CRYSTAL CHEMISTRY OF COMPLEX SULFIDES (SULFOSAL TS) AND ITS CHEMICAL APPLICATION
EMIL MAKOVICKY Geological Institute University of Copenhagen 0ster Voldgade 10 DK-1350 Copenhagen K Denmark
ABSTRACT. Crystal structures of many complex sulfides can be obtained by recombination of fragments of archetypal structures by the action of various structure-building operators. Chemical categories, coordination polyhedra, archetypes and recombination operators for these structures are described in the paper. These structures often occur as members of accretional or variable-fit homologous series or as homologous pairs of various kinds. Three case studies supplemented by exercises are described in some detail: the lillianite homologous series, the misfit layer structures and those structures that represent combination of the accretional and variable-fit principles.
1. Crystal Structures of Complex Sulfides and Sulfosalts
Crystal structures of a large number of complex sulfides can be described as composed of rods, blocks or layers of simple, archetypal structures that are joined (recombined) in various ways by the action of one or more structure-building operators, such as the various kinds of unit-cell twinning. The resulting structures were denoted as the recombination structures by the Subcommittee on the Nomenclature of Inorganic Structure Types (Lima-de Faria et aI., 1990). It is this kind of complex sulfides we deal with in the present review; those with complex chemistry but simple structural principles (e.g., the substitution-, interstitial- or omission derivatives of archetypal structures) are not included.
1.1. SULFIDE FAMILIES WITH RECOMBINATION STRUCTURES
The principal sulfide family with recombination structures are the so-called "sulfosalts". They are defined as complex sulfides (rarely also selenides and tellurides) AxBySz with A = Pb, Cu, Ag, Hg, TI, Fe, Mn, Sn2+, Sn4+ and other metals; these may include alkali metals, alkaline earths and lanthanides in some synthetic sulfosalts; B = formally trivalent cations As, Sb and Bi (rarely also Te4+) in non-planar, (fundamentally) three-fold coordination BS3; and S = S, Se and Te2-. Structurally very close to sulfosalts are on the one hand the complex sulfides of (primarily) octahedral In with Pb, Sn, Bi and other
metals and on the other, the layer misfit sulfides, based on a combination of layers that contain large cations (La to Sn, Y, Sn2+, Pb and Bi3+) with layers that contain smaller cations with octahedral (Cr, Ti, V) or trigonal-prismatic (Nb and Ta) coordinations. Yet another group of complex sulfides with recombination structures are the lanthanide sulfides in which large coordination polyhedra of light lanthanides combine with smaller polyhedra of heavier lanthanides.
1.2. TYPICAL COORDINATION POLYHEDRA
At the small-scale level, the crystal chemistry of sulfide recombination structures is determined by the easy and variable combinations of fundamental coordination polyhedra, yielding both the blocks of archetypal structures and the configurations on
Pb Pb
~~ 6 6 Sb Bi
Bi Bi
Bi Ag
Figure 1. Selected coordination polyhedra in complex sulfides: tri, bi- and 'lying' monocapped trigonal prisms of Pb and metalloids; a quasi-octahedral and octahedral coordination of Bi, [2+4] coordination of Ag. Small circles: cations, large circles: sulfur; void and filled circles denote atoms at 2 distinct height levels, -2 A apart.
their boundaries. With the exception of largest cations such as TI and K that have high coordination numbers and mostly irregular coordination polyhedra, the large cations typically assume trigonal prismatic coordination with additional anions capping all three, two, or one of the prismatic faces (Fig. 1). Such prisms-may be "standing", i.e. parallel to, or "lying", i.e. perpendicular to the short (-4A) repetition period that is characteristic for
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the majority of these structures. Such trigonal prisms are typical for the most characteristic chemical element of recombination sulfosalt structures, lead. Moreover, they are also the typical coordination polyhedron for As3+, Sb3+, Sn2+ and a large part of Bi3+. In these cases it is the lone electron pair of the cations that occupies the volume of the prism whereas the cation resides in (or just inside of) one of the approximately square-shaped prismatic faces. All short bonds are concentrated in the s~uare pyramid that caps this face. The length differences between the three shortest (Q: ) bonds in the pyramid, two additional, longer bonds (Q,2) in the pyramidal base and the remaining distances, to the S atoms shielded away by the lone electron pair 22 of the cation, decrease from As to Bi. For bismuth, cases of irregular, quasi-octahedral coordinations BiS(3+2)+1 grading into fairly regular octahedral coordinations BiS6 are frequent; in fact, all intermediate stages between the limiting coordinations BiS3 and BiS6 are observed.
The prism caps of Pb coordination prisms are fairly well commensurable with the dimensions of square pyramids of As, Sb and Bi, the latter (or the groups BS3) are in tum commensurable with the coordination octahedra around Pb, Mn, In, Fe, Ag, (Hg), etc., or with tetrahedral and trigonal coordinations of primarily Ag and Cu. The latter may fill interspaces in the above arrays or form polyhedral arrangements of their own, with only subdued amounts of prismatic, pyramidal or octahedral cations present.
The interplay of variously capped (and oriented) trigonal coordination prisms with coordination octahedra is essential also for the complex lanthanide sulfides and the layer misfit compounds. Through the entire range of structures and compositions the relative plasticity (ability for minor adjustments) of large coordination polyhedra created around large or lone-electron-pair cations is of primary importance in the interlayer and interblock adjustments.
