Natural monoclinic AgPb(Bi₂Sb)₃S₆, an Sb-rich gustavite

The crystal structure of the Sb-rich variety of the mineral gustavite, silver lead tris­(dibismuth/antimony) hexa­sulfide, AgPb(Bi₂Sb)₃S₆, consists of blocks of diagonal chains of four octa­hedra, viz. M1a (Bi), M2a (Sb/Bi), M2b (Bi/Sb) and M1b (Ag), separated by Pb atoms in a trigonal prismatic coordination. Two marginal octa­hedral sites, M1a and M1b, where the gustavite substitution Ag⁺ + Bi³⁺ = 2Pb²⁺ takes place, are formed by Bi and Ag, respectively. Two central octa­hedra, M2a and M2b, where the Bi³⁺ = Sb³⁺ substitution takes place, are formed by two mixed Bi/Sb sites with different occupancies of Bi and Sb. The alternating occupation of the M1 site by Bi and Ag atoms (which thus creates two distinct sites M1a and M1b) results in the monoclinic space group P2₁/c. A statistical distribution of Ag/Bi in the M1 position (one mixed Ag/Bi site) was reported for synthetic gustavite, resulting in the ortho­rhom­bic space group Cmcm.

S1. Comment

Gustavite, AgPbBi₃S₆, a member of the lillianite homologous series, was discovered in the cryolite deposit at Ivigtut, Greenland, and described and named by Karup-Møller (1970). The mineral crystallized in an orthorhombic centred cell with dimensions a = 13.55 (3), b = 19.44 (3) and c = 4.11 (1) Å. Harris & Chen (1975) found that natural gustavite from Bernic Lake and Camsell River deposits in Canada was monoclinic. They reported a P2₁/c cell with a = 7.077 (7), b = 19.566 (12), c = 8.272 (8) Å and β = 107.18 (9)°. They observed that if in this cell the relatively weak l = 2n + 1 reflections were ignored, the remaining reflections could be indexed in a pseudo-cell with space-group symmetry Bbmm and cell parameters almost identical to those of Karup-Møller. The geometric features of the lillianite structures and the crystal chemistry of the lillianite homologues were described in detail by Makovicky & Karup-Møller (1977a). They concluded that what was originally interpreted as a supercell of the centred orthorhombic cell described by Karup-Møller was a twin of the monoclinic true cell, identical to that described by Harris & Chen (1975). Attempts at structure determination of monoclinic natural gustavite by Steins et al. (1991) were unsatisfactory. Later, Bente et al. (1993) published the structure of a synthetic gustavite which, unlike natural gustavite, is orthorhombic, space group Cmcm with cell parameters a = 4.077 (2), b = 13.477 (7) and c = 19.88 (2) Å.

The lillianite homologous series is an example of the accretional extensive series (Ferraris et al., 2004). Members of this series are Pb–Bi–Ag sulfosalts with structures consisting of alternating layers of PbS (NaCl) archetype cut parallel to (311)_PbS. The overlapping octahedra of adjacent mirror-related layers are replaced by bicapped trigonal coordination prisms of PbS₆₊₂, with the Pb atoms positioned on the mirror planes (Otto & Strunz, 1968). Distinct homologues differ in the thickness of the PbS-like layers. This is expressed as the number N of octahedra in the chain that runs diagonally across an individual archetypal layer (block) (Makovicky & Karup-Møller, 1977a.) Each lillianite homologue can be denoted as ^N1,N2L, where N1 and N2 are the values of N for two alternating (neighbouring) sets of layers (chains of octahedra) separated by trigonal prisms of Pb. Natural lillianite homologues start at N = 4. In the Ag-free subsystem Pb–Bi–S, ^4,4L (Pb₃Bi₂S₅, lillianite, xilingolite) and ^7,7L (Pb₆Bi₂S₉, heyrovskyite, aschalmamite) are known. With the Ag⁺+Bi³⁺ = 2Pb²⁺ substitution in octahedral layers, apart from ^4,4L (gustavite), there are also ^4,7L (vikingite), ^4,8L (treasurite), ^5,9L (eskimoite) and ^11,11L (ourayite), as well as the disordered combination of different proportions of slabs ⁴L and ⁷L (a so-called schirmerite) (Makovicky & Karup-Møller, 1977b).

The structure of the gustavite reported here consists of 11 independent sites. Five metal positions include one Pb site (M3), one Ag site (M1a), one Bi site (M1b), two mixed Bi/Sb sites (M2a with dominant Sb and M2b with dominant Bi) and six S sites (S1a, S1b, S2, S3, S4a and S4b). The numbering scheme of the sites used here is that used by Berlepsch et al. (2001) for the structure of xilingolite (monoclinic Pb₃Bi₂S₆) and by Pinto et al. (2006) for Ag-free lillianite (orthorhombic Pb₃Bi₂S₆). All of the atoms are in a general Wyckoff position 4e of the space group P2₁/c (Fig. 1). The structure of gustavite is built of alternating slabs of PbS archetype cut parallel to (311)_PbS, and each layer is N = 4 octahedra thick (Fig. 2). Each layer contains a sequence of central atoms M1a (Bi) - M2a(Sb/Bi) - M2b (Bi/Sb) - M1b (Ag). The slabs are separated by rods of Pb atoms in a bicapped-trigonal prismatic coordination approximately parallel to (010) at y ≈ 0.25 and 0.75.

