the basis of the diffraction pattern. Let b and c be the vectors giving the periodicities of the single layer (a 0, the third basis vector, is not translation vector). A, B e C (vectors of the ‘family structure’) are related to a 0, b e c as in (1): A = pa 0 (p will be determined later) B = b (1) C = c/2 The scrutiny of the family reflections indicates an orthorhombic symmetry (2/m2/m2/m) with the following systematic absences: reflections H0L are present only for H+L=2n reflections 0KL are present only for K=2n reflections HK0: no systematic absence These absences point to the following space group for the ‘family structure’: P 2 1 /b 2 1 /n 2 1 /m (2) Solution of exercise 4 (b)

28. The Table compiled by Dornberger-Schiff and Fichtner (1973) is very useful for the next steps. That Table presents all the OD groupoid families of monoclinic and orthorhombic symmetries, for OD structures built up with equivalent layers. The symmetries of the ‘family structure’ reflect the and  symmetries of the corresponding OD family. Two observations seem proper: i) The Table by Dornberger-Schiff and Fichtner (1973) was built assuming a and b as translation vectors of the single layer and c 0 in the direction of ‘missing periodicity’; ii) in passing from the and  operations of the layers to the isogonal operations of the ‘family structure’, the translation components of screws and glides have to be modified to take account of the different periodicities in the various directions, as indicated in (1) [A = pa 0, B = b, C = c/2] Solution of Exercise 4 (c)

29. As regards observation i) it seems useful to change the reference system, so that the direction of ‘missing periodicity’ be c 0, not a 0 ; what may be obtained through a cyclic transformation A  C  B  A. Relations (1) now become: A = a B = b/2 (3) C = pc 0 and the space group of the ‘family structure’ is now: P 2 1 /n 2 1 /m 2 1 /a (4) The correct solution may be found by looking for those OD groupoid families which have, in all the three positions of the symbol, both a general twofold axis (2, 2 1, 2 s,....) and a general mirror plane (m, n, a, n 2,r,....), for example: P 2 m (m)  n s,2 2 s (2 2 )  (5) Solution of Exercise 4 (d)

30. Observation ii) gives a clue for the third position. In space group (4) there is, in the third position, a glide [.. a], namely [.. n 1,2 ] (general symbol for a): the or  operator of the single layer corresponding to it is n 1,1, namely [.. n], taking into account the relations (3). Therefore only two symbols, among all the symbols of kind (5), survive: P 2 1 m (n)  n s,2 2 s-1 (2 2 )  (6) and P 2 a (n)  n s,2 2 s-1 (2 2 )  (7) A = a B = b/2 (3) C = pc 0 P 2 1 /n 2 1 /m 2 1 /a Space group of the family structure Solution of Exercise 4 (e)

31. P 2 1 m (n) P 2 a (n)  n s,2 2 s-1 (2 2 )  (6)  n s,2 2 s-1 (2 2 )  (7) P 2 1 /n 2 1 /m 2 1 /a A = a, B = b/2, C = pc 0 It is now possible to determine the value of p in (3), namely the number of layers (width c 0 ) for each translation C of the ‘family structure’. Dornberger-Schiff and Fichtner (1973) indicate how to obtain the values of p 1, p 2 e p 3 (p=p 1 p 2 p 3 ). Category p 1 Isogonality of the corresponding p 2 Bravais symbol of the p 3 and  operations ‘family structure’ I, II 1 yes 1 P, C 1 III 2 no 2 A, B, I, F 2 Both (6) and (7) correspond to category I (p 1 =1). In both cases the corresponding and  operations are not isogonal (p 2 =2). As p 3 =1, we obtain p=2, namely there are two layers for each translation C. Consequently the operator 2 2 in the third position of (6) and (7) becomes 2 1 in the third position of (4). Solution of Exercise 4 (f)

32. P 2 1 m (n) P 2 a (n)  n s,2 2 s-1 (2 2 )  (6)  n s,2 2 s-1 (2 2 )  (7) P 2 1 /n 2 1 /m 2 1 /a A = a, B = b/2, C = 2c 0 The choice between the two survived symbols (6) and (7) may be easily done just looking at their symbols and at the symbol of the layer group of the family structure. The correct solution is given by (6), due to the presence of [2 1 - -] and [- m -] which characterize the symmetry of the family structure. At the same time we may determine – with the same procedures we have previously developed - the value of s, which is s=1/2. In conclusion the symbol of the ‘OD groupoid family’ under study is: P 2 1 m (n)  n 1/2,2 2 -1/2 (2 2 )  If we like to go back to the orientation originally assumed for brochantite we obtain: P (n) 2 1 m  (2 2 ) n 1/2,2 2 -1/2  Solution of Exercise 4 (g)