1.3. LARGE-SCALE CRYSTAL CHEMISTRY: ARCHETYPES AND STRUCTURE BUILDING PRINCIPLES
For the recombination structures aggregation is typical of a number of coordination polyhedra into rods, blocks or layers with internal structure similar to a simple 'archetypal' structure.
The structure archetype occurs as an independent structure and encompasses all the geometric as well as bonding properties of a given atomic array. In the case of weakly expressed steric lone electron pair activity of cations (e.g. in the case of Bi3+or Pb2+) the internal structure of blocks (etc.) represents little deformed PbS (i.e., B1) archetyRe (Fig. 2a). In the structures with stereochemically active lone electron pairs (As-, Sb-, Sn +-, and some Bi-containing structures) this archetype is often replaced by the SnS archetype (Fig. 2b) with the coordinations BS(3+2)+1+1 or the Til archetype (Fig. 2c) with the coordinations BS5+2· The relationship between the PbS and Til (approx. also SnS) archetypes can be expressed as 2A-shear between double layers M2S2 along the -4A direction. This shear alters coordination octahedra into monocapped trigonal coordination prisms, offering thus better accommodation for the lone electron pairs of cations.
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a
o b ,---
b
c
Figure 2. Three principal archetype structures for complex sulfides: (a) PbS, (b) SnS, and (c) Til. Drafting conventions as in Fig. 1.
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The rods, blocks or layers of archetypal structures are limited by surfaces that can be indexed in terms of the archetypal structure. For the majority of cases these surfaces are (100)PbS' (100)SnS (with pseudotetragonal motifs of metal and S atoms; the so-called Q surfaces), as well as (111)PbS (with pseudohexagonal, H, motif of S atoms) and the analogous planes (210)SnS and (301 )SnS (sheared H surfaces).
The more complicated surfaces (e.g. (101)SnS' (501)SnS or (311)PbS) can be interpreted as combinations of stripes of two fundamental, simplest surfaces. The two atomic planes thick (1 OO)PbS layers can be interpreted as belonging to either PbS or Til (SnS) archetype.
For the large part, the structure-building principles (operations) according to which the rods, blocks or layers of archetypal structure are recombined into a complex sulfide structure are various types of unit-cell twinning: reflection-(Le. mirror-)twinning that acts either on a full set of atoms (Andersson & Hyde, 1974) or on a "contracted" (Le. partial) set of atoms out of the larger one of the two mirror-related portions (Takeuchi, 1978), glide-reflection twinning (I.c.) that can consecutively be applied on different (hkl) planes of the archetypal structure (swinging twinning of Bovin & Andersson, 1977), and cyclic twinning (Hyde et al. 1974). For complex sulfides non-commensurability between adjacent building blocks, etc. (the vemier principle) comes next in the order of importance (Makovicky & Hyde, 1981), followed by antiphase and out-of-phase boundaries crystallographic shear (Hyde et aI., 1974), the 2A-shear (or slip) in the structures with 4A periodicity, and the intergrowth of two different structure types on a unit cell scale (ibid.). Detailed definitions of these operations were made by other contributors to this volume.
Coordination states and, in the case of cations, often also the chemical species of atoms on the block surfaces, rarely also in the interfaces, differ from those inside the building blocks.
2. The homologous series
A large number of sulfosalts and other complex sulfides represent members of homologous series. There are two principal types of homologous series which the present author defined as the accretional and the variable-fit series, respectively (Makovicky, 1989).
2.1. THE ACCRETIONAL SERIES
The accretional series is a series of structures in which the type(s) of building blocks (rods, layers) and the prinCiples that define their mutual relationships remain preserved but the size of these blocks varies incrementally by varying the number of fundamental coordination polyhedra in them. The order N of a homologue in this type of series can be defined by the number of coordination polyhedra (polyhedral layers) across a suitably defined diameter of the building block (rod, layer). The ratio of atoms on the block surfaces (interfaces) to those inside the blocks varies with N as also does the overall MIS ratio. -
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Every member of the accretional series has its own chemical formula, unit cell parameters and symmetry; a general chemical formula for the entire series can be devised as well. A given homologue can represent a single compound or a (dis)continuous solid solution which can exsolve into a series of structurally ordered phases with well defined compositions (or compositional ranges) but with the same N. The ideal space group can then be reduced to subgroups in the process of (cation) ordering.
Thompson (1978) described accretional homologous series using a different name (polysomatic series) and different approach: the coordination polyhedra on the surfaces and in the interfaces of the blocks (in his case, layers) are treated as one type of (layer) module whereas the incrementally accreted polyhedral layers in the layer (block) interior as another type of layer modules. Thus, all accretional homologues are treated as ordered intergrowths of two structure types which occur in different proportions in different homologues (polysomes). The two descriptions are equally valid; the polysomatic approach presents difficulties for either rudimentary or very elaborate interfaces (Makovicky, 1989) as well as for rod structures with two intersecting systems of interfaces (Veblen, 1991).