The standing bicapped trigonal prism (site M3) is occupied by Pb and is situated on the plane of the unit-cell twinning of the PbS-type (octahedral) structure. It is characterized by bond distances between 2.830 (5) and 3.311 (8) Å, typical of Pb. The shortest bonds of 2.830 (5) and 2.841 (7) Å are to the S3 atoms, which form the vertices of the marginal octahedral sites M1a and M2b where the Ag–Bi substitution takes place, and are similar to the equivalent distances in heyrovskyite, xilingolite and lillianite. The distances to the capping atoms S1a and S1b, which are also part of the marginal octahedra, are 3.101 (8) and 3.141 (8) Å, which are very similar to the same distances in xilingolite [3.093 (12) Å] but differ significantly from that in lillianite [3.416 (4) Å]. The longest distances [3.311 (8) and 3.307 (6) Å] are to the S4a and S4b atoms forming the base edge of the most distorted site M1b (Ag). The calculated bond valence (Brown & Altermatt, 1985) for this pure Pb site is 1.974 (15) v.u. The anisotropic displacement parameters attain the largest values along [010].

The remaining four M sites are all in a slightly distorted octahedral coordination. M1a and M1b are marginal octahedral sites closest to the plane of unit-cell twinning, where the Ag–Bi substitution takes place. M1a is the least distorted octahedral site and is occupied by Bi, with M1a—S distances between 2.609 (8) and 3.177 (8) Å. The shortest distance is to the vertex of the octahedron pointing to S3 coordinating to the Pb atom in the trigonal prismatic site, and the longest distance is the opposite vertex to the S2 atom coordinating to the Bi/Sb mixed site. The base of the octahedron, running parallel to (150), is almost rectangular, with two distances to atoms S4a and S4b of 2.725 (8) and 2.739 (6) Å, and two longer distances to atoms S1a and S1b of 2.975 (6) and 2.980 (8) Å, respectively, with the Bi atom displaced away from the centre of the base towards the S4 atoms. The angle between the central atom and the two S atoms to the vertices is 175.2 (2)°, much closer to 180° than in the case of the more distorted site M1b. The calculated bond valence for Bi is 2.89 (3) v.u. The atomic displacement parameters of the Bi atom show no positional disorder and are nearly isotropic, unlike the other marginal octahedral site M1b.

M1b is the most distorted octahedral site and is occupied by Ag. The M1b—S distances are between 2.495 (8) and 3.359 (8) Å. Similar to the M1a site, the shortest distance is to the vertex of the octahedron towards the S3 atom coordinating to the Pb atom in the trigonal-prismatic site (towards the margin of the slab of octahedra). The distance to the opposite vertex to the S2 atom coordinating to the Bi/Sb mixed site is 2.606 (9) Å. The base of the octahedron shows the largest distortion. It has a trapezoidal shape, with two shorter bonds of 2.708 (6) and 2.795 (9) Å to atoms S4a and S4b, respectively, and two long bonds of 3.359 (8) and 3.314 (6) Å to atoms S1a and S1b, respectively, with the Ag atom displaced away from the centre of the base towards the S4 atoms. The angle between the central atom and the two S atoms to the vertices is 165.4 (2)°, significantly different from the ideal value of 180° and considerably less than in the case of the M1b site. The calculated bond valence for Ag is 1.162 (12) v.u. The atomic displacement parameters of the Ag atom are large (more than double of those of Bi in the M1a site) and highly anisotropic, showing strong positional disorder, roughly along the line between atoms S1a and S4b across the base of the octahedron. Augmented atomic displacement parameters of Ag have been observed in other sulfosalt structures, such as diaphorite (Armbruster et al., 2003).

The octahedra in the central part of the octahedral layer are formed by two mixed Sb/Bi sites, M2a and M2b. The mixed Sb/Bi site M2a, in which the Sb fraction is 0.65 (1), has a slightly distorted Sb-like coordination with three short bonds [2.540 (8), 2.579 (6) and 2.608 (8) Å] opposed by much longer Sb—S distances [3.139 (9), 3.159 (6) and 3.337 (8) Å], revealing an active and symmetrically placed lone electron pair. The shortest distance of 2.540 (8) Å to the vertex formed by atom S4a is counterbalanced across the lone electron pair micelle by the distance of 3.337 (8) Å to the opposite vertex formed by atom S1a, which is the capping atom of the trigonal prismatic Pb. The base of the octahedron is nearly rectangular, with two distances of 2.579 (6) and 2.608 (8) Å to atoms S1b and S1a, respectively, and two much longer distances of 3.139 (9) and 3.159 (6) Å to atoms S2, with the central atom displaced from the centre of the rectangle towards atoms S1a and S1b. The angle between the central atom and the two S atoms to the vertices is 174.56 (19)°. The calculated bond valence for Sb is 2.54 (2) v.u., for Bi it is 3.32 (3) v.u., and for this mixed site it is 2.93 v.u., which is close to the theoretical value of 3 (Reference?). The atomic displacement parameters of this Sb/Bi site show the smallest displacement of all metal sites, with a slight anisotropy in the direction perpendicular to (010).

The last octahedral site, M2b, is a mixed Bi/Sb site, with a Bi fraction of 0.74 (1), and [Which?] distances between 2.563 (8) and 3.251 (8) Å. The lone electron pair of Bi is less active than that of Sb, which is reflected in the shorter bond of 3.251 (8) Å across the lone electron pair micelle to the vertex to atom S1b, compared with 3.337 (8) Å to the equivalent S atom of the Sb-dominant M2a site. Accordingly, the opposite vertex towards atom S4b shows the shortest bond of 2.563 Å. The base is nearly rectangular, with the central atom displaced from the centre with two shorter bonds of 2.746 (7) and 2.776 (9) Å to atoms S2, and two longer bonds of 2.936 (6) and 2.931 (8) Å to atoms S1a and S1b, respectively. The angle between the central atom and the two S atoms to the vertices is 177.94 (19)°. The calculated bond valence for Bi is 2.96 (3) v.u., for Sb it is 2.26 (2) v.u. and the calculated valence of this mixed site is 2.78 v.u. This is less than the theoretical value of 3 and, similarly to other sulfosalt structures, this could mean that we are not dealing with a single position but with at least two overlapping fractional sites (Makovicky, personal communication)