Some accretional series are extensive series with N varying over a range of values. Besides the members with equal widths lli1 = J::k) across the unit-cell twinning plane or interface of another kind, those with unequal widths (N1 *- N2) can occur. A number of accretional series are limited to only pairs of homologues (N1, and N2 = N1 + 1) because of various local or global crystal chemical reasons. These pairs can be extended into combinatorial series, the members of which represent regular intergrowths of the above two accretional homologues: N1 N2N1 N2· .. , N1 N1 N2N1 N1 N2· .. , N1 N2N2N2N1 N2N2N2· .. , etc.
2.2. THE VARIABLE-FIT SERIES
This type of homologous series occurs for crystal structures that are composed of two kinds of alternating, mutually non-commensurate layers (rarefy of such columns or of a matrix/infilling combination). Each kind of layer has its own short-range (sub)periodicity and it takes m periods of one layer and .!l periods of the other layer before they meet in the same configuration as at the origin. These are so-called semi-commensurate cases for which the two layer (sub)periodicities comprise a ratio of two not very large integers. Besides them, incommensurate cases exist for which these short-range periodicities comprise irrational fractions or ratio of very large integers. Non-commensurability of layers may occur in one or two interplanar directions. With minor compositional changes in the cations, the min ratio and the MIS ratio vary within certain, rather narrow limits, leading to a series of closely related compounds.
Those cases for which the pile of the two alternating non-commensurate layers is periodically sheared, kinked or modified by antiphase boundaries, represent combination of the accretional (separately for each layer) and variable-fit (for the interlayer match) principles. The geometrical constraints in these structures are much more severe than in the pure variable-fit structures and result in a very reduced number of homologues (usually homologous pairs).
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3. Case study I: The lillianite homologous series
The Iillianite homologous series (Makovicky & Karup-M0I1er, 1977) is an easily understandable extensive accretional series, eminently suitable for illustrating all the above defined problems and categories. Members of this series are the Pb-Bi-Ag sulfosalts with the structures consisting of altemating layers of PbS-like structure, cut parallel to (311 )PbS. These planes also represent the reflection- and contact planes of unit-cell twinning. The overlapping octahedra of the adjacent, mirror-related layers are replaced by bicapped trigonal coordination prisms PbSS+2 with the Pb atoms positioned on the mirror planes (Otto & Strunz, 1968; Takeuchi et aL, 1974, etc.).
Distinct homologues differ in the thickness of the PbS-like layers. This is conveniently expressed as the number.N of octahedra in the chain of octahedra that runs diagonally across an individual PbS-like layer and is parallel to [011]PbS (Fig. 3). Each lillianite homologue can be denoted as N1,N2L where N1 and li2 are the (not necessarily equal) values of N for the two alternating sets of layers (Fig. 4). Its chemical formula is PbN-1-2xBi2+xAgxSN+2 (Z = 4) where N = (N 1+N2)/2 and ~ is the coefficient of the Ag+Bi = 2Pb substitution. If the trigonal coordination prisms of Pb are not substitutable (which is very close to the real situation), xmax = (N-2)/2. This structure type is quite frequent also outside the Pb-Bi-Ag compositional space, e.g. for a number of complex lanthanide sulfides; the general formula then becomes M2+ N-1 M3+ 2SN+2. For a somewhat different generalized approach the reader should consult Hyde et aL (1979).
The existence of lillianite homologues (sensu lato) depends on the suitable sizes of coordination polyhedra (trigonal prisms vs. octahedra), satisfactory valence balance and feasibility of close-to-regular octahedral (Le., ccp or PbS-like) arrays. The case with N = 1 is typified by, for example, NdYbS3 (Fig. 5) or UFeS3, N = 2 by MnY2S4 or CrEr2S4, whereas N1;N2 = 1;2 by FeHo4S7, MnY4S7 and Er3Sc2S7 (lanthanide atoms occupy the trigonal coordination prisms as well as they mix statistically with the other component in the octahedral positions). For the MnS-Y 2S3 system, Bakker & Hyde (1978) found that the homologous pair MnY2S4 and "MnYS3" (that occurs only as a layer in MnY4S7) form a combinatorial series that comprises MnY 4S7 (1,2L), Mn2 Y SS11 (1,2,2L), Mn4YlOS1ge,2,2,2,2L), etc. Both the cases with ideal symmetry and those with subgroup symmetry (Problem 1 ) are present; the latter caused either by deformations of coordination polyhedra or by the asymetric position of cations in the trigonal coordination prisms. The tetrahedral voids on the mirror planes of unit cell twinning are occupied only in exceptional cases (Eu2CuS3,1, 1 L) [for references on individual compounds the reader should consult Structure Reports].
Higher homologues start at N = 4; they allow more pronounced departures from the galena-like array, especially in the form of locally "inflated" interspaces that accommodate lone electron pairs of quasi-octahedral Bi or Sb (so-called lone electron pair micelles, Makovicky & Mumme, 1983). On~ 4,4L and 7,7L are known for the Ag-free subsystem PbBi-S. Reduction of symmetry in ,7L from the usual orthorhombic to monoclinic, caused by Pb/Bi ordering that proceeded differently for the two ~originally) mirror-related slabs was exceptionally observed in one natural occurrence of 7, L (Mumme, unpubL). Allowing for the Ag+Bi = 2Pb substitution in octahedral layers, also the cases 4,7L, 4,8L, 5,9L and
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Figure 3. The crystal structure of lillianite, Pb3Bi:p6' N1;N2 = 4;4 (Takagi & Takeuchi, 1972). In order of decreasing size circles denote 8, Pb, (Pb,Bi). "Galena-like" layers are ruled, bicapped trigonal coordination prisms of Pb are stippled. Void and filled circles denote atoms at ~ = 112 and 0, respectively, along the vertical, 4.1 A axis.
Figure 4. The crystal structure of vikingite, Pb1OA94Bi1:P30 (Makovicky, Mumme & Madsen, in prep.), a lillianite homologue N1;N2 = 4;7. The same conventions apply as for Fig. 3; the smallest circles represent (Ag,Bi).
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Figure 5. The crystal structure of NdYbSiCam§ & Laruelle, 1974), a lillianite homologue N1;N2 = 1;1. Nd occupies trigonal prisms, Yb the octahedral positions of the (rudimentary) PbS-like layers. For other conventions see Fig. 3.
Figure 6. The crystal structure of gustavite, PbAgBi:p6 (Kupcfk & Steins 1990). In order of decreasing size circles denote S, Pb, Bi and (Ag,Bi); two atomic levels of the orthorhombic subcell, -2 A apart, are indicated by void and filled circles. Stippling indicates cores of large lone electron pair micelles in which the ~ pairs of Bi are concentrated.
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Figure 7. The crystal structure of ramdohrite, Ag6Pb12Sb2:P48 (Makovicky & Mumme, 1983). In the order of decreasing size the circles indicate S, Pb, Ag (and mixed positions) and Sb. All atoms are situated about four levels along the 8.7 A (vertical) axis, -2.2 A apart. Stippling indicates cores of large and small lone electron pair micelles in which the ~ pairs of Sb are concentrated.
Figure 8. The crystal structure of benjaminite, CuO.SPbOAA92.3Bi6.8S12 (i.e. -Ag3Bi.p12) (Makovicky & Mumme, 1979), the 7th homologue of the pavonite homologous series. Conventions are analogous to Fig. 6 but the PbS-like portions are ruled and the BiS3+2 caps of distorted trigonal prisms on the boundaries of galena-like layers with N 1 = 1 and N2 = 7 are stippled.
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11,11 L (Makovicky & Karup-M0I1er, 1977) are found, as well as the disordered combinations based on N1 = 4 and N2 = 7 in different proportions (ibid., Skowron & Tilley, 1990). The structures with close-to-ideal PbS-like arrays (Fig. 3) and those with extensive lone electron pair micelles (Fig. 6) do not form continuous solid solutions inspite of the same Ii or/and they separate by exsolution at low temperatures (e.g. Pb3Bi2SS-
PbAgBi3SS' Figs. 3 and 6). The match of coordination polyhedra is better in the Pb-Ag-Bi sulfosalts and yields an extensive accretional series. In the parallel system Pb-Ag-Sb-S the mismatch of Pb, Ag and Sb coordination polyhedra appears serious; only members with N = 4 were found and they are known only for;:: 50% of substitution (Ag+Sb) for 2Pb; still, they form a string of intermediate (exsolved) phases with interesting distribution of lone electron pair micelles and superperiods of the 4A dimension (Fig. 7). The existence of phases close to 50% substitution often depends on incorporation of smaller M2+ instead of octahedral Pb; e.g. AgPb3MnSb5S12 (Moelo et aI., 1988).
The symmetry or asymmetry of the trigonal prismatic cation site varies widely for the lanthanide-based lillianite homologues with lower N values. The difference in the symmetry or asymmetry of this position becomes substantial, and to the large extent unsurmountable by cation substitutions, between the lillianite and pavonite homologous series in the Pb-Bi-Ag-Cu system (Makovicky et aI., 1977; Makovicky, 1981, 1989). In the pavonite homologues (Fig. 8) all members have N1;N2 = 1;Npav where Npav = 2 to 8, possibly even 11. The trigonal prisms on the planes of contracted-set unit-cell twinning are occupied by square-pyramidal Bi and its lone electron pair. The extensive PbS-like portions (N2) contain quasi-octahedral to octahedral Bi combined with Ag,Cu and some Pb. The sole, skewed octahedra in the narrow portions (N1 = 1) represent AgS2+4 or contain [3] and [4] coordinated Cu. Also two other Iillianite analogues, BaCu4S3 (N = 1) and Pb1_xBi2+xCu4_xS512 (x = 0.88; 2-A sheared, with N '" 5) contain Cu atoms in similar coordinations.
Two more homologous· series, the extensive meneghinite homologous series and the combinatorial sarto rite homologous series, both being based on glide-reflection twinned arrays, are examined in Problems 2 and 3.
4. Case study II: Sulfides of heavy and transition metals with misfit layer structures
These sulfides consist of two types of altemating, mutually non-commensurate layers. One type (0 layers) has pseudotetragonal symmetry and belongs to the PbS or SnS archetype. The other type (H) are octahedral or trigonal prismatic layers.