An attempt was made to bring the structural formula closer to that obtained by microprobe analysis, especially with regard to the content of Ag and Pb. The occupancies of the mixed sites were adjusted to fit the chemical composition given by electron-probe microanalysis and refined. All Ag content is found in the marginal octahedral sites. Thus, special attention was paid to the M1a site (Bi), which can be refined with the same R factors as a pure Bi site with occupancy 0.909 (4), resulting in the current structural formula, or as a mixed site with a Bi fraction of 0.81 and an Ag fraction of 0.19. The former model was chosen because (i) an Ag content of 1.00 atom per formula unit (apfu) is closer to that established from the microprobe analysis (1.08 apfu, c.f. 1.22 apfu with the latter model); (ii) the sum of Bi+Sb from the refinement is closer to the measured value of 3.10 (2.91 apfu in the former model, compared with 2.80 apfu in the latter); (iii) the charge balance of the cations is closer to the ideal value of 12 (11.77 in the former model and 11.69 in the latter); (iv) in the latter model, the bond valence of Bi in M1a is 3.62 (3). The M3 site is occupied only by Pb and the occupancy of the site does not refine below 100% of Pb.

Comparison of the structures of natural monoclinic gustavite and synthetic orthorhombic gustavite shows that the main features of the two structures are very similar, with the same type of coordination. The main difference is in the octahedral sites. Monoclinic gustavite contains two symmetry-independent sites M1a and M1b and two symmetry-independent sites M2a and M2b, instead of one equivalent site M1 and one site M2 in synthetic orthorhombic gustavite. It is the occupancy of the octahedral sites M1 and M2 that results in a monoclinic or orthorhombic cell. The marginal octahedral site M1 in the orthorhombic structure, with a statistical occupancy of 50% Bi and 50% Ag, splits into two sites M1a and M1b in the monoclinic structure, with one site occupied by Ag and the other by Bi. The same applies to the M2 site. Thus, if there are approximately equal amounts of Bi and Sb in the formula, there is only one site M2 which always has equal occupancies of Bi and Sb and there are no M2a and M2b sites with different occupancies of the two atoms, resulting in monoclinic symmetry.

S2. Experimental

The sample containing the title mineral was found during mineralogical investigation of primary ores from mediaeval mine dumps in the Kutná Hora polymetallic deposit, Czech Republic. A crystal fragment (sample ST 63) was extracted from a polished section of a homogenous single-phase grain pre-analyzed by electron microprobe with the following composition in wt.%: Ag 11.36, Pb 16.90, Fe 0.01, Cd 0.01, Bi 42.96, Sb 11.39, S 18.51, Se 0.08, Σ = 101.23. The gustavite fragment was found to be homogeneous within analytical error. The formula from the microprobe calculated to 11 atoms is Ag_1.08Pb_0.84(Bi_2.11Sb_0.96)_Σ=3.07(S_5.93Se_0.01)_Σ5.94, Z = 4, N_chem = 4.02, Bi/(Bi+Sb) = 0.69, gustavite substitution L% = 107.2.

The measured crystal was a strongly absorbing material with a linear absorption coefficient µ ~55 mm^-1. The absorption correction was carried out in several steps. A highly redundant data set was collected with average redundancy 4.7 for P2/msymmetry. R_int for the space group P2₁/c before absorption correction was 0.302 for all reflections. The crystal shape was indexed using the program CrysAlis RED (Oxford Diffraction, 2006), which lowered R_int to 0.253. The distances and angles of the indexed faces were then refined using the program X-SHAPE (Stoe & Cie, 1998), which lowered R_int to 0.169. The resulting shape was checked against the photographs of the sample used for the numerical absorption correction and followed by multi-scan correction. The resulting R_int was 0.1078. In the final step, 58 reflections were excluded for which the intensity differed by more than 20 estimated standard deviations from the average value in the relevant symmetry-equivalent group, and the final R_int for the space group P2₁/c was 0.918 for all reflections and 0.747 for observed reflections.

The structural model was solved by the charge-flipping program SUPERFLIP (Palatinus & Chapuis, 2007) and refined in the program JANA2006 (Petříček et al., 2006). Convergence was attained for an anisotropic model.

Bond valences were calculated using the formula s = exp[(r₀ - r)/B], where r₀ is an element-pair-specific bond-valence parameter, r is the bond distance and B is a constant equal to 0.37 (Brown & Altermatt, 1985).

S3. Refinement

The atom sites are labelled as follows: M3 = Pb3, M1a = Bi1a, M1b = Ag1b, mixed site M2a = Sb2a + Bi2a, mixed site M2b = Bi2b + Sb2b.

Computing details

Data collection: CrysAlis CCD (Oxford Diffraction, 2006); cell refinement: CrysAlis RED (Oxford Diffraction, 2006); data reduction: CrysAlis RED (Oxford Diffraction, 2006); program(s) used to solve structure: SUPERFLIP (Palatinus & Chapuis, 2007); program(s) used to refine structure: JANA2006 (Petříček et al., 2006); molecular graphics: DIAMOND (Brandenburg & Putz, 2005); software used to prepare material for publication: JANA2006 (Petříček et al., 2006).

Figures

	Fig. 1. The crystal structure of gustavite, a monoclinic ^4,4L homologue of the lillianite homologous series, viewed along the c axis. M3 is Pb in a bicapped trigonal-prismatic coordination, coordination number 8. M1a is Bi, M1b is Ag, M2a is Sb/Bi and M2b is Bi/Sb, all in octahedral coordination, coordination number 6. S are S atoms.
	Fig. 2. The arrangement of the parallel layers of diagonal chains of four octahedra in the structure of gustavite, viewed along the c axis. Neighbouring blocks of octahedral layers are separated by Pb atoms in trigonal-prismatic coordination.