These compounds can display one of the three following types of interlayer match. If the o layers are described as (100) and the match is expressed in terms of centered pseudotetragonal and orthohexagonal <£H = -v3.l4i) subcells, the first, "cylindrite-like" match (Fig. 15) implies (exactly or approximately).l2a 1I.l4i and fa II £H with.l2a >.l4i and fa < £H (e.g. in cylindrite -FePb3Sn4Sb2S14' 19.12a '" 30.l4i and 13fa = 12£H, common modulation of both layers II £); the second, ILaCrS3-like" match implies fa = £H and .l2a > .l4i (Q. is the semi- or incommensurate modulation direction, e.g. 3.l2a = 5QH for -LaCrS3). The third match involves 0 layers tumed by 45° so that.l4i II (and equal to) .l2aprimitive (=
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Figure 9. The crystal structure of meneghinite, CuPb1:pb.,824 (Euler & Hellner, 1960), homologue N = 5 of the meneghinite homologous series. Circles in order of decreasing size indicate S, Pb, (Pb,Sb), Sb. The statistically occupied, unrefined positions of Cu in the interfaces of SnS-like blocks were omitted. Filled and empty circles indicate atoms at two £ levels, 2. 1 A apart.
Figure 10. The crystal structure of BixBb2_xBn2SS (0.4 > x> 0.2) (Kupcfk & Wendshuh, 1982), N = 4. Metal atoms are not differentiated, remaining conventions as in Fig. 8.
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Figure 11. The crystal structure of aikinite, CuPbBiS3 (Ohmasa & Nowacki, 1970; Kohatsu & Wuensch, 1971 ). In the order of decreasing size circles indicate S, Pb, Bi and Cu; atoms are at two ~ levels, 2 A apart.
Figure 12. The crystal structure of sartorite, PbAs2S 4 (/itaka & Nowacki, 1961 ), homologue N = 3 of the sartorite homologous series. Only the 4.2 A substructure, with atoms at two levels, has been determined. Circles in order of decreasing size indicate S, Pb and As.
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Figure 13. The crystal structure of rathite I Pb11.1T1o.~s1iAs1.eAg2.;)S40 (Marumo & Nowacki; 1965), homologue N = 4 of the sartorite homologous series. Projection of the 8.4 A structure onto (001). In the order of decreasing size circles indicate S, Pb (and TI), As (and Ag). Overlapping atoms are 4.2 A apart in the.f direction; statistically occupied Ag and As positions are on the same level.
Figure 14. The crystal structure of baumhauerite, Pb11.eAS15.~gO.£P36 (Engel & Nowacki, 1969), a member of the combinatorial sartorite series. The same conventions apply as for Figs. 12 and 25.
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1/2 diagonal of Q centred) and ~ > £oprimitive (e.g., cannizzarite -Pb24Bi2SS66 with 12£0 "" 7~). The LaCrSs-like match occurs tor the largest Q/H cation radius ratio whereas, as the other extreme, the cannizzarite match is typical for about equally large average cations in the Q and H layers (Fig. 16).
The Q layers, MeS, can be 2 to 4 atomic layers thick; the H layers, MeS2, represent single, double- or seldom even triple-octahedral layers; the trigonal prismatic layers occur between two consecutive Q layers as single layers or in pairs. Fine changes in the chemistry of the two layer types lead to variable-fit homologous series for all three match types (Problem 4). Geometric modulation of layers along the non-commensurate direction is ascribed solely to layer interactions for the "ABXs" cases whereas it might be connected with the constant presence of minor cations, Fe and Sb, in the structures of cylindrite family (complex Pb-Sn2+-Sn4+-Sb-Fe sulfosalts). Recent detailed reviews were written by Makovicky & Hyde (1981,1992).
5. Case study III: Complex sulfides with combination of the accretional and variable-fit principle
Stresses and strains as well as the charge balance problems in the non-commensurate layer structures can be relieved by breaking them up by means of glide planes, antiphase or out-ot-phase boundaries, or by exposing alternatively the Q and the H surfaces of the same layer (Fig. 17). With respect to the rest of the structure, these planes of break-up may be composition-and-structure-conservative or non-conservative (Fig. 17). Problem 5 deals with several basic types of this category.
For the majority of these phases, cannizzarite (Fig. 18) acts as a parent structure. Similar to cannizzarite, they represent piles of Q- and H-type layers of different thicknesses (not always the thicknesses observed in cannizzarite itself); they have similar match modes as cannizzarite and similar average cation radii.
6. Epilogue
The rationale behind this contribution is to outline the basic structural features of complex sulfides with recombination structures. Its volume prevents exhaustive discussion of details for the structural families for which detailed reviews have already been written (referenced in Makovicky 1989 and Makovicky & Hyde 1992). Detailed exhaustive descriptions of individual structure-building principles are given by other contributors to this crystal chemistry volume.
7. Acknowledgments
Kind assistance of the editor of this volume, Prof. E. Parthe, (University of Geneve), the typing assistance of Mrs. M.L. Johansen and the help of generations of draftsmen and -women are gratefully acknowledged.
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Il II t Iq I-- H II- ~ I~ I · Il ilc. ,I I I J II
Il l- • ~( 11;11 h I j l
1 11 I 11
lur Ie .lIs :-1-I I 1-1 .....
I~[-T
1-:= I-r
t+ ltoe
Ie" a ~
=
II I-'-
1==1= iJ. I r~ I c
b - b. b.