Silver lead tris(dibismuth/antimony) hexasulfide

Crystal data

AgPb(Bi2Sb)3S6	F(000) = 1749
Mr = 1035.9	Dx = 6.282 Mg m^−3
Monoclinic, P21/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ycb	Cell parameters from 2340 reflections
a = 7.0455 (6) Å	θ = 2.8–28.3°
b = 19.5294 (17) Å	µ = 54.58 mm^−1
c = 8.3412 (11) Å	T = 293 K
β = 107.446 (10)°	Platelet, grey
V = 1094.9 (2) Å^3	0.25 × 0.2 × 0.1 mm
Z = 4

Data collection

Oxford [Model?] CCD diffractometer	2408 independent reflections
Radiation source: X-ray tube	1290 reflections with I > 3σ(I)
Graphite monochromator	Rint = 0.092
ϕ & ω scans	θmax = 27.8°, θmin = 2.8°
Absorption correction: numerical followed by multi-scan, both corrections implemented in CrysAlis RED (Oxford Diffraction, 2006)	h = −9→9
Tmin = 0.022, Tmax = 0.159	k = −24→25
10947 measured reflections	l = −10→10

Refinement

Refinement on F	0 restraints
R[F^2 > 2σ(F^2)] = 0.059	18 constraints
wR(F^2) = 0.060	Weighting scheme based on measured s.u.'s w = 1/(σ^2(F) + 0.0001F^2)
S = 1.98	(Δ/σ)max = 0.007
2408 reflections	Δρmax = 3.80 e Å^−3
103 parameters	Δρmin = −4.13 e Å^−3

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
Pb3	0.36678 (16)	0.24985 (8)	0.34306 (11)	0.0350 (4)
Bi1a	0.82041 (18)	0.13967 (7)	0.20339 (13)	0.0238 (4)	0.909 (4)
Ag1b	0.8571 (4)	0.36619 (15)	0.2141 (3)	0.0493 (11)
Sb2b	0.26197 (18)	0.45056 (7)	0.06313 (13)	0.0242 (5)	0.261 (12)
Bi2b	0.26197 (18)	0.45056 (7)	0.06313 (13)	0.0242 (5)	0.739 (12)
Bi2a	0.2793 (2)	0.05451 (8)	0.07319 (15)	0.0219 (6)	0.350 (11)
Sb2a	0.2793 (2)	0.05451 (8)	0.07319 (15)	0.0219 (6)	0.650 (11)
S3	0.6512 (10)	0.2599 (4)	0.1642 (7)	0.025 (2)
S4b	0.0924 (11)	0.3331 (4)	0.0084 (8)	0.026 (3)
S1a	0.5016 (10)	0.4015 (4)	0.3948 (8)	0.029 (3)
S4a	0.0838 (11)	0.1658 (4)	0.0293 (8)	0.025 (3)
S1b	0.5089 (10)	0.0969 (4)	0.3582 (8)	0.027 (3)
S2	0.9871 (12)	0.4915 (4)	0.2228 (8)	0.030 (3)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb3	0.0334 (7)	0.0416 (7)	0.0301 (6)	−0.0021 (7)	0.0096 (5)	−0.0006 (7)
Bi1a	0.0225 (7)	0.0244 (8)	0.0246 (6)	−0.0011 (6)	0.0073 (4)	0.0006 (5)
Ag1b	0.058 (2)	0.0400 (19)	0.0456 (14)	−0.0217 (14)	0.0090 (13)	−0.0017 (13)
Sb2b	0.0272 (8)	0.0213 (8)	0.0229 (7)	0.0000 (6)	0.0055 (5)	−0.0002 (5)
Bi2b	0.0272 (8)	0.0213 (8)	0.0229 (7)	0.0000 (6)	0.0055 (5)	−0.0002 (5)
Bi2a	0.0168 (9)	0.0258 (11)	0.0224 (8)	0.0023 (7)	0.0049 (6)	−0.0008 (6)
Sb2a	0.0168 (9)	0.0258 (11)	0.0224 (8)	0.0023 (7)	0.0049 (6)	−0.0008 (6)
S3	0.025 (4)	0.029 (5)	0.023 (3)	−0.001 (3)	0.009 (3)	−0.001 (3)
S4b	0.030 (4)	0.021 (5)	0.028 (3)	0.003 (3)	0.010 (3)	0.000 (3)
S1a	0.015 (4)	0.031 (5)	0.041 (4)	−0.006 (3)	0.009 (3)	0.003 (3)
S4a	0.026 (4)	0.020 (4)	0.029 (3)	0.007 (3)	0.012 (3)	0.006 (3)
S1b	0.011 (4)	0.034 (5)	0.034 (4)	−0.006 (3)	0.003 (3)	0.005 (3)
S2	0.030 (4)	0.030 (5)	0.032 (4)	0.010 (3)	0.013 (3)	0.001 (3)

Geometric parameters (Å, º)