Figure 15. Inter layer match in cylindrite, -FePb3Sn4Sb2S14 (Makovicky, 1974). The two component layers, 0 and H, are indicated by their submesh (pseudotetragonal and orthohexagonal; denoted by the two grids with their centrations partly omitted) and by the two unit cells. The 12. x f (incommensurate x semicommensurate) coincidence mesh is stippled; modulation vector is [001J.
,:~ .. bo ..
a
Figure 16. Inter layer match of pseudotetragonal (0) and orthohexagonal (H) submesh in (a) layer misfit compound (LaS}1.20CrS2 (i.e., "-LaCrS3") and (b) in the lanthanide sulfide Er9La1oS27 with a match analogous to that in cannizzarite.
o --"----------------....!... ••....••••.•••••••••••••••••.••.••••••...•.....••.••..•..•••.•.....•••••.•.••......•. ..!---"- c -...!.... ••••••••••••••••••••••••••••••••••••••••••••.•.••••••••••••••••••••••••..•..•••••••• ..!-
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I, ',','.,',.,.,',',','.'.ij: ................... H : i''''''''''''''''''', Hl i :HI .................. , i--__ :1u1 1 i:ih .................. ·1 lill .................. jlii
iv C8)VI····::::::····:~~VI····:::::::···:~~VI ..... :::::: ... :~8j I. ••••••••• ~ ••••.••••• ~ ••••••••••• ~
I "·1 ". 1 ". I I '. '. '.
Figure 17. A non-commensurate, layer misfit structure (a), composed of alternating layers of two types, and the recombination structures (i-iii) derived from (0) by means of (i) composition-conservative and of (ii-iii) two kinds of composition non-conservative antiphase boundaries (or glide planes). (iv) A layered non-commensurate structure with corrugated layers that are composed of alternating strips of two kinds that exhibit layer misfit on interfaces.
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8. Problems
Problem 1 (a) Determine ideal space groups for lillianite homologues with N1 = N2 (Fig. 3) and N1 '* N2 (Fig. 4). Derive one or more subgroups by displacing various atoms (groups) from their 'ideal' positions or by ordered cation substitutions in the 4A or sA cell.
~bJ Calculate chemical formulae of Ag-free and (Ag+Bi)-substituted homologues 4,4L, 7,7L, , L, 11,11L, 4,7L as well as of the hypothetical homologue 2,2L.
(c) U nit cell parameters of Pb3Bi2S6(4,4L) are g 13.54 A, Q 20.45 A and .£ 4.10 A,; those of Pb6Bi2S9 e,7L) are g 13.70 A, Q 31.21 A and.£ 4.13 A. Calculate the increment Ab for AN = 1 and estimate the Q parameters for the hypothetical homologues 2,2L, 5,5L, 6,S'land 11,11L. What parameter do we obtain for 4,7L?
Problem 2 Figures 9-11 show three distinct members of the meneghinite homologous series: meneghinite CuPb13S~S24' synthetic BixSb2.XSn2S5 and CuPbBiS3.
(a) Outline the slabs (50±1 )SnS in the SnS-like portions of each structure.
(b) Determine the value of .!:i for each of these structures using the number of square coordination pyramids in the SnS-like regions.
(c) Show the statistically occupied Cu site in meneghinite using the structure of CuPbBi3S6 as a model.
(d) Calculate chemical formulae of meneghinite homologues N = 2 to 5 using the general formula you have derived from Figs. 9-11. Calculate these formulae separately for Cu-free and Cu-saturated cases. What is the site occupancy for Cu in real meneghinite, Cu Pb13Sb7S24?
(e) Unit-cell parameters of CuPb13Sb7S.24 are g 11.36 A, Q 24.06 A and.£ 4.13 A. Those of CuPbBiS3 are g 11.32 A, Q 11.64 A and .£ 4.04 A. Determine Q parameters of the hypothetical phases with N = 3 and 4. What is the (ideal) space group for all these structures? Compare the calculated data with the incremental parameter Q (19.S1 A) of CUo.2Pb2.2(Sb,Bi)1.aS5 and with that of the closely related compound FeSb2S4 (berthierite, 14.12 A).
Problem 3 Figures 12 and 13 show the two known simple members of the sartorite homologous series, N = 3 and 4 (i.e., more exactly, 3; 3 and 4; 4). Their structures are based on the SnS archetype, cut along (301 )SnS' In this family, the SnS archetype is pretwinned on (011 )SnS what does not influence our interpretation. This series is of combinatorial character.
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(a) Outline slabs (301) of SnS archetype in each structure.
(b) Demonstrate the derivation of N for each of the illustrated structures.
(c) The general formula of sarto rite homologues is Me2+4N_8_2 Me3\+XMe\S4N+4 (Makovicky, 1985) where Me+ enters the composition as 2Pb2+ = Ag~ +AS3.f. Derive ideal formulae for the combinations (3;3), (4;4), (3;4), (3;4;4) and the yet hypothetical case (3;3;4); all formulae should be Ag-free.
(d) The incremental direction ~ is 19.62 A for sartorite and 25.74 A for Pb2As2SS (dufrenoysite, N = 4 without Ag). Derive Q100 for the above (Ag-free) slab combinations.
(e) Analyse the crystal structure of baumhauerite in Fig. 14 and find its (N1;N2) combination.