Pb3—S3	2.841 (7)	Sb2b—S2^vii	2.776 (9)
Pb3—S3^i	2.830 (5)	Sb2b—S2^viii	2.746 (7)
Pb3—S4b	3.307 (6)	Bi2b—S4b	2.563 (8)
Pb3—S4b^i	3.141 (8)	Bi2b—S1a	2.936 (6)
Pb3—S1a	3.101 (8)	Bi2b—S1b^vi	3.251 (8)
Pb3—S4a	3.224 (6)	Bi2b—S1b^iii	2.931 (8)
Pb3—S4a^i	3.311 (8)	Bi2b—S2^vii	2.776 (9)
Pb3—S1b	3.141 (8)	Bi2b—S2^viii	2.746 (7)
Bi1a—S3	2.609 (8)	Bi2a—Sb2a	0
Bi1a—S4b^ii	2.739 (6)	Bi2a—S1a^ix	3.337 (8)
Bi1a—S1a^iii	2.975 (6)	Bi2a—S1a^iii	2.608 (8)
Bi1a—S4a^iv	2.725 (8)	Bi2a—S4a	2.540 (8)
Bi1a—S1b	2.980 (8)	Bi2a—S1b	2.579 (6)
Bi1a—S2^v	3.177 (8)	Bi2a—S2^ix	3.139 (9)
Ag1b—S3	2.495 (8)	Bi2a—S2^x	3.159 (6)
Ag1b—S4b^iv	2.795 (9)	Sb2a—S1a^ix	3.337 (8)
Ag1b—S1a	3.359 (8)	Sb2a—S1a^iii	2.608 (8)
Ag1b—S4a^ii	2.708 (6)	Sb2a—S4a	2.540 (8)
Ag1b—S1b^iii	3.314 (6)	Sb2a—S1b	2.579 (6)
Ag1b—S2	2.606 (9)	Sb2a—S2^ix	3.139 (9)
Sb2b—Bi2b	0	Sb2a—S2^x	3.159 (6)
Sb2b—S4b	2.563 (8)	S3—S1a	3.698 (11)
Sb2b—S1a	2.936 (6)	S4b—S4a	3.275 (11)
Sb2b—S1b^vi	3.251 (8)	S1a—S4a^i	3.692 (11)
Sb2b—S1b^iii	2.931 (8)	S4a—S1b	3.658 (9)
S3—Pb3—S3^i	95.22 (19)	Pb3—S1a—Bi2a^vi	169.0 (3)
S3—Pb3—S4b	81.30 (19)	Pb3—S1a—Bi2a^i	100.5 (2)
S3—Pb3—S4b^i	152.9 (2)	Pb3—S1a—Sb2a^vi	169.0 (3)
S3—Pb3—S1a	76.8 (2)	Pb3—S1a—Sb2a^i	100.5 (2)
S3—Pb3—S4a	88.32 (19)	Pb3—S1a—S3	48.43 (16)
S3—Pb3—S4a^i	146.2 (2)	Pb3—S1a—S4a^i	57.55 (18)
S3—Pb3—S1b	79.6 (2)	Bi1a^i—S1a—Ag1b	82.05 (17)
S3^i—Pb3—S4b	154.2 (2)	Bi1a^i—S1a—Sb2b	167.2 (3)
S3^i—Pb3—S4b^i	84.50 (19)	Bi1a^i—S1a—Bi2b	167.2 (3)
S3^i—Pb3—S1a	80.3 (2)	Bi1a^i—S1a—Bi2a^vi	87.80 (17)
S3^i—Pb3—S4a	145.5 (2)	Bi1a^i—S1a—Bi2a^i	91.3 (2)
S3^i—Pb3—S4a^i	86.81 (18)	Bi1a^i—S1a—Sb2a^vi	87.80 (17)
S3^i—Pb3—S1b	76.78 (19)	Bi1a^i—S1a—Sb2a^i	91.3 (2)
S4b—Pb3—S4b^i	110.08 (18)	Bi1a^i—S1a—S3	89.4 (2)
S4b—Pb3—S1a	74.03 (17)	Bi1a^i—S1a—S4a^i	96.0 (2)
S4b—Pb3—S4a	60.17 (18)	Ag1b—S1a—Sb2b	86.55 (19)
S4b—Pb3—S4a^i	82.49 (18)	Ag1b—S1a—Bi2b	86.55 (19)
S4b—Pb3—S1b	126.90 (17)	Ag1b—S1a—Bi2a^vi	80.24 (18)
S4b^i—Pb3—S1a	129.4 (2)	Ag1b—S1a—Bi2a^i	168.4 (3)
S4b^i—Pb3—S4a	77.37 (18)	Ag1b—S1a—Sb2a^vi	80.24 (18)
S4b^i—Pb3—S4a^i	60.94 (19)	Ag1b—S1a—Sb2a^i	168.4 (3)
S4b^i—Pb3—S1b	73.9 (2)	Ag1b—S1a—S3	41.07 (16)
S1a—Pb3—S4a	133.52 (17)	Ag1b—S1a—S4a^i	146.4 (3)
S1a—Pb3—S4a^i	70.2 (2)	Sb2b—S1a—Bi2b	0
S1a—Pb3—S1b	145.29 (18)	Sb2b—S1a—Bi2a^vi	84.5 (2)
S4a—Pb3—S4a^i	108.76 (18)	Sb2b—S1a—Bi2a^i	99.0 (2)
S4a—Pb3—S1b	70.15 (17)	Sb2b—S1a—Sb2a^vi	84.5 (2)
S4a^i—Pb3—S1b	133.2 (2)	Sb2b—S1a—Sb2a^i	99.0 (2)
S3—Bi1a—S4b^ii	96.2 (2)	Sb2b—S1a—S3	85.82 (19)
S3—Bi1a—S1a^iii	86.4 (2)	Sb2b—S1a—S4a^i	96.7 (2)
S3—Bi1a—S4a^iv	97.1 (2)	Bi2b—S1a—Bi2a^vi	84.5 (2)
S3—Bi1a—S1b	86.5 (2)	Bi2b—S1a—Bi2a^i	99.0 (2)
S3—Bi1a—S2^v	175.2 (2)	Bi2b—S1a—Sb2a^vi	84.5 (2)
S4b^ii—Bi1a—S1a^iii	173.0 (2)	Bi2b—S1a—Sb2a^i	99.0 (2)
S4b^ii—Bi1a—S4a^iv	93.5 (2)	Bi2b—S1a—S3	85.82 (19)
S4b^ii—Bi1a—S1b	93.0 (2)	Bi2b—S1a—S4a^i	96.7 (2)
S4b^ii—Bi1a—S2^v	81.8 (2)	Bi2a^vi—S1a—Bi2a^i	90.1 (2)
S1a^iii—Bi1a—S4a^iv	92.7 (2)	Bi2a^vi—S1a—Sb2a^vi	0
S1a^iii—Bi1a—S1b	80.58 (19)	Bi2a^vi—S1a—Sb2a^i	90.1 (2)
S1a^iii—Bi1a—S2^v	95.12 (19)	Bi2a^vi—S1a—S3	121.0 (3)
S4a^iv—Bi1a—S1b	172.2 (2)	Bi2a^vi—S1a—S4a^i	133.3 (3)
S4a^iv—Bi1a—S2^v	87.4 (2)	Bi2a^i—S1a—Sb2a^vi	90.1 (2)
S1b—Bi1a—S2^v	89.3 (2)	Bi2a^i—S1a—Sb2a^i	0
S3—Ag1b—S4b^iv	97.3 (2)	Bi2a^i—S1a—S3	148.9 (3)
S3—Ag1b—S1a	76.8 (2)	Bi2a^i—S1a—S4a^i	43.44 (17)
S3—Ag1b—S4a^ii	96.6 (2)	Sb2a^vi—S1a—Sb2a^i	90.1 (2)
S3—Ag1b—S1b^iii	78.1 (2)	Sb2a^vi—S1a—S3	121.0 (3)
S3—Ag1b—S2	165.4 (2)	Sb2a^vi—S1a—S4a^i	133.3 (3)
S4b^iv—Ag1b—S1a	169.09 (18)	Sb2a^i—S1a—S3	148.9 (3)
S4b^iv—Ag1b—S4a^ii	104.9 (2)	Sb2a^i—S1a—S4a^i	43.44 (17)
S4b^iv—Ag1b—S1b^iii	85.21 (19)	S3—S1a—S4a^i	105.6 (3)
S4b^iv—Ag1b—S2	88.0 (3)	Pb3—S4a—Pb3^iii	79.31 (17)
S1a—Ag1b—S4a^ii	85.1 (2)	Pb3—S4a—Bi1a^vii	90.97 (19)
S1a—Ag1b—S1b^iii	84.60 (18)	Pb3—S4a—Ag1b^x	161.2 (3)
S1a—Ag1b—S2	95.6 (2)	Pb3—S4a—Bi2a	98.17 (19)
S4a^ii—Ag1b—S1b^iii	169.2 (2)	Pb3—S4a—Sb2a	98.17 (19)
S4a^ii—Ag1b—S2	95.1 (2)	Pb3—S4a—S4b	61.18 (17)
S1b^iii—Ag1b—S2	88.9 (2)	Pb3—S4a—S1a^iii	94.2 (2)
Bi2b—Sb2b—S4b	0	Pb3—S4a—S1b	53.86 (15)
Bi2b—Sb2b—S1a	0	Pb3^iii—S4a—Bi1a^vii	160.9 (3)
Bi2b—Sb2b—S1b^vi	0	Pb3^iii—S4a—Ag1b^x	85.5 (2)
Bi2b—Sb2b—S1b^iii	0	Pb3^iii—S4a—Bi2a	96.7 (2)
Bi2b—Sb2b—S2^vii	0	Pb3^iii—S4a—Sb2a	96.7 (2)
Bi2b—Sb2b—S2^viii	0	Pb3^iii—S4a—S4b	56.97 (19)
S4b—Sb2b—S1a	88.9 (2)	Pb3^iii—S4a—S1a^iii	52.22 (17)
S4b—Sb2b—S1b^vi	177.94 (19)	Pb3^iii—S4a—S1b	93.6 (2)
S4b—Sb2b—S1b^iii	86.5 (2)	Bi1a^vii—S4a—Ag1b^x	100.2 (2)
S4b—Sb2b—S2^vii	89.3 (2)	Bi1a^vii—S4a—Bi2a	100.9 (3)
S4b—Sb2b—S2^viii	94.1 (2)	Bi1a^vii—S4a—Sb2a	100.9 (3)
S1a—Sb2b—S1b^vi	89.22 (19)	Bi1a^vii—S4a—S4b	104.0 (3)
S1a—Sb2b—S1b^iii	99.9 (2)	Bi1a^vii—S4a—S1a^iii	145.9 (3)
S1a—Sb2b—S2^vii	85.9 (2)	Bi1a^vii—S4a—S1b	93.8 (2)
S1a—Sb2b—S2^viii	171.9 (2)	Ag1b^x—S4a—Bi2a	94.5 (2)
S1b^vi—Sb2b—S1b^iii	92.9 (2)	Ag1b^x—S4a—Sb2a	94.5 (2)
S1b^vi—Sb2b—S2^vii	91.5 (2)	Ag1b^x—S4a—S4b	101.1 (2)
S1b^vi—Sb2b—S2^viii	87.8 (2)	Ag1b^x—S4a—S1a^iii	85.1 (2)
S1b^iii—Sb2b—S2^vii	172.78 (18)	Ag1b^x—S4a—S1b	139.1 (3)
S1b^iii—Sb2b—S2^viii	87.7 (2)	Bi2a—S4a—Sb2a	0
S2^vii—Sb2b—S2^viii	86.7 (2)	Bi2a—S4a—S4b	147.5 (3)
Sb2b—Bi2b—S4b	0	Bi2a—S4a—S1a^iii	44.93 (18)
Sb2b—Bi2b—S1a	0	Bi2a—S4a—S1b	44.81 (16)
Sb2b—Bi2b—S1b^vi	0	Sb2a—S4a—S4b	147.5 (3)