Problem 4 (a) Derive chemical formulae of the following series of -ABX3 compounds with 2 atomic planes thick Q layers MeS and single trigonal prismatic H layers MS2 (see text for details of the interlayer match). These compounds are described as truly incommensurate. Normalize therefore to 1 orthohexagonal cation.
Cations ~ (A) .Q (A) £ (A)
Y-Nb Q 22.28 5.393 5.660 H 11.13 3.322 5.660
Sm-Ta Q 22.56 5.562 5.648 H 22.56 3.292 5.648
Ce-Ta Q 11.44 5.737 5.750 H 22.89 3.293 5.750
La-Ta Q 11.53 5.813 5.778 H 23.06 3.295 5.778
(b) Misfit layer structures of the cylindrite family have a semicommensurate Q/H match (and common modulation) in the interplanar Q-direction that changes from 12 centred Q subcells: 11 ortho-hexagonal H subcells (valid for Sn2+-rich Q layers in this variable-fit series) to about 16 Q subcells: 15 H subcells (for the Pb-rich Q layers; the Pb-Sn2+ substitution in the Q layers is the principal chemical variation in this series; it does not concern the H layers that are based on Sn4+). If we assume (in agreement with available data) that the Q/H match in the - presumably incommensurate - Q-direction is reasonably constant and can be simplified to 7ba "" 11 bH, try to calculate the generalized formulae MaMHStotal for the cylindrite homologues 12Q/11H, 13Q/12H, 14Q/13H, 15Q/14H, and 16Q/15H and examine the development of different atomic ratios. Normalize the formulae to 1 H subcell. N.B. Cylindrite has 2 atomic planes thick Q layers and single-octahedral H layers.
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Figure 18. The crystal structure of cannizzarite, Pb46Bi54S127 (Matzat.1979). In order of decreasing size the circles indicate S, Pb, (Pb,Bi) and Bi; filled and open circles indicate atoms at two discrete levels along the 4 A axis, 2 A apart. Double-octahedral H layers are intercalated by double Q layers.
Figure 19. The crystal structure of weibullite, A9o.:3sPbs.33Bi8.3iS,Se)18 (Mumme. 1980). Small circles: undifferentiated Me atoms, large circles: undifferentiated Sand Se atoms. Two height levels, 2 A apart, are populated. Remaining features (signatures) are to be determined in Problem 5.
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Figure 20. The crystal structure of Pb41n3Bi~18 (Kramer & Reis, 1986). In order of decreasing size circles represent $, Pb, In and Bi. Remaining features (signatures) are to be determined in Problem 5.
Figure 21. The crystal structure of galenobismutite PbBi2$4 (litaka & Nowacki,1962). Circles in order of decreasing size signify $, Pb and Bi. The two types of shading (a) and (b) relate to the structures in Figs. 19 and 20.
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Figure 22. The crystal structure of junoite, CU2Pb3BiiS,Se)16 (Mumme, 1975), a sheared layer structure. Circles in order of decreasing size indicate (S,Se),Pb and Bi, and Cu. Atoms are at y = 0 and 112 of the 4 A axis. Remaining features and signatures are subject of Problem 5.
Figure 23. The crystal structure of cosalite, -Pb2Bi2SS (Srikrishnan & Nowacki, 1979). Circles in order of decreasing size: S, Pb, Bi and Cu, empty and full circles are at two levels, 2 A apart.
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Problem 5 (a) Figures 19 and 20 show the crystal structures of weibullite, A9o.gPb5.gBis.g(S,Se)1S' and of Pb41ngBi7S1S' Describe their structural relationship to cannizzarite (Fig. 18), define the sizes and types of building blocks, interlayer match and recombination operators. Find and characterize their relationship to the crystal structure of galenobismutite (Fig. 21).
(b) Define the building blocks, interlayer match and recombination operator for the crystal structure of junoite Cu2PbgBis(S,Se)1S in Fig. 22.
(c) Characterize the building blocks and their sizes, as well as the extent of the two types (H and Q) of their surfaces (in terms of Hand Q subcells/subperiodicites) for an individual, periodically constricted layer in the crystal structure of cosalite -Pb2Bi2S5 in Fig. 23. Find the interlayer match between two consecutive breakpoints on the surfaces of two adjacent layers. (Hint: resolve ambiguities by preserving the constant thickness of the interface).
(d) Try layer selection and descriptions as sub(c) for the crystal structure of jamesonite FePb4SbsS14 (Fig. 24).
8.1. SOLUTIONS OF THE PROBLEMS
Problem 1
(a) For the choice of axes in Fig. 3 the space groups are Bbmm and B2Im, respectively. For gustavite, PbAgBigSS (Fig. 6) with regular sequences Ag-Bi-Ag-Bi in the mixed octahedral positions, Ag (and Bi)in the consecutive Ag-Bi strings that follow each other in the orthorhombic [100] direction are staggered in a monoclinic fashion; the resulting space group is P2/c for 8 A cell (Harris & Chen, 1975). The 8 A cell of ramdohrite AgsPb12Sb22S4S is P 1 2/n 1 (Fig. 7). Space group for NdYbSg is B2212 due to minor shifts in atomic positions.