Sb2b—Bi2b—S1b^iii	0	Sb2a—S4a—S1a^iii	44.93 (18)
Sb2b—Bi2b—S2^vii	0	Sb2a—S4a—S1b	44.81 (16)
Sb2b—Bi2b—S2^viii	0	S4b—S4a—S1a^iii	108.1 (3)
S4b—Bi2b—S1a	88.9 (2)	S4b—S4a—S1b	112.5 (2)
S4b—Bi2b—S1b^vi	177.94 (19)	S1a^iii—S4a—S1b	63.19 (19)
S4b—Bi2b—S1b^iii	86.5 (2)	Pb3—S1b—Bi1a	89.0 (2)
S4b—Bi2b—S2^vii	89.3 (2)	Pb3—S1b—Ag1b^i	88.33 (19)
S4b—Bi2b—S2^viii	94.1 (2)	Pb3—S1b—Sb2b^ix	167.7 (2)
S1a—Bi2b—S1b^vi	89.22 (19)	Pb3—S1b—Sb2b^i	95.0 (2)
S1a—Bi2b—S1b^iii	99.9 (2)	Pb3—S1b—Bi2b^ix	167.7 (2)
S1a—Bi2b—S2^vii	85.9 (2)	Pb3—S1b—Bi2b^i	95.0 (2)
S1a—Bi2b—S2^viii	171.9 (2)	Pb3—S1b—Bi2a	99.4 (2)
S1b^vi—Bi2b—S1b^iii	92.9 (2)	Pb3—S1b—Sb2a	99.4 (2)
S1b^vi—Bi2b—S2^vii	91.5 (2)	Pb3—S1b—S4a	55.99 (16)
S1b^vi—Bi2b—S2^viii	87.8 (2)	Bi1a—S1b—Ag1b^i	83.32 (17)
S1b^iii—Bi2b—S2^vii	172.78 (18)	Bi1a—S1b—Sb2b^ix	87.0 (2)
S1b^iii—Bi2b—S2^viii	87.7 (2)	Bi1a—S1b—Sb2b^i	169.9 (2)
S2^vii—Bi2b—S2^viii	86.7 (2)	Bi1a—S1b—Bi2b^ix	87.0 (2)
Sb2a—Bi2a—S1a^ix	0	Bi1a—S1b—Bi2b^i	169.9 (2)
Sb2a—Bi2a—S1a^iii	0	Bi1a—S1b—Bi2a	91.7 (2)
Sb2a—Bi2a—S4a	0	Bi1a—S1b—Sb2a	91.7 (2)
Sb2a—Bi2a—S1b	0	Bi1a—S1b—S4a	96.6 (2)
Sb2a—Bi2a—S2^ix	0	Ag1b^i—S1b—Sb2b^ix	79.69 (15)
Sb2a—Bi2a—S2^x	0	Ag1b^i—S1b—Sb2b^i	87.48 (18)
S1a^ix—Bi2a—S1a^iii	89.9 (2)	Ag1b^i—S1b—Bi2b^ix	79.69 (15)
S1a^ix—Bi2a—S4a	174.56 (19)	Ag1b^i—S1b—Bi2b^i	87.48 (18)
S1a^ix—Bi2a—S1b	93.8 (2)	Ag1b^i—S1b—Bi2a	170.8 (3)
S1a^ix—Bi2a—S2^ix	86.7 (2)	Ag1b^i—S1b—Sb2a	170.8 (3)
S1a^ix—Bi2a—S2^x	88.68 (18)	Ag1b^i—S1b—S4a	144.3 (3)
S1a^iii—Bi2a—S4a	91.6 (3)	Sb2b^ix—S1b—Sb2b^i	87.1 (2)
S1a^iii—Bi2a—S1b	95.9 (2)	Sb2b^ix—S1b—Bi2b^ix	0
S1a^iii—Bi2a—S2^ix	176.0 (2)	Sb2b^ix—S1b—Bi2b^i	87.1 (2)
S1a^iii—Bi2a—S2^x	84.4 (2)	Sb2b^ix—S1b—Bi2a	92.3 (2)
S4a—Bi2a—S1b	91.2 (2)	Sb2b^ix—S1b—Sb2a	92.3 (2)
S4a—Bi2a—S2^ix	91.6 (2)	Sb2b^ix—S1b—S4a	136.0 (2)
S4a—Bi2a—S2^x	86.3 (2)	Sb2b^i—S1b—Bi2b^ix	87.1 (2)
S1b—Bi2a—S2^ix	86.4 (2)	Sb2b^i—S1b—Bi2b^i	0
S1b—Bi2a—S2^x	177.5 (2)	Sb2b^i—S1b—Bi2a	96.8 (2)
S2^ix—Bi2a—S2^x	93.5 (2)	Sb2b^i—S1b—Sb2a	96.8 (2)
Bi2a—Sb2a—S1a^ix	0	Sb2b^i—S1b—S4a	93.3 (2)
Bi2a—Sb2a—S1a^iii	0	Bi2b^ix—S1b—Bi2b^i	87.1 (2)
Bi2a—Sb2a—S4a	0	Bi2b^ix—S1b—Bi2a	92.3 (2)
Bi2a—Sb2a—S1b	0	Bi2b^ix—S1b—Sb2a	92.3 (2)
Bi2a—Sb2a—S2^ix	0	Bi2b^ix—S1b—S4a	136.0 (2)
Bi2a—Sb2a—S2^x	0	Bi2b^i—S1b—Bi2a	96.8 (2)
S1a^ix—Sb2a—S1a^iii	89.9 (2)	Bi2b^i—S1b—Sb2a	96.8 (2)
S1a^ix—Sb2a—S4a	174.56 (19)	Bi2b^i—S1b—S4a	93.3 (2)
S1a^ix—Sb2a—S1b	93.8 (2)	Bi2a—S1b—Sb2a	0
S1a^ix—Sb2a—S2^ix	86.7 (2)	Bi2a—S1b—S4a	43.96 (15)
S1a^ix—Sb2a—S2^x	88.68 (18)	Sb2a—S1b—S4a	43.96 (15)
S1a^iii—Sb2a—S4a	91.6 (3)	Bi1a^xi—S2—Ag1b	170.6 (3)
S1a^iii—Sb2a—S1b	95.9 (2)	Bi1a^xi—S2—Sb2b^iv	92.1 (2)