(c) The average increment of Q for i1N = 1 in the case of Ag-free lillianite homologues is
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Figure 24. The crystal structure of jamesonite, FePb4Sb6S14 (Niizeki & Buerger, 1957). In order of decreasing size circles denote S, Pb, Sb and Fe. All atoms are at ~ -0 and ~ -112, respectively. Remaining features are the subject of Problem 5.
Figure 25. Slabs (501 }SnS of SnS archetype in the crystal structures of meneghinite homologues (ruled). (a) CUPb13SbP24 (N= 5); (b) Bi~b2_~b~5 (N = 4), and (c) CuPbBiS3 (N = 2). Asterisks denote some of the statistically occupied Cu tetrahedral sites. Compare with Figs 9-11.
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3.587 A. The estimated 12. parameters are 24.04 A for s,sL, 27.62 A for 6,6L and 45.56 A for 11,11L. ~10 for 4,7L is 25.83 A. For comparison,.Q of Ag-substituted 11,11L (ourayite) is 44.04 A; that of Ag-substituted 4,7L (vikingite) is 25.25 A.
Problem 2
(a) Slabs of SnS archetype and hatched in Fig. 25.
(b) N is determined by the number of square pyramids on either side of a tightly bonded double layer running diagonally across the slab (501 )SnS.
CUPb13S~S24 N = 5
BixSb2_xSb2SS N = 4
CuPbBiS3 N = 2
(c) The statistically occupied Cu site (Euler & Hellner, 1960) is indicated by an asterisk in Fig. 25.
(d) Chemical formulae of meneghinite homologues, CU\Me2+2N_4+xMe3\_XS2N+2
Natural meneghinite CUPb13S~S24 displays 25% saturation by Cu.
(e) Increment of 12. for AN = 1 is 4.14 A. This value includes the changes caused by the Bi/Sb exchange which correlates in nature with the decrease in the value of N. Therefore 12. = 15.78 A for N = 3 (compare with only 14.12 A for related berthierite FeSb2S4 with N = 3) and 19.92 A for N = 4 (compare with 19.81 for jaskolskiite CUo.2Pb2.2(Sb,BikaSs). The ideal space group of meneghinite homologues is Pbnm.
Problem 3
(a) Slabs of SnS archetype are outlined in Fig. 26.
(b) N is defined as the number of coordination pyramids on one side of the tightly-bonded double layer, diagonal across a slab of SnS archetype (Figs. 26 and 14).
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Figure 26. Slabs (301 )SnS of SnS archetype (ruled) in the crystal structures of two simple sartorite homologues; (a) PbAs2S4 (N = 3) and (b) Pb11.1 TlO.gAS16 (Asu)Ag2.sJS4CY Compare with Figs 12-14.
Figure 27. Definition of layers (indicated by ruling) for the crystal structure of jamesonite, FePb4Sb6S14. Layers represent periodically twinned SnS archetype; the twinned and constricted portions are populated by octahedrally coordinated Fe; the non-commensurate interlayer spaces are unshaded. Compare with Figs. 23 and 24.
(c,d) Sarto rite combinatorial series
N1;N2 (3;3) (3;3;4) (3;4) (3;4;4) (4;4)
Ideal a or Q100 19.62A 32.49 A 22.68 A 35.55 A 25.74 A
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(e) Baumhauerite is a combined homologue (3;4) with ~ 22.8 A and ~ 97.3 A (Fig. 14).
Problem 4
(b) Cylindrite homologues (7.Qa '" 11~)
Subcell Q component H component Composition MeafMeH Me/S match m£aln~ 7x 11x Mea/MewS
Weibullite: alternation of two-planes-thick a layers and double-octahedral H layers, respectively 5 1/2 a subcells and 3 1/2 H subcells broad; interlayer match 6a /3 1/2 H; composition/structure non-conservative boundaries are composed of trigonal prisms represent glide planes in the layer pile.
Pb41n3Bi7S1S: alternation of four-planes-thick a layers and single-octahedral H layers, respectively 4a and 3H subcells broad; interlayer match 3Q/2H. The composition- and structure non-conservative boundaries are based on a combination of 'vertical' and 'horizontal' trigonal prisms, respectively MeS6+2 and MeS6+1, and represent out-of-phase boundaries.
PbBi2S4 (galenobismutite) represents the lowermost homologue for both of the above compounds; these relationships are schematically shown in Fig. 21 ~ and Q.
Junoite: alternation of two-planes thick a layers and single H layers, the layer fragments are respectively 3a and 21/2 H subcells broad; interlayer match is 31/2 a /2H subcell; (out-of-phase) shear planes are almost perpendicular to layers, they display doubling of
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the H layer and Cu atoms in the Q layer.
Cosalite -Pb2Bi2SS: a continuous layer of PbS archetype; building blocks are rods with cross-sections 4Q x 1 1/2 H, with interconnecting layer intervals that have octahedral, .!:! character. Surfaces: respectively 3Q subcells and 2H subcells broad; interlayer match 3Q/2H.
Jamesonite FePb4SbsS14: a continuous layer of (in constricted portions periodically twinned) SnS archetype (Fig. 27); building blocks are 3Q x 1 1/2 (sheared) H rods that are in contact via the Fe octahedra in the constricted portions; rod surfaces and interlayer match: 2Q subcells 1 1 1/2 (sheared) H subcells.
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