S1a^iii—Sb2a—S2^ix	176.0 (2)	Bi1a^xi—S2—Sb2b^viii	85.3 (2)
S1a^iii—Sb2a—S2^x	84.4 (2)	Bi1a^xi—S2—Bi2b^iv	92.1 (2)
S4a—Sb2a—S1b	91.2 (2)	Bi1a^xi—S2—Bi2b^viii	85.3 (2)
S4a—Sb2a—S2^ix	91.6 (2)	Bi1a^xi—S2—Bi2a^vi	80.06 (19)
S4a—Sb2a—S2^x	86.3 (2)	Bi1a^xi—S2—Bi2a^ii	87.57 (17)
S1b—Sb2a—S2^ix	86.4 (2)	Bi1a^xi—S2—Sb2a^vi	80.06 (19)
S1b—Sb2a—S2^x	177.5 (2)	Bi1a^xi—S2—Sb2a^ii	87.57 (17)
S2^ix—Sb2a—S2^x	93.5 (2)	Ag1b—S2—Sb2b^iv	90.4 (3)
Pb3—S3—Pb3^iii	94.7 (2)	Ag1b—S2—Sb2b^viii	103.6 (2)
Pb3—S3—Bi1a	103.9 (3)	Ag1b—S2—Bi2b^iv	90.4 (3)
Pb3—S3—Ag1b	115.9 (3)	Ag1b—S2—Bi2b^viii	103.6 (2)
Pb3—S3—S1a	54.73 (18)	Ag1b—S2—Bi2a^vi	97.0 (3)
Pb3^iii—S3—Bi1a	103.0 (2)	Ag1b—S2—Bi2a^ii	83.4 (2)
Pb3^iii—S3—Ag1b	115.1 (3)	Ag1b—S2—Sb2a^vi	97.0 (3)
Pb3^iii—S3—S1a	108.1 (2)	Ag1b—S2—Sb2a^ii	83.4 (2)
Bi1a—S3—Ag1b	120.5 (3)	Sb2b^iv—S2—Sb2b^viii	93.3 (2)
Bi1a—S3—S1a	143.0 (2)	Sb2b^iv—S2—Bi2b^iv	0
Ag1b—S3—S1a	62.2 (2)	Sb2b^iv—S2—Bi2b^viii	93.3 (2)
Pb3—S4b—Pb3^iii	80.55 (17)	Sb2b^iv—S2—Bi2a^vi	171.7 (3)
Pb3—S4b—Bi1a^x	161.5 (3)	Sb2b^iv—S2—Bi2a^ii	90.4 (2)
Pb3—S4b—Ag1b^vii	84.24 (18)	Sb2b^iv—S2—Sb2a^vi	171.7 (3)
Pb3—S4b—Sb2b	99.58 (19)	Sb2b^iv—S2—Sb2a^ii	90.4 (2)
Pb3—S4b—Bi2b	99.58 (19)	Sb2b^viii—S2—Bi2b^iv	93.3 (2)
Pb3—S4b—S4a	58.64 (16)	Sb2b^viii—S2—Bi2b^viii	0
Pb3^iii—S4b—Bi1a^x	92.5 (2)	Sb2b^viii—S2—Bi2a^vi	88.8 (2)
Pb3^iii—S4b—Ag1b^vii	160.9 (3)	Sb2b^viii—S2—Bi2a^ii	172.0 (3)
Pb3^iii—S4b—Sb2b	103.0 (3)	Sb2b^viii—S2—Sb2a^vi	88.8 (2)
Pb3^iii—S4b—Bi2b	103.0 (3)	Sb2b^viii—S2—Sb2a^ii	172.0 (3)
Pb3^iii—S4b—S4a	62.1 (2)	Bi2b^iv—S2—Bi2b^viii	93.3 (2)
Bi1a^x—S4b—Ag1b^vii	98.4 (2)	Bi2b^iv—S2—Bi2a^vi	171.7 (3)
Bi1a^x—S4b—Sb2b	98.7 (2)	Bi2b^iv—S2—Bi2a^ii	90.4 (2)
Bi1a^x—S4b—Bi2b	98.7 (2)	Bi2b^iv—S2—Sb2a^vi	171.7 (3)
Bi1a^x—S4b—S4a	102.9 (2)	Bi2b^iv—S2—Sb2a^ii	90.4 (2)
Ag1b^vii—S4b—Sb2b	90.8 (2)	Bi2b^viii—S2—Bi2a^vi	88.8 (2)
Ag1b^vii—S4b—Bi2b	90.8 (2)	Bi2b^viii—S2—Bi2a^ii	172.0 (3)
Ag1b^vii—S4b—S4a	100.0 (3)	Bi2b^viii—S2—Sb2a^vi	88.8 (2)
Sb2b—S4b—Bi2b	0	Bi2b^viii—S2—Sb2a^ii	172.0 (3)
Sb2b—S4b—S4a	154.0 (3)	Bi2a^vi—S2—Bi2a^ii	86.53 (18)
Bi2b—S4b—S4a	154.0 (3)	Bi2a^vi—S2—Sb2a^vi	0
Pb3—S1a—Bi1a^i	89.0 (2)	Bi2a^vi—S2—Sb2a^ii	86.53 (18)
Pb3—S1a—Ag1b	88.9 (2)	Bi2a^ii—S2—Sb2a^vi	86.53 (18)
Pb3—S1a—Sb2b	96.60 (19)	Bi2a^ii—S2—Sb2a^ii	0
Pb3—S1a—Bi2b	96.60 (19)	Sb2a^vi—S2—Sb2a^ii	86.53 (18)

Symmetry codes: (i) x, −y+1/2, z+1/2; (ii) x+1, −y+1/2, z+1/2; (iii) x, −y+1/2, z−1/2; (iv) x+1, y, z; (v) −x+2, y−1/2, −z+1/2; (vi) −x+1, y+1/2, −z+1/2; (vii) x−1, y, z; (viii) −x+1, −y+1, −z; (ix) −x+1, y−1/2, −z+1/2; (x) x−1, −y+1/2, z−1/2; (xi) −x+2, y+1/2, −z+1/